*4.6. Transformation of A and L to the New Basis States*

We must transform *<sup>A</sup>* as given in Equation (89) and *<sup>L</sup>* <sup>=</sup> *<sup>r</sup>* <sup>×</sup> *<sup>p</sup>* when we change our basis states from eigenstates of the energy to eigenstates of the inverse coupling constant. The correct transformation may be derived by requiring that the transformed generators produce the same linear combination of new states as the original generators produced of the old states. Thus, because

$$\mathcal{A}|nlm\rangle = \sum\_{l',m'} |nl'm'\rangle A^{lm}\_{l'm'} \tag{117}$$

where the coefficients *Alm <sup>l</sup> <sup>m</sup>* are the matrix elements of *<sup>A</sup>*, we require that the transformed generator *<sup>a</sup>* satisfies the equation

$$\mathfrak{a}\left|nlm\right\rangle = \sum\_{l'm'} \left|nl'm'\right\rangle A^{lm}\_{l'm'}.\tag{118}$$

In other words, since the Runge–Lenz vector *A* is a symmetry operator of the original energy eigenstates, *a* will be a symmetry operator of the new states with precisely the same properties and matrix elements. Because *A* is Hermitian, *a* is Hermitian.

To obtain a differential expression for *a* acting on the new states, we need to transform the generator using Equation (114):

$$\mathfrak{a} = D^{-1}(\lambda\_n) \left( \sqrt{\rho(a\_n)} A \frac{1}{\sqrt{\rho(a\_n)}} \right) D(\lambda\_n). \tag{119}$$

The effect of the scale change on the quantity in large parenthesis is to replace *an* everywhere by *a*. By explicit calculation, we find

$$a = \frac{1}{2a} \left( \frac{rp^2 + p^2r}{2} - r \cdot pp - pp \cdot r \right) - \frac{ar}{2} \tag{120}$$

for *ρ*(*a*)=(*p*<sup>2</sup> + *a*2)/2*a*2. And we obtain

$$a = \frac{1}{2a} \left( \frac{rp^2 + p^2r}{2} - r \cdot pp - pp \cdot r - \frac{r}{4r^2} \right) - \frac{ar}{2} \tag{121}$$

for *ρ*(*a*) = *n*/*ar*.

Both of these expressions for *a* are manifestly Hermitian. In addition, since there is no dependence on the principal quantum number these expressions are valid in the entire Hilbert space, and not just in a subspace spanned by the degenerate states, as was the case when we used the energy eigenstates as a basis (Equation (89)).

The angular momentum operator is invariant under scale changes and it commutes with scalar operators. Therefore *<sup>L</sup>* is invariant under the similarity transformation √ <sup>1</sup> *ρ*(*an*) *D*(*λn*) and the expression for the angular momentum operator with respect to the eigenstates of (*Zα*)−<sup>1</sup> is the same as the expression with respect to the eigenstates of the energy.

#### *4.7. The U* | *Representation*

The *U* coordinates provide the natural representation for the investigation of the symmetries of the hydrogenlike atom in quantum mechanics, as in classical mechanics [125]. Therefore, we briefly consider the relevant features of this representation and, in particular, its relationship to the momentum representation. The eigenstate < *U* | of *Ub*, *b* = 1,2,3,4, is defined by

$$
\langle \mathcal{U}' | \, \mathcal{U}\_b = \mathcal{U}'\_b \, \langle \mathcal{U}' | \,\tag{122}
$$

These states are complete on the unit hypersphere in four dimensions:

$$\int |\mathcal{U}'\rangle\langle\mathcal{U}'|\,d^3\Omega'=1\tag{123}$$

where Ω refers to the angles (*θ*4, *θ*, *φ*) defined in Equation (54). The *U* variables are defined in terms of the momentum variables and the quantity *a* in Equation (52). Therefore, the momentum and the U operators commute

$$[p\_{i\prime} \, \mathcal{U}\_b] = 0\tag{124}$$

and the state *U* | is proportional to a momentum eigenstate *p* |:

$$
\langle \mathcal{U}' | = \langle p' | \sqrt{J(p)} \tag{125}
$$

where the momentum eigenstate is defined by *p* <sup>|</sup>*<sup>p</sup>* <sup>=</sup> *<sup>p</sup>- p*| and

$$\int d^3 p' |p'\rangle\langle p'| = 1.\tag{126}$$

The function *J*(*p* ) may be determined by equating the completeness conditions and substituting Equation (125):

$$1 = \int d^3 p' |p'\rangle\langle p'| = \int d^3 \Omega' |\mathcal{U}'\rangle\langle \mathcal{U}'| = \int d^3 \Omega' |f(p')|p'\rangle\langle p'| \tag{127}$$

which leads to the identification of the differential quantities

$$d^3 p' = d^3 \Omega' f(p') \tag{128}$$

demonstrating that *J*(*p* ) is the Jacobian of the transformation from the p- to the U- space. Noting that on the unit sphere

$$
\hbar I\_4^2 = 1 - \mathcal{U}\_i \mathcal{U}\_i \tag{129}
$$

we can compute the Jacobian

$$J(p') = \left[\frac{p'^2 + a^2}{2a}\right]^3. \tag{130}$$

Therefore from Equation (125) we have the important result

$$
\langle \mathcal{U}' | = \langle p' | \left[ \frac{p^2 + a^2}{2a} \right]^{3/2} . \tag{131}
$$

We can use this result to compute the action of *<sup>r</sup>* on *<sup>U</sup>* | in terms of the differential operators. Using the equation

$$<\langle p'|r = i\nabla\_{p'}\langle p'|\tag{132}$$

we obtain

$$
\langle \mathsf{U}' | r = \left( i \nabla\_{p'} - \frac{3i p'}{p'^2 + a^2} \right) \langle \mathsf{U}' | . \tag{133}
$$

#### Action of *<sup>a</sup>* and *<sup>L</sup>* on *<sup>U</sup>* |

Using Equation (133) for the action of *<sup>r</sup>* on *<sup>U</sup>* | and using the expression Equation (120) for *<sup>a</sup>* since *<sup>U</sup>* | is proportional to *p* |, we immediately find that when acting on *U* <sup>|</sup>, *<sup>a</sup>* has the differential representation

$$\mathbf{a}' = \frac{\dot{\mathbf{i}}}{2a} \left( (p'^2 - a^2) \nabla\_{p'} - 2p' \mathbf{p'} \cdot \nabla\_{p'} \right) \tag{134}$$

where

$$<\langle \mathcal{U}' | \mathfrak{a} = \mathfrak{a}' \langle \mathcal{U}' |.\tag{135}$$

We can also write *a* in terms of the *U* variables by using the relationship Equation (52) between the *p* and *U* variables:

$$a' = \mathcal{U}\_4' i\nabla\_{\mathcal{U}'} - \mathcal{U}' i\partial / \partial\_4' \tag{136}$$

where the spatial part of the four vector *U* is *U* = (*U*1, *U*2, *U*3) and *U*<sup>4</sup> is the fourth component. This is the differential representation of a rotation operator mixing the spatial and the fourth components of *U <sup>a</sup>*. When acting on the state *U* <sup>|</sup> , clearly *<sup>e</sup>ia*·*<sup>ν</sup>* generates a four-dimensional rotation that produces a new eigenstate *U*"|. To explicitly derive the form of the finite transformation, we compute

$$[a'\_{j'} \mathcal{U}'\_j] = i \mathcal{U}'\_4 \delta\_{ij} \qquad [a'\_{j'} \mathcal{U}'\_4] = -i \mathcal{U}'\_i. \tag{137}$$

For a finite transformation *aia*·*n<sup>ν</sup>* with *n*<sup>2</sup> = 1, we have

$$\begin{split} \mathbf{U}^{\prime\prime} &= \varepsilon^{ia' \cdot m\nu} \mathbf{U}^{\prime} \varepsilon^{-ia' \cdot m\nu} \\ &= \mathbf{U}^{\prime} - nm \cdot \mathbf{U}^{\prime} + nm \cdot \mathbf{U}^{\prime} \cos \nu - n \mathbf{U}\_{4}^{\prime} \sin \nu \end{split} \tag{138}$$

and

$$\begin{split} \, \|\boldsymbol{\mathcal{U}}\|\_{4} &= \varepsilon^{\dot{\boldsymbol{a}}\boldsymbol{a}' \cdot \boldsymbol{u}\boldsymbol{v}} \, \|\boldsymbol{\mathcal{U}}\boldsymbol{a}' \boldsymbol{e}^{-\dot{\boldsymbol{a}}\boldsymbol{a}' \cdot \boldsymbol{u}\boldsymbol{v}} \\ &= \, \|\boldsymbol{\mathcal{U}}\_{4}' \cos \boldsymbol{\nu} + \boldsymbol{n} \cdot \boldsymbol{\mathcal{U}}' \sin \boldsymbol{\nu} \end{split} \tag{139}$$

These equations of transformation are like those for a Lorentz transformation of a four-vector (r, it). We can illustrate the equations for *eia*2*<sup>ν</sup>* (cf Equation (58)), which mixes the two and four components of *U* :

$$\begin{aligned} \mathcal{U}l\_1'' &= \mathcal{U}l\_1' & \mathcal{U}\_3'' &= \mathcal{U}l\_3'\\ \mathcal{U}l\_2'' &= \mathcal{U}l\_2' \cos \nu - \mathcal{U}l\_4' \sin \nu & \mathcal{U}\_4'' &= \mathcal{U}l\_2' \sin \nu + \mathcal{U}l\_4' \cos \nu \end{aligned} \tag{140}$$

When *<sup>L</sup>* acts on *<sup>U</sup>* |, it has the differential representation

$$L' = \mathcal{U}' \times i\nabla\_{\mathcal{U}'} \tag{141}$$

This result follows directly, since *U <sup>i</sup>* equals *p <sup>i</sup>* times a factor that is a scalar under rotations in three dimensions. When *<sup>e</sup>iL*·*<sup>ω</sup>* acts on *<sup>U</sup>* | it produces a new state *U*"| , where the spatial components of *U* have been rotated to produce *U*".

In summary, we see that *U* is a four-vector under rotations generated by *a* and *L* . Therefore the states *U* | provide a vector representation of the group of rotations in four dimensions SO(4), with the generators *a* and *L*.

#### **5. Wave Functions for the Hydrogenlike Atom**

In this section, we analyze the wave functions of the hydrogenlike atom, working primarily in the *U* <sup>|</sup> representation and using eigenstates of the inverse of the coupling constant (*Zα*)−<sup>1</sup> for the basis states. In this representation the wave functions are spherical harmonics in four dimensions. We derive the relationship of the usual energy eigenfunctions in momentum space to the spherical harmonics and discuss the classical limits in momentum and configuration space.

#### *5.1. Transformation Properties of the Wave Functions under the Symmetry Operations*

We can show that the wave functions *Ynlm*(*U* ) in the *U* | representation with respect to the eigenstates of (*Zα*)−<sup>1</sup>

$$\mathcal{Y}\_{nlm}(\mathcal{U}') \equiv \langle \mathcal{U}' | nlm \rangle \tag{142}$$

transform as four-dimensional spherical harmonics under the four-dimensional rotations generated by the Runge-Lenz vector *a* and the angular momentum *L*. We note that the quantity *a* is implicit in both the bra and the ket in Equation (142). For our basis states we employ the set of (*Zα*)−<sup>1</sup> eigenstates <sup>|</sup>*nlm*) of the inverse coupling constant that are convenient for momentum space calculations (*<sup>ρ</sup>* <sup>=</sup> *<sup>p</sup>*2+*a*<sup>2</sup> <sup>2</sup>*a*<sup>2</sup> ). We choose these states rather than those convenient for configuration space calculations, because the *U* | eigenstates are proportional to the *p* | eigenstates.

If we transform our system by the unitary operator *<sup>e</sup>i<sup>θ</sup>* where *<sup>θ</sup>* <sup>=</sup> *<sup>L</sup>* · *<sup>ω</sup>* <sup>+</sup> *<sup>a</sup>* · *<sup>ν</sup>*, then the wave function in the new system is

$$Y\_{n\text{Im}}'(\mathcal{U}') = \langle \mathcal{U}' | e^{i\theta} | nlm \rangle. \tag{143}$$

There are two ways in which we may interpret this transformation, corresponding to what have been called the active and the passive interpretations. In the passive interpretation we let *ei<sup>θ</sup>* act on the coordinate eigenstate *U* | . As we have seen in Section 3.6, this produces a new eigenstate *U*"|, where the four-vector *U*" is obtained by a four-dimensional rotation of *U* (Equations (138) and (139)). Thus, we have

$$\mathcal{Y}\_{nlm}'(\mathcal{U}') = \langle \mathcal{U}'' | nlm \rangle = \mathcal{Y}\_{nlm}(\mathcal{U}''). \tag{144}$$

In the active interpretation, we let *<sup>e</sup>i<sup>θ</sup>* act on the basis state <sup>|</sup>*nlm*). Because *<sup>L</sup>* and *<sup>a</sup>* are symmetry operators of the system, transforming degenerate states into each other, it follows that *<sup>e</sup>i<sup>θ</sup>* <sup>|</sup>*nlm*) must be a linear combination of states with principal quantum number equal to n. Therefore, we have

$$\left(Y\_{nlm}^{\prime}(\mathcal{U}^{\prime}) = \sum\_{l'm'} \langle \mathcal{U}^{\prime} | R\_{nl'm'}^{nlm} | nl'm'\rangle \right. \\ \left. \quad = \sum\_{l'm'} R\_{nl'm'}^{nlm} Y\_{nl'm'}(\mathcal{U}^{\prime}). \tag{145}$$

The wave functions for degenerate states with a given *n* transform irreducibly among themselves under the four-dimensional rotations, forming a basis for an irreducible representation of SO(4) of dimensions *n*<sup>2</sup> . Equating the results of the two different interpretations gives

$$\mathcal{Y}\_{nlm}(\mathcal{U}'') = \sum\_{l'm'} \mathcal{R}^{nlm}\_{nl'm'} \mathcal{Y}\_{nl'm'}(\mathcal{U}'). \tag{146}$$

The transformation properties Equation (146) of *Ynlm* are precisely analogous to those of the three-dimensional spherical harmonic functions. It follows that the *Ynlm* are four-dimensional spherical harmonics [35,124].

#### *5.2. Differential Equation for the Four Dimensional Spherical Harmonics Ynlm*(*U* )

The differential equation for the *Ynlm*(*U* ) may be obtained from the equation

$$(\mathcal{L}'^2 + \mathcal{a}'^2)\mathcal{Y}\_{nlm}(\mathcal{U}') = (n^2 - 1)\mathcal{Y}\_{nlm}(\mathcal{U}') \tag{147}$$

which follows from *<sup>C</sup>*<sup>2</sup> <sup>=</sup> *<sup>n</sup>*<sup>2</sup> <sup>−</sup> 1 and the definition of *<sup>C</sup>*2, Equation (81). Substituting in the differential expressions Equations (134) and (141) for *a* and *L* we find that *L-*<sup>2</sup> + *a-*<sup>2</sup> equals <sup>∇</sup><sup>2</sup> *<sup>U</sup>* <sup>−</sup> (*U-* · *<sup>∇</sup><sup>U</sup>* )2, which is the angular part of the Laplacian operator in four dimensions (cf in three dimensions, *<sup>L</sup>*2/*r*<sup>2</sup> <sup>=</sup> *<sup>p</sup>*<sup>2</sup> <sup>−</sup> *<sup>p</sup>*<sup>2</sup> *<sup>r</sup>*). Thus, Equation (147) is the differential equation for four-dimensional spherical harmonics with the degree of homogeneity equal to *<sup>n</sup>* <sup>−</sup> 1, which means *<sup>n</sup>*<sup>2</sup> such functions exist, in agreement with the know degree of degeneracy.

#### *5.3. Energy Eigenfunctions in Momentum Space*

We want to determine the relationship between the usual energy eigenfunctions in momentum space *ψnlm*(*p* ) ≡ *p* |*nlm* (with *a* = *an*) and the four-dimensional spherical harmonic eigenfunctions *Ynlm*(*U* ; *a*) = *U* |*nlm*; *a*).

We choose the RMS momentum *a* to have the value *an*. If we use the expression Equation (131) for *U* | in terms of *p* |

$$
\langle \mathcal{U}' | = \langle p' | \left( \frac{p^2 + a\_n^2}{2a\_n} \right)^{3/2} \tag{148}
$$

and the expression Equation (104) for the eigenstates of (*Zα*)−<sup>1</sup> in terms of the energy eigenstates

$$|nlm;a\_{\rm n}\rangle = \sqrt{\frac{p^2 + a\_n^2}{2a\_n^2}}|nlm\rangle\tag{149}$$

we find the desired result

$$\mathcal{Y}\_{nlm}(\mathcal{U}';a\_n) = \left(\frac{p^2 + a\_n^2}{2a\_n}\right)^2 \frac{1}{\sqrt{a\_n}} \psi\_{nlm}(p'). \tag{150}$$

The usual method of deriving this relationship between the wave function in momentum space and the corresponding spherical harmonics in four dimensions involves transforming the Schrodinger wave equation to an integral equation in momentum space [18,35]. As in the classical case, to do this we first replace *p* by *p*/*a* and perform a stereographic projection from the hyperplane corresponding to the three- dimensional momentum space to a unit hypersphere in a four-dimensional space. The resulting integral equation manifests a four-dimensional invariance. When the wave functions are normalized as in Equation (150), the solutions are spherical harmonics in four dimensions. As another alternative to this procedure, we can Fourier transform the configuration space wave functions directly [126].

#### *5.4. Explicit Form for the Spherical Harmonics*

The spherical harmonics in four dimensions can be expressed as [127]:

$$\mathcal{Y}\_{nlm}(\Omega) = \mathcal{N}\_{1}(n,l)(\sin\theta\_{4})^{l}\mathcal{C}\_{n-1-l}^{l+1}(\cos\theta\_{4}) \cdot \left[\mathcal{N}\_{2}(l,m)(\sin\theta)^{m}\mathcal{C}\_{l-m}^{m+1/2}(\cos\theta)\frac{\varepsilon^{im\phi}}{\sqrt{2}\pi}\right].\tag{151}$$

The factor in brackets is equal to *Y<sup>m</sup> <sup>l</sup>* (*θ*, *φ*), the usual spherical harmonic in three-dimensions [127]. The Gegenbauer polynomials *C<sup>λ</sup> <sup>n</sup>* of degree *n* and order *λ* are defined in terms of a generating function:

$$\frac{1}{(1 - 2t\mathbf{x} + t^2)^{\lambda}} = \sum\_{n=0} t^n \mathbb{C}\_n^{\lambda}. \tag{152}$$

*N*1(*n*, *l*) and *N*2(*l*, *m*) are chosen to normalize the *Ynlm* on the surface of the unit sphere:

$$\int |Y\_{nlm}(\Omega)|^2 d^3 \Omega\_{ll} = 1 \tag{153}$$

where *d*3Ω*<sup>U</sup>* = sin<sup>2</sup> *θ*<sup>4</sup> sin *θdθdφ*. We find

$$N\_1(n,l) = \sqrt{\frac{2^{2l+1}}{\pi} \frac{n(n-l-1)!(l!)^2}{(n+l)!}} \tag{154}$$

$$N\_2(l,M) = \sqrt{\frac{2^{2m}}{\pi} (l+\frac{1}{2}) \frac{(l-m)!}{(l+m)!} [\Gamma(m+\frac{1}{2})]^2} \,. \tag{155}$$

In the next section, we discuss the asymptotic behavior of *Ynlm* for large quantum numbers and compare it to the classical results of Section 3.

#### *5.5. Wave Functions in the Classical Limit*

#### 5.5.1. Rydberg Atoms

Advances in quantum optics, such as the development of ultra short laser pulses, microwave spectroscopy, and atom inteferometery, have opened new possibilities for experiments with atoms and Rydberg states, meaning hydrogenlike atoms in states with very large principal quantum numbers and correspondingly large diameter electron orbits. The pulsed electromagnetic fields can be used to modify the behavior of the orbital electrons. Semi-classical electron wave packets in hydrogenlike atoms were first generated in 1988 by ultrashort laser pulses, and today are often generated by unipolar teraherz pulses [128–130]. Over the last few decades, there has been interest in the classical limit of the hydrogenlike atom for *n* very large, Rydberg states, for a number of reasons [131]: 1. Rydberg states are at the border between bound states and the continuum, and any process which leads to excited bound states, ions or free electrons usually leads to the production of Rydberg states. This includes, for example, photo-ionization or the application of microwave fields. The very large cross section for scattering is unique. 2. Rydberg states can be used to model atoms with a higher atomic number that have an excited valence electron that orbits beyond the core. 3. In Rydberg states, the application of electric and magnetic fields breaks the symmetry of the atom and allows for the study of different phenomena, including the transition from classical to quantum chaos [132]. 4. Rydberg atoms can be used to study coherent transient excitation and relaxation, for example, the response to short laser pulses creating coherent quantum wave packets that behave like a classical particle.

The square of the wave function for a given quantum state gives a probability distribution for the electron that is independent of time. If we want to describe the movement of an electron in a semiclassical state, with a large radius, going around the nucleus with a classical time dependence, then we need to form a wave packet. The wave packet is built as a superposition of many wave functions with a band of principal quantum numbers.

A variety of theoretical methods have been used to derive expressions for the hydrogen atom wave functions and wave packets for highly excited states. There is general agreement on the wave functions for large n, and that the wave functions display the expected classical behavior, elliptical orbits in configurations space, and great circles in the four dimensional momentum space [133–136].

Researchers have proposed a variety of wave packets to describe Rydberg states. There are general similarities in the wave packets that describe electrons going in circular or elliptical orbits with a classical time dependence for some characteristic number of orbits, and it is maintained that the quantum mechanical wave packets provide results that agree with the classical results [129,130,133–139]. Most of the approaches exploit the SO(4) or SO(4,2) symmetry of the hydrogen atom which is used to rotate a circular orbital to an elliptical orbit. The starting orbital is often taken as a coherent state, which is usually considered a classical like state with minimum uncertainty. The most familiar example of a coherent state is for a one dimensional harmonic oscillator characterized by creation and annihilation operators *<sup>a</sup>*† and *<sup>a</sup>*. The coherent state <sup>|</sup>*α* is a superposition of energy eigenstates that is an eigenstate of *a* where *a*|*α* = *α*|*α* for a complex *α*. This coherent state will execute harmonic motion like a classical particle [140]. To obtain a coherent state for the hydrogenlike atom, eigenstates of the operator than lowers the principal quantum number n (which will be discussed in Section 7.4) have been used [141], as well as lowering operators based on the equivalence of the four dimensional harmonic oscillator representation of the hydrogen atom [134,135,142].

In either case, this coherent eigenstate is characterized by a complex eigenvalue, which needs to be specified. Several constraints have been used to obtain the classical wave packet that presumably obeys Kepler's Laws, such as requiring that the orbit lie in a plane so *z* = 0 for the orbital, or that *r* − *rclassical* be a minimum, or that some minimum uncertainty relationship is obtained. In addition, there are issues regarding the approximations used, in particular, those that relate to time. For times

characteristic of the classical hydrogen atom, the wavepackets act like a classical system. For longer times, the wavepacket spreads in the azimuthal direction and after some number of classical revolutions of order 10 to 100 the spread is 2*π*, so the electron is uniformly spread over the entire orbit. The spread arises because the component wave functions forming the wavepacket have different momenta. In two derivations, still longer times were considered, and recoherence was predicted to occur after about *n*/3 (where *n* is the approximate principal quantum number) revolutions, although there is some difference in the predicted amount of recoherence [131,136]. Because of the conservation of *L* and *A*, the spread of the wave packets is inhibited, except in the azimuthal direction.

This is a system with very interesting physics. For example, one can view the transition from a well defined wave packet representing the electron to a 2*π* spread and back again as a dynamical illustration of the wave-particle duality.

Brown took a different approach to develop a wavepacket for a circular orbit [139]. He first developed the asymptotic wave function for large *n* and then optimized the coefficients in a Gaussian superposition to minimize the spread in *φ*, obtaining a predicted characteristic decoherence time of about 10 minutes, considerable longer than any other predicted decoherence time.

Other authors have explored the problem from the perspective of classical physics and the correspondence principle [133,137,143,144]. The results from the different methods are similar with the basic conclusion that the wave functions are peaked on the corresponding classical Kepler trajectories: "atomic elliptic states sew the wave flesh on the classical bones" [129].

With the variety of experimental methods used to generate Rydberg states, a variety of Rydberg wave packets are created, and it is not clear which theoretical model, if any, is preferred [131]. We take a very simple approach to forming a wavepacket and simply use a Gaussian weight for the different frequency components. This does not give an intentionally optimized wave packet, but it is a much simpler approach and the result has all of the expected classical behavior that is very similar to that obtained from much more complicated derivations. We start with a circular orbit and then do a SO(4) rotation to secure an elliptical orbit. We show that it has the classical period of rotation.

#### 5.5.2. Wave Functions in the Semi-Classical Limit

We need to derive the semiclassical limit of the wave functions that correspond to circular orbits in configuration space. For this case, sin *ν*, which we interpret as the expectation value of the eccentricity, vanishes. We derive expressions for the wave functions in momentum space and then form a wavepacket. To obtain corresponding expressions for elliptical orbits, we perform a rotation by *eia*·*ν*, which does not alter the energy but changes the eccentricity and the angular momentum.

#### Case 1: Circular orbits, sin *ν* = *e* = 0

We derive the asymptotic form of *Ynlm* for large quantum numbers, where for simplicity we choose the quantum numbers *n* − 1 = *l* = *m* corresponding to a circular orbit in the 1–2 plane. From Equation (151) we see we encounter Gegenbauer polynomials of the form *C<sup>λ</sup>* <sup>0</sup> , which, by Equation (152), are unity. For a very large *l*, sin*<sup>l</sup> θ* will have a very strong peak at *θ* = *π*/2, so we make the expansion [103]

$$\sin \theta = \sin \left(\frac{\pi}{2} + \left(\theta - \frac{\pi}{2}\right)\right) = 1 - \frac{1}{2}\left(\theta - \frac{\pi}{2}\right)^2 + \dotsb \quad \approx e^{-(1/2)(\theta - \pi/2)^2} \tag{156}$$

to obtain

$$\sin^l \theta \approx e^{-(1/2)l(\theta - \pi/2)^2}.\tag{157}$$

The asymptotic forms for *N*<sup>1</sup> and *N*<sup>2</sup> can be computed using the properties of Γ functions:

$$\begin{aligned} \Gamma(2z) &= \left(\frac{1}{\sqrt{2\pi}}\right) 2^{2z - \frac{1}{2}} \Gamma(z) \Gamma\left(z - \frac{1}{2}\right) \\\\ \lim\_{z \to \infty} \Gamma(az + b) &\simeq \sqrt{2\pi} e^{-az} (az)^{az + b - \frac{1}{2}} .\end{aligned} \tag{158}$$

We finally obtain [145]

$$\mathcal{Y}\_{n,n-1,n-1}(\Omega) = \sqrt{\frac{n}{2\pi^2}} e^{-\frac{1}{2}n\left(\theta\_4 - \frac{\pi}{2}\right)^2} \cdot e^{-\frac{1}{2}n\left(\theta - \frac{\pi}{2}\right)^2} e^{i(n-1)\theta} \,, \tag{159}$$

which gives the probability density

$$\left|\left|\boldsymbol{Y}\_{n,n-1,n-1}(\Omega)\right|\right|^2 = \frac{n}{\left(2\pi^2\right)} e^{-n\left(\theta\_4 - \frac{\pi}{2}\right)^2} \cdot e^{-n\left(\theta - \frac{\pi}{2}\right)^2}.\tag{160}$$

We have Gaussian probability distributions in *θ*<sup>4</sup> and *θ* about the value *π*/2. The distributions are quite narrow with widths Δ*θ*<sup>4</sup> ≈ Δ*θ* ≈ 1/ <sup>√</sup>*<sup>n</sup>* and the spherical harmonic essentially describes a circle (*θ*<sup>4</sup> = *θ* = *π*/2) on the unit sphere in the 1–2 plane. As *n* becomes very large, *U*<sup>4</sup> = cos *θ*<sup>4</sup> ≈ (*r* − *rc*)/*r* (Equations (54) and (72)) and *U*<sup>3</sup> = sin *θ*<sup>4</sup> cos *θ*, which is proportional to *p*3, both go to zero as 1/ <sup>√</sup>*n*. The distribution approaches the great circle *U*<sup>2</sup> <sup>1</sup> + *<sup>U</sup>*<sup>2</sup> <sup>2</sup> = 1 that we found in Section 3.6 for a classical particle moving in a circular orbit in the 1–2 plane in configuration space. Note that this state is a stationary state with a constant probability density. To get the classical time dependence we need to form a wavepacket.

#### Forming a Wavepacket

We form a time dependent wavepacket for circular orbits by superposing circular energy eigenstates:

$$\chi(\Omega, t) = \sum\_{n} e^{itE\_n} Y\_{n, n-1, n-1} A\_{n-N} \tag{161}$$

where *An*−*<sup>N</sup>* is an amplitude peaked about *n* = *N* >> 1. For *n* >> 1 we expand *En* about *EN*:

$$E\_n = E\_N + \left. \frac{\partial E}{\partial n} \right|\_N s + \left. \frac{\partial^2 E}{\partial n^2} \right|\_N s^2 + \dots \tag{162}$$

where *<sup>s</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> *<sup>N</sup>*. From the equation for the energy levels, *<sup>E</sup>* <sup>=</sup> <sup>−</sup>*m*(*Zα*)2/(2*n*2) we compute

$$\frac{\partial E\_{\rm tr}}{\partial n}\Big|\_{N} = \frac{m(Z\alpha)^2}{N^3} = \sqrt{\frac{-8E\_N^3}{m(Z\alpha)^2}}.\tag{163}$$

In agreement with the Bohr Correspondence Principle, the right-hand side of this equation is just the classical frequency *ωcl* as given in Equation (38). For the second order derivative, we have

$$\frac{\partial^2 E}{\partial n^2}\Big|\_{N} = -\frac{3}{N}\omega\_{cl} \equiv \beta\tag{164}$$

which gives

$$\chi(\Omega, t) = e^{-it\mathbb{E}\_N} e^{i\phi(N-1)} \sum\_{\varepsilon=-N+1}^{\infty} e^{-i\left(\omega\_{i1}\varepsilon t - \left(\frac{\beta}{2}\right)s^2 t - \varepsilon\phi\right)} \cdot A\_{\delta} |\boldsymbol{\gamma}\_{N+\varepsilon, N+\varepsilon-1, N+\varepsilon-1}| \tag{165}$$

We choose a Gaussian form for *As*

$$A\_{\rm s} = \frac{1}{\sqrt{2\pi N}} e^{-\rm s^2/(2N)}.\tag{166}$$

Brown used *As* = *Ce*−*s*23*ωclt*/*<sup>N</sup>* which minimizes the diffusion in *<sup>φ</sup>* at time *<sup>t</sup>* [139]. Since |*YN*+*s*,*N*+*s*−1,*N*+*s*−1| varies slowly with *s* for *N* >> 1 we can take it outside the summation in Equation (165). We now replace the sum by an integral over *s*. Because *As* is peaked about *N* we can integrate from *s* = −∞ to *s* = +∞. We perform the integral by completing the square in the usual way. The final result for the probability amplitude for a circular orbital wave packet is

$$\begin{split} \left| \mathbf{x} (\Omega, t) \right|^2 &= \left| Y\_{N, N-1, N-1} \right|^2 \left( 1 + \beta^2 t^2 N^2 \right)^{-1} \\ &\cdot \exp \left[ -\left( \phi - \omega\_{cl} t \right)^2 \frac{N}{1 + (\beta t N)^2} \right]. \end{split} \tag{167}$$

This represents a Gaussian distribution in *φ* that is centered about the classical value *φ* = *ωclt*, meaning that the wavepacket is traveling in the classical trajectory with the classical time dependence. The width of the *φ* distribution is

$$
\Delta\Phi = (N)^{(-1/2)} (1 + \beta^2 t^2 N^2)^{1/2} = (N)^{(-1/2)} (1 + 9\omega\_{cl}^2 t^2)^{1/2}.\tag{168}
$$

The distribution in *φ* at *t* = 0 is very narrow, proportional to 1/ <sup>√</sup>*N*, but after several orbits <sup>Δ</sup>*<sup>φ</sup>* is increasing linearly with time.

The distributions in *θ*<sup>4</sup> and *θ* are Gaussian and centered about *π*/2 in each case as for the circular wave function (cf. Equation (160)) with widths equal to (*N*)−1/2. The spreading of these distributions in time is inhibited because of the conservation of angular momentum and energy. The detailed behavior of the widths depend on our use of the Gaussian distribution. Other distributions will give different widths, although the general behavior is expected to be similar.

As a numerical example, consider a hydrogen atom that is in the semiclassical region when the orbital diameter is about 1 cm. The corresponding principal quantum number is about 104, the mean velocity is about 2.2 <sup>×</sup> <sup>10</sup><sup>4</sup> cm/s and the period about 1.5 <sup>×</sup> <sup>10</sup>−<sup>4</sup> s. After about 34 revolutions or <sup>5</sup> <sup>×</sup> <sup>10</sup>−<sup>3</sup> s, the spread in *<sup>φ</sup>* is about 2*π*, meaning the electron is spread uniformly about the entire circular orbit. This characteristic spreading time can be compared to 1.6 <sup>×</sup> <sup>10</sup>−<sup>3</sup> s for a fully optimized wave packets formed from coherent SO(4,2) states [136,146]. In order to make predictions about significantly longer times, we would need to retain more terms in the expansion Equation (162) of *En*.

Case 2: Elliptical orbits sin *ν* = *e* = 0

We can obtain the classical limit of the wave function for elliptical orbits by first writing our asymptotic form Equation (159) for *Yn*,*n*−1,*n*−<sup>1</sup> in terms of the *U* variables instead of the angular variables by using definitions Equation (52), and setting *a* = *an*. Retaining only the lowest order terms in (*θ*<sup>4</sup> − *π*/2) and (*θ* − *π*/2), we find

$$\mathcal{Y}\_{n,n-1,n-1}(\mathcal{U}) = \left(\frac{n}{2\pi^2}\right)^{\frac{1}{2}} e^{i(n-1)\tan^{-1}\left(\frac{l\mathcal{U}\_2}{\mathcal{U}\_1}\right)} \cdot e^{-\frac{1}{2}n(l\mathcal{U}\_4)^2} e^{-\frac{1}{2}n(l\mathcal{U}\_3)^2} \tag{169}$$

For large n, this represents a circular orbit in the 1–2 plane. We now perform a rotation by *A*2*ν*. which will change the eccentricity to sin *ν*, and change the angular momentum, but will not change the energy or the orbital plane. Using Equation (140) to express the old coordinates in terms of the new coordinates, we find to lowest order

$$Y\_{n,n-1,n-1}^{\prime}(lI) = \left(\frac{n}{2\pi^2}\right)^{1/2} e^{i(n-1)\tan^{-1}\left(\frac{lL\_2}{lI\_1\cos\nu}\right)} \cdot e^{-\frac{1}{2}n\left\{lL\_2\sin\nu - lL\_4\cos\nu\right\}^2} \cdot e^{-\frac{1}{2}n\left(lL\_3\right)^2} \cdot \tag{170}$$

In Section 3.6, we found that the vanishing of the term in braces 0 = *U*<sup>2</sup> sin *ν* − *U*<sup>4</sup> cos *ν* specifies the classical great hypercircle orbit (Equation (56)) corresponding to an ellipse in configuration space with eccentricity *e* = sin *ν* and lying in the 1–2 plane. The probability density |*Y <sup>n</sup>*,*n*−1,*n*−1(*U*)<sup>|</sup> <sup>2</sup> vanishes except within a hypertorus with a narrow cross section of radius approximately <sup>√</sup><sup>1</sup> *<sup>n</sup>* , which is centered about the classical distribution. Because the width <sup>√</sup><sup>1</sup> *<sup>n</sup>* of the distribution is constant in U space, it will not be constant when projected onto p space.

In terms of the original momentum space variables, the asymptotic spherical harmonic is

$$Y\_{n,n-1,n-1}(p) = \left(\frac{\
u}{2\pi^2}\right)^{\frac{1}{2}} e^{i(n-1)\tan^{-1}\left(\frac{p\_2}{p\_1\cos\upsilon}\right)}$$

$$\cdot \exp\left\{-\left(\frac{\eta}{2}\right)\left[p\_1^2 + (p\_2 - a\tan\upsilon)^2 - \frac{a^2}{\cos^2\upsilon}\right]^2 \left(\frac{\cos\upsilon}{p^2 + a^2}\right)\right\}^2 \quad . \tag{171}$$

$$\cdot \exp\left\{-\left(\frac{\eta}{2}\right)\left(\frac{2p\_3a}{p^2 + a^2}\right)\right\}^2 \Big| a = a\_{\text{fl}}$$

The expression in brackets corresponds to the momentum space classical orbit equation we found previously (Equation (49)). As we expect, *p*<sup>3</sup> is Gaussian about zero since the classical orbit is in the 1–2 plane. We can simplify the expressions for the widths by observing that to lowest order we can use Equation (48), which implies *p*<sup>2</sup> + *a*<sup>2</sup> = 2*a*<sup>2</sup> + 2*ap*<sup>2</sup> tan *ν* in the exponentials. The widths of both distributions therefore increase linearly with *p*2. We also note that, since classically there exists a one-to-one correspondence between each point of the trajectory in momentum space and each point in configuration space, we may interpret the widths of the distributions using Equation (32) *<sup>p</sup>*2+*a*<sup>2</sup> *<sup>a</sup>*<sup>2</sup> <sup>=</sup> <sup>2</sup>*rc r* . Accordingly, the widths increase as the momentum increases or as the distance to the force center decreases (Figure 8).

**Figure 8.** Wave function probability distribution |*Y <sup>n</sup>*,*n*−1,*n*−1(*p*)<sup>|</sup> <sup>2</sup> in momentum space for large n, showing the variation in the width of the momentum distribution about the classical circular orbit. The center of the distribution is at *p*<sup>2</sup> = *a* tan *ν*. The classical orbit is in the 1–2 plane.

#### Forming a Wave packet for Elliptical Motion

We may form a time dependent wavepacket superposing the wave functions of Equation (170). Care must be taken to include the first order dependence (through *an*) of tan−1(*U*2/*U*<sup>1</sup> cos *ν*) on the principal quantum number when integrating over the Gaussian weight function. The result for the probability density is the same as before (Equation (167)), except |*Y <sup>N</sup>*,*N*−1,*N*−1| <sup>2</sup> (given in Equation (170)) replaces |*YN*,*N*−1,*N*−1| <sup>2</sup> and

$$
\omega\_{cl}t = \tan^{-1}\left(\frac{lI\_2}{lI\_1\cos\nu}\right) + \sin\nu\left(lI\_1\right) \tag{172}
$$

where replaces *ωclt* = *φ* . The result is exactly the same as the classical time dependence Equation (73). The spreading of the wave packet will be controlled by the same factor as for the circular wave packet.

#### Remark on the Semiclassical Limit in Configuration Space

The time dependent quantum mechanical probability density follows the classical trajectory in momentum space meaning that the probability is greatest at the classical location of the particle in momentum space. Because the configuration space wave function is the Fourier transform of the momentum space wave function, the classical limit must also be obtained in configuration space. That this limit is obtained is made explicit by observing that the momentum space probability density is large when

$$(\mathcal{U}\_2 \sin \nu - \mathcal{U}\_4 \cos \nu)^2 \approx 0. \tag{173}$$

However from Section 3 we can show that

$$\mathcal{U}l\_2 \sin \nu - \mathcal{U}\_4 \cos \nu = \cos \nu \left(\frac{r - r\_{classical}}{r\_{classical}}\right) \tag{174}$$

where *rclassical* is given by the classical orbit Equation (33). Accordingly, we see that the configurations space probability will be large when

$$\left[\frac{r - r\_{classical}}{r\_{classical}}\right]^2 \approx 0.\tag{175}$$

#### *5.6. Quantized Semiclassical Orbits*

It is convenient at times to have a semiclassical model for the orbitals of the hydrogenlike atom. Historically this was first done by Pauling and Wilson [147]. We can obtain a model by interpreting the classical formulae for the geometrical properties of the orbits as corresponding to the expectation values of the appropriate quantum mechanical expressions. Thus, when the energy *<sup>E</sup>* <sup>=</sup> <sup>−</sup>*a*2/2*<sup>m</sup>* appears in a classical formula, we employ the expression for *a* for the quantized energy levels *a* = <sup>1</sup> *nr*<sup>0</sup> where *r*<sup>0</sup> = (*mZα*)−1, which is the radius 0.53 Angstrom of the ground state. Similarly if *L*<sup>2</sup> appears in a classical formula, we substitute *l*(*l* + 1), where *l* is quantized *l* = 0, 1, 2..*n* − 2, *n* − 1; and *m* , the component of *<sup>L</sup>* along the 3-axis is quantized: *<sup>m</sup>*<sup>=</sup> <sup>−</sup>*l*, <sup>−</sup>*<sup>l</sup>* <sup>+</sup> 1, <sup>−</sup>*<sup>l</sup>* <sup>+</sup> 2, ..*l*.

Orbits in Configuration Space

Recalling Equation (28) *rc* = *mZ<sup>α</sup> <sup>a</sup>*<sup>2</sup> , we see *arc* <sup>=</sup> *<sup>n</sup>*, which gives a semimajor axis of length *rc* <sup>=</sup> *<sup>n</sup>*2*r*0, where *r*<sup>0</sup> is the radius for the circular orbit of the ground state and *rc* is for a circular orbit for a state with principal quantum number *n*. The equations for the magnitude of *L* and *A* are

$$L = r\_c a \cos \upsilon = n \cos \upsilon = \sqrt{l(l+1)}\tag{176}$$

$$A = r\_c a \sin \upsilon = n \sin \upsilon = \sqrt{n^2 - l(l+1)}\tag{177}$$

This gives an eccentricity sin *ν* equal to

$$x = \sin v = \sqrt{1 - \frac{l(l+1)}{n^2}}\tag{178}$$

and a semi-major axis equal to

$$b = r\_{\mathcal{C}} \sin \nu = n \sqrt{l(l+1)}.\tag{179}$$

Note that the expression for *e* is limited in its meaning. For an *s* state, it always gives *e* = 1, and for states with *<sup>l</sup>* <sup>=</sup> *<sup>n</sup>* <sup>−</sup> 1 it give *<sup>e</sup>* <sup>=</sup> <sup>√</sup>1/*n*, not the classically expected 0 for a circular orbit.

#### Orbits in U-Space

The corresponding great hypercircle orbits (*ν*, Θ) in U-space are described by giving the quantized angle *ν*, between the three-dimensional hyperplane of the orbit and the 4-axis, and the quantized angle Θ, between the hyperplane of the orbit and the 3-axis:

$$\cos \nu = \sqrt{\frac{l(l+1)}{n^2}}\tag{180}$$

$$\cos \Theta = \sqrt{\frac{m^2}{l(l+1)}}.\tag{181}$$

Note the similarity in these two equations, suggesting that *m* relates to *l* the same way that *l* relates to *n*, which suggests a generalization of the usual vector model of the atom which only describes the precession of *L* about the *z* axis.

The results for orbits in configuration and momentum space illustrate some interesting features:

1. The equation *ar* = *n* illustrates that the characteristic dimensions of an orbit in configuration space and the corresponding orbit in momentum space are inversely proportional, as expected, since they are related by a Fourier transform, consistent with the Heisenberg Uncertainty Principle.

2. If *l* = 0, then no classical state exists. The orbit in configuration space degenerates into a line passing through the origin while the corresponding circular orbit in momentum space attains an infinite radius and an infinite displacement from the origin. Although this seems peculiar from the pure classical viewpoint, quantum mechanically it follows, since for S states there is a nonvanishing probability of finding the electron within the nucleus.

In order to interpret these statements about quantized semiclassical elliptical orbits we observe that for quantum mechanical state of the hydrogenlike atom with definite *n*, *l*, *m*, the probability density is (1) independent of *φ<sup>r</sup>* or *φ<sup>p</sup>* and (2) it does not confine the electron to some orbital plane. Because the quantum mechanical distribution for such a state specifies no preferred direction in the 1–2 plane, we must imagine this distribution as corresponding in some way to an average over all possible orientations of the semiclassical elliptical orbit. This interpretation is supported by the fact that the region within which the quantum mechanical radial distribution function differs largely from zero is included between the values of *r* corresponding to the semiclassical turning points *rc*(1 ± sin *ν*).

#### *5.7. Four-Dimensional Vector Model of the Atom*

In configuration space or momentum space" the angle between the classical plane of the orbit and the 3-axis is Θ, which is usually interpreted in terms of the vector model of the atom in which we imagine *L* to be a vector of magnitude *l*(*l* + 1) precessing about the 3-axis, with *m* as the component along the 3-axis. This precession may be linked to the *φ<sup>r</sup>* independence of the probability and the absence of an orbital plane as mentioned at the end of the preceding section. The precession constitutes a classical mechanism which yields the desired average over all possible orientations of the semiclassical elliptical orbit. Because the angle Θ is restricted to have only certain discrete values one can say that there is a quantization of space.

The expression for cos *ν* = *l*(*l* + 1)/*n*<sup>2</sup> is quite analogous to that for Θ and so suggests a generalization of the vector model of the atom to four dimensions. The projection of the four-dimensional vector model onto the physical three-dimensional subspace must give the usual vector model. We can achieve this by imagining that a four-dimensional vector of length *n*, where *n* is the principal quantum number, is precessing in such a way that its three and four components are constants, while the one and two components vary periodically. The projection onto the 1–2–3 hyperplane is a vector of constant magnitude *l*(*l* + 1) precessing about the 3-axis. The component along the 3-axis is *<sup>m</sup>*. The component along the 4-axis is *<sup>A</sup>* <sup>=</sup> *n*<sup>2</sup> <sup>−</sup> *<sup>l</sup>*(*<sup>l</sup>* <sup>+</sup> <sup>1</sup>) the magnitude of the vector *A*. The vectors *L* and *A* are perpendicular to each other. Thus, the precessing *n* vector makes a constant angle Θ with the 3-axis and a constant angle *π*/2 − *ν* with the 4-axis. Because both angles are restricted to certain values, we may say that we have a quantization of four-dimensional space.

#### **6. The Spectrum Generating Group SO(4,1) for the Hydrogenlike Atom**

We consider the Schrodinger hydrogen atom and its unitary "noninvariance" or spectrum generating operators *eiDiβ<sup>i</sup>* where *Di* is a generator and *β<sup>i</sup>* is a real parameter, using eigenstates of (*Zα*)−<sup>1</sup> for our basis of our representation. These operators transform an eigenstate of the kernel K with a definite value of the coupling constant (or principal quantum number) into a linear combination of eigenstates with different values of the coupling constant (or different principal quantum numbers), and different *l* and *m*. Unlike the invariance generators *L* and *A*, the noninvariance generators clearly do not generally commute with the kernel K, [*Di*, *K*] = 0, so they change the principal quantum number.

The set of all invariance and noninvariance operators forms a group with which we may generate all eigenstates in our complete set from a given eigenstate. We show that this group, called the Spectrum Generating Group of the hydrogenlike atom, is SO(4,1), the group of orthogonal transformations in a 5-dimensional space with a metric *gAB* = (−1, 1, 1, 1, 1), where *A*, *B* = 0, 1, 2, 3, 4. The complete set of eigenstates of (*Zα*)−<sup>1</sup> for the hydrogenlike atom forms a unitary, irreducible, infinite-dimensional representation of SO(4,1) which, we shall find, can be decomposed into an infinite sum of irreducible representations of SO(4), each corresponding to the degeneracy group for a particular principal quantum number. A unitary representation means all generators are unitary operators. An irreducible representation does not contain lower dimensional representations of the same group. In Section 6.3, we discuss the isomorphism between the spectrum generating group SO(4,l) and the group of conformal transformations in momentum space. An isomorphism means the groups have the same structure and can be mapped into each other.

#### *6.1. Motivation for Introducing the Spectrum Generating Group Group SO(4,1)*

We have examined the group structure for the degenerate eigenstates of (*Zα*)−<sup>1</sup> for the Schrodinger hydrogenlike atom: the *n*<sup>2</sup> degenerate states form an irreducible representation of SO(4). The next question we might ask is: Do all or some of the states with different principal quantum numbers form an irreducible representation of some larger group which is reducible into SO(4) subgroups? If such a group exists then it clearly is not an invariance group of the kernel K (Equation (106)). If we want our noninvariance group to include just some of the states then it will be a compact group, since unitary representations of compact groups can be finite dimensional. If we want to include all states then it will be a noncompact group since there are an infinite number of eigenstates of (*Zα*)−<sup>1</sup> and all unitary representations of noncompact groups are infinite dimensional [9].

We can find a compact noninvariance group for the first N levels of the coupling constant, *n* = 1, 2, ...*N*. The dimensionality of our representation is

$$\sum\_{n=1}^{N} n^2 = \frac{N(N+1)(2N+1)}{6}.\tag{182}$$

Mathematical analysis of the group SO(5) shows that this is the dimensionality of the irreducible symmetrical tensor representation of SO(5), given by the tensor with five upper indices *Tabc*..., where *a*, *b*, .. = 1, 2, 3, 4 or 5 [11]. Reducing this representation of SO(5) into its SO(4) components gives

$$\begin{aligned} (\text{symm} \cdot \text{tensor } N)\_{SO(5)} &= (0,0) \oplus \left(\frac{1}{2}, \frac{1}{2}\right) \oplus \dots \quad \oplus \left(\frac{N-1}{2}, \frac{N-1}{2}\right) \\ &= (\text{symm} \cdot \text{ tensor } n=1)\_{SO(4)} \oplus \\ (\text{symm} \cdot \text{ tensor } n=2)\_{SO(4)} \oplus \\ &\dots \oplus \quad (\text{symm} \cdot \text{ tensor } n=N)\_{SO(4)} \end{aligned} \tag{183}$$

which is precisely the structure of the first N levels of a hydrogenlike atom. If we want to include all levels then we guess that the appropriate noncompact group is SO(4,1), whose maximal compact subgroup is SO(4). Thus, we conjecture that all states form a representation of SO(4,1).

Consider the Lie algebra of O(4,1) and the general structure of its generators in terms of the canonical variables. The algebra of O(n) has *<sup>n</sup>*(*n*−1) <sup>2</sup> generators so to extend the algebra of O(n) to O(n+1) takes n generators, which can be taken as the components of a n-vector. To extend the Lie algebra from O(4) to O(5) or O(4,1) we can choose the additional generators *Ga* to be components of a four-vector G under O(4):

$$\left[\left[S\_{ab}, G\_c\right] = i \left(G\_b \delta\_{ac} - G\_a \delta\_{bc}\right) \quad a, b, c = 1, 2, 3, 4. \tag{184}$$

If we apply Jacobi's identity to *Sab*, *Ga*, and *Gb* and use Equation (184) we find

$$\mathbb{E}\left[\mathbb{S}\_{ab\prime}\left[\mathbb{G}\_{a\prime},\mathbb{G}\_{b}\right]\right] = 0.\tag{185}$$

We require that the Lie algebra closes, so [*Ga*, *Gb*] must be a linear combination of the generators, clearly proportional to *Sab* and we choose the normalization, such that

$$\left[\mathcal{G}\_{\mathbf{a}\prime}\mathcal{G}\_{\mathbf{b}}\right] = -i\mathcal{S}\_{\mathbf{a}\prime}.\tag{186}$$

If we define

$$G\_4 = S\_{40} = S; \quad G\_i = S\_{i0} = B\_i \tag{187}$$

and recall Equation (79)

$$L\_i = e\_{ijk} S\_{jk} \quad A\_i = S\_{i4}.$$

then the additional commutation relations that realize SO(4,1) may be written in terms of *L*, *A*, *B*, and *<sup>S</sup>*:

$$\begin{aligned} \left[L\_i, B\_j\right] &= i\varepsilon\_{ijk} B\_k & \left[L\_i, S\right] &= 0\\ \left[S\right., A\_j\right] &= iB\_j & \left[S\,, B\_j\right] &= iA\_j\\ \left[A\_j, B\_k\right] &= i\delta\_{jk} S & \left[B\_i, B\_j\right] &= -i\varepsilon\_{ijk} L\_k. \end{aligned} \tag{188}$$

The top two commutators show that *B* transforms as a three-vector under O(3) rotations and that *S* is a scalar under rotations. Alternatively, we can write the commutation relations in terms of the generators *SAB* , *A*, *B* = 0, 1, 2, 3, 4 :

$$\left[\mathcal{S}\_{AB}, \mathcal{S}\_{\mathbb{CD}}\right] = i \left(\mathcal{g}\_{AC}\mathcal{S}\_{BD} + \mathcal{g}\_{BD}\mathcal{S}\_{AC} - \mathcal{g}\_{AD}\mathcal{S}\_{BC} - \mathcal{g}\_{BC}\mathcal{S}\_{AD}\right) \tag{189}$$

where *g*<sup>00</sup> = −1, *gaa* = 1.

The commutators above follow directly from the mathematical theory of SO(4,1), but the theory does not tell us what these generators represent, just their commutation properties. We now investigate the general features of the representations of SO(4,1) provided by the hydrogenlike atom and how to represent the generators in terms of the canonical variables.

#### *6.2. Casimir Operators*

The two Casimir operators of SO(4,1) are [120]

$$Q\_2 = -\frac{1}{2} S\_{AB} S^{AB} = S^2 + \mathcal{B}^2 - \mathcal{A}^2 - L^2 \tag{190}$$

and

$$Q\_4 = -w\_A w^A = (SL - A \mathbf{x} \mathbf{B})^2 - \frac{1}{4} [L \cdot (A + \mathbf{B}) - (A + \mathbf{B}) \cdot L]^2 \tag{191}$$

where *wA* = <sup>1</sup> <sup>8</sup> *ABCDESBCSDE* .

For SO(4), we recall that for the SO(4) representations the structure of the generators in terms of the canonical variables led to the vanishing of one Casimir operator *<sup>C</sup>*<sup>1</sup> <sup>=</sup> *<sup>L</sup>* · *<sup>A</sup>* and, consequently, the appearance of only symmetrical tensor representations. We will find *Q*<sup>4</sup> vanishes for analogous reasons.

If *B* is a pseudovector, it is proportional to *L*, which is the only independent pseudovector that can be constructed from the dynamical variables. The coefficient of proportionality, a scalar, *X* need not commute with *H*:

$$B = XL \quad [X,L] = 0 \quad [X,H] \neq 0 \tag{192}$$

Because [*Bi*, *Bj*] = <sup>−</sup>*ieijkLk* it follows that *<sup>X</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>1 and *<sup>B</sup>* would therefore be a constant multiple of *L* and not an independent generator. Thus *B* must be a vector and expressible as

$$B = f\mathbf{r} + h\mathbf{p} \tag{193}$$

where f and h are scalar functions of *<sup>r</sup>*, *<sup>p</sup>*2, and *<sup>r</sup>* · *<sup>p</sup>*. Accordingly we find

$$\mathbf{B} \cdot \mathbf{L} = \mathbf{L} \cdot \mathbf{B} = 0 \tag{194}$$

Further, since *<sup>B</sup>* is a vector and *<sup>A</sup>* is a vector, *<sup>A</sup>* <sup>×</sup> *<sup>B</sup>* is a pseudovector and therefore is proportional to *L* :

$$\mathbf{A} \times \mathbf{B} = \mathbf{Y}\mathbf{L} \quad , \quad [\mathbf{Y}, \mathbf{L}] = \mathbf{0} \tag{195}$$

For this equation to be consistent with the SO(4,1) commutation relations we find *Y* = *S* and, therefore

$$A \times \mathcal{B} = SL.$$

It follows from substituting *<sup>L</sup>* · *<sup>A</sup>* <sup>=</sup> 0 and Equations (194) and (196) into Equation (191) that for the SO(4,1) representations realized by the hydrogenlike atom

$$Q\_4 = 0.\tag{197}$$

As with the SO(4) symmetry, the dynamics of the hydrogen atom require that only certain representations of SO(4,1) appear. From the mathematical theory of irreducible infinite dimensional unitary representations of SO(4,1), we have the following results:

$$\begin{aligned} \text{Class I:} \quad & Q\_4 = 0; \quad Q\_2 \text{ real, } > 0\\ & \text{SU(2) } \times \text{SU(2) content:} \\ & (Q)^I = (0, 0) \oplus \left(\frac{1}{2}, \frac{1}{2}\right) \oplus (1, 1) \oplus \dots \end{aligned} \tag{198}$$

Class II: *Q*<sup>4</sup> = 0, *Q*<sup>2</sup> = −(*s* − 1)(*s* + 2),*s* = integer > 0

$$\operatorname{SU}(2)\times\operatorname{SU}(2)\text{ content:}\tag{199}$$

$$\mathbb{P}(\mathbb{Q})^{II}=\left(\frac{s}{2},\frac{s}{2}\right)\oplus\left(\frac{s+1}{2},\frac{s+1}{2}\right)\oplus\dotsb\tag{190}$$

The class I representations are realized by the complete set of eigenstates of (*Zα*)−<sup>1</sup> for the hydrogenlike atom. Note, however, that we have an infinite number of such class I representations since *Q*<sup>2</sup> may have any positive real value. We shall find that for *Q*<sup>2</sup> = 2 we may extend our group from SO(4,l) to SO(4,2). The class II representations are realized by the eigenstates of (*Zα*)−<sup>1</sup> with principal quantum numbers from *n* = *s* + 1 to *n* becomes infinite. The first *s* levels could, if we desire, be described by SO(5).

In this section, we have analyzed the group structure and the representations using the complete set of eigenstates of (*Zα*)−<sup>1</sup> for our basis. We might ask: what if we used energy eigenstates instead as a basis for the representations? From Section 4.3 we know that the quantum numbers and multiplicities of the (*Zα*)−<sup>1</sup> eigenstates are precisely the same as those of the bound energy eigenstates. Thus, with the energy eigenstates as our basis, we would reach the same conclusions about the group structure as before but we would be including only the bound states in our representations and we would be ignoring all scattering states.

#### *6.3. Relationship of the Dynamical Group SO(4,1) to the Conformal Group in Momentum Space*

We can give a more complete analysis of the hydrogenlike atom in terms of SO(4,1) by considering the relationship between the four-dimensional rotations of the four-vector *U <sup>a</sup>*, with *a* = *l*, 2, 3, 4, which we discussed in Section 4.6, and the group of conformal transformations in momentum space. Conformal transformations preserve the angles between directed curves, but not necessarily lengths. The rotations generated by the Runge-Lenz vector *a* and the angular momentum *L* leave the scalar product *UaV<sup>a</sup>* of four-vectors invariant and, therefore, are conformal transformations. The stereographic projection we employed is also a conformal transformation. Since the product of two conformal transformations is itself a conformal transformation, we must conclude that *a* generates a conformal transformation of the momentum three-vector *p*.

We must introduce two additional operators that correspond to the operators *B* and *S* introduced in Section 6.1 in order to express the most general conformal transformation. By employing the isomorphism between the generators *L*, *a*, *B*, and *S* of SO(4,1) and the generators of conformal transformations in momentum space we can immediately obtain expressions for the additional generators *B* and *S* in terms of the canonical variables, which is our objective. We need these additional generators to complete our SO(4,1) group for the hydrogen atom.

To derive the isomorphism we use the most convenient representation, namely that based on eigenstates of (*Zα*)−<sup>1</sup> convenient for momentum space calculations (*<sup>ρ</sup>* <sup>=</sup> (*p*2+*a*2) <sup>2</sup>*a*<sup>2</sup> ). Once established, the isomorphism becomes a group theoretical statement and it is independent of the particular representation.

The Conformal Group in Momentum Space

An arbitrary infinitesimal conformal transformation in momentum three-space may be written as

$$
\delta p\_j = \delta a\_j + \delta \omega\_{jk} p\_k + \delta \rho p\_j + \left(p^2 \delta c\_j - 2p\_j p \cdot \delta \mathbf{c}\right) \tag{200}
$$

where *δωjk* = −*δωkj*.

The terms in *δpj* arise as follows:

$$\begin{aligned} \delta a\_{\rangle} \text{ translation generated by } \mathbf{R} \cdot \delta \mathbf{a} \\ \delta \omega\_{jk} \text{ rotation generated by } I \cdot \delta \omega\_{\prime} J\_{\parallel j} &= \varepsilon\_{ijk} J\_{k} \\ \delta \rho \text{ dilation generated by } D \delta \rho \\ \delta c\_{j} \text{ special conformal transformation generated by } \mathbf{K} \cdot \delta \mathbf{c} \end{aligned} \tag{201}$$

This is a ten parameter group with the generators (*R*, *J*, *D*, *K*) which obey the following commutation relations:

$$\begin{array}{ll} \left[D, R\_{j}\right] \cdot = iR\_{j} & \left[D, J\_{i}\right] = 0\\ \left[D, K\_{j}\right] = -iK\_{j} & \left[R\_{i}, J\_{k}\right] = i\epsilon\_{ijk}R\_{k} \\ \left[K\_{n}, R\_{m}\right] = 2i\epsilon\_{nmr}J\_{r} - 2i\delta\_{mn}D & \left[I\_{i}, J\_{k}\right] = i\epsilon\_{ikm}J\_{m} \\ \left[R\_{i}, R\_{j}\right] = 0 & \left[K\_{i}, J\_{k}\right] = i\epsilon\_{ikm}K\_{m} \\ \left[K\_{j}, K\_{j}\right] = 0 \end{array} \tag{202}$$

There is an isomorphism between the algebra of the generators of conformal transformations and the dynamical noninvariance algebra of SO(4,1) of the hydrogen atom. Because *Ji* is the generator of spatial rotations we make the association *Li* = *Ji*. Comparing the differential change in *pi* from a transformation generated by *<sup>a</sup>* · *<sup>δ</sup><sup>ν</sup>* (in the representation with *<sup>ρ</sup>* <sup>=</sup> (*p*2+*A*2) <sup>2</sup>*a*<sup>2</sup> )

$$\begin{split} \delta p\_i &= i \left[ \mathfrak{a} \cdot \delta \nu, p\_i \right] \\ &= -\frac{1}{2a} \left[ \left( p^2 - a^2 \right) \delta \nu\_i - 2 \mathfrak{p} \cdot \delta \nu p\_i \right] \end{split} \tag{203}$$

to the differential change in *pi* from a conformal transformation leads to the association

$$a\_{\bar{l}} = \frac{1}{2} \left( \frac{K\_{\bar{l}}}{a} - aR\_{\bar{l}} \right). \tag{204}$$

To confirm the identification we can use the commutation relations of the conformal group to show that the O(4) algebra of *L* and *a* corresponds precisely to that of *J* and <sup>1</sup> 2 ( *K <sup>a</sup>* <sup>−</sup> *<sup>a</sup>R*). This result alone suggests that our SO(4) degeneracy group should be considered as a subgroup of the larger group SO(4,1). It suggests introducing the operators

$$\mathbf{B} = \frac{1}{2} \left( \frac{\mathbf{K}}{a} + a\mathbf{R} \right) \qquad \mathbf{S} = D. \tag{205}$$

The commutation relations of *S* and *B* which follow from Equations (205) and the commutation relations Equation (202) are identical to the commutation relations given for *S* and *B* in Section 6.1. Thus, by considering the *a* and *L* transformations in momentum space as conformal transformations, we were led to introduce the generators *B* and *S* and obtain the dynamical algebra SO(4,l). Further, we are led to the expressions for these generators in terms of the canonical variables.

By comparing the expression for *a* in terms of the conformal generators with our known expressions for *a*, Equation (120) or Equation (121), we obtain expressions for *Ki* and *Ri* in terms of the canonical variables. If we use the eigenstates convenient for configuration space calculations (*ρ* = *n*/*ar*) we make the identifications

$$\begin{aligned} \mathcal{K} &= \frac{1}{2}(rp^2 + p^2r) - r \cdot pp - pp \cdot r - r\frac{1}{4r^2} \\ \mathcal{R} &= r. \end{aligned} \tag{206}$$

Substituting these results in the equation for *B* we find

$$B = \frac{1}{2a} \left( \frac{p^2r + rp^2}{2} - r \cdot pp - pp \cdot r - \frac{r}{4r^2} \right) + \frac{ar}{2} \tag{207}$$

which is a manifestly Hermitean operator valid throughout Hilbert space. From Equations (204) and (205) we note that

$$B - a = ar. \tag{208}$$

To compute *D*, we substitute the expressions for *K* and *R* into the commutation relation

$$D = \frac{i}{2}[K\_i, R\_i]$$

$$S = \frac{1}{2}(p \cdot r + r \cdot p) = D \tag{209}$$

obtaining the result

which is identical to the generator of the scale change transformation *D*(*λ*) defined in Equation (109) in Section 4.3.

The significance of the generator *D* = *S* of the scale change in terms of SO(4,1) is apparent if we compute

$$e^{i\lambda D} \left(\frac{\mathbf{K}}{a} \pm a\mathbf{R}\right) e^{-i\lambda D} = \frac{\mathbf{K}}{a'} \pm a'\mathbf{R} \tag{210}$$

where *a* = *eλa*.

The unitary transformation *eiλ<sup>D</sup>* may be viewed as generating an inner automorphism of SO(4,1) which is an equivalent representation of SO(4) that is characterized by a different value of the quantity *a* or the energy. In other words, under the scale change *eiλD*, the basis states for our representation of SO(4,1), <sup>|</sup>*nlm*; *<sup>a</sup>*), transform to a new set, <sup>|</sup>*nlm*;*eλa*) in agreement with our discussion in Section 4.3.

Because the algebra of our generators closes, we may also view *eiλ<sup>D</sup>* as transforming a given generator into a linear combination of the generators. With the definitions of *a* and *B* (Equation (204) and (205)), we can easily show that Equation (210), with the upper sign, may also be written

$$
\epsilon^{i\lambda D} \mathcal{B} \epsilon^{-i\lambda D} = \mathcal{B} \cosh \lambda + \mathfrak{a} \sinh \lambda. \tag{211}
$$

#### **7. The Group SO(4,2)**

#### *7.1. Motivation for Introducing SO(4,2)*

We would like to express Schrodinger's equation as an algebraic equation in the generators of some group [37,38]. As we are unable to do this with our SO(4,l) generators *SAB* we again expand the group. To guide us, we recall that to expand SO(3) to SO(4) we added a three-vector of generators *A*, and to expand SO(4) to SO(4,1), we added a four-vector of generators (*S*, *B*). In both cases, this type of expansion produced a set of generators convenient for the study of the hydrogenlike atom. We guess that the appropriate expansion of SO(4,1) is obtained by adding a five-vector (under SO(4,1)) of generators Γ*<sup>A</sup>* to obtain SO(4,2) [37,38]. We can provide additional motivation for this choice by considering Schrodinger's equation. The generators in terms of which we want to express this equation must be scalars under *Li* rotations. Additionally we know *S* = *S*<sup>40</sup> (Equation (187)) generates scale changes of Schrodinger's equation. The fact that *S*<sup>40</sup> mixes the zero and four components of a five-vector suggests that Schrodinger's equation may be expressed in terms of the components Γ<sup>0</sup> and Γ<sup>4</sup> which are scalars under *Li*, of the five vector Γ*A*. Since Γ*<sup>A</sup>* is a five-vector under SO(4,1), it must satisfy the equation

$$\left[\mathcal{S}\_{AB}, \Gamma\_{\mathbb{C}}\right] = i \left(\Gamma\_B \mathcal{g}\_{AC} - \Gamma\_A \mathcal{g}\_{BC}\right). \tag{212}$$

The spatial components of Γ*<sup>A</sup>* which are (Γ1, Γ2, Γ3) = **Γ** transform as a vector under rotations generated by *L*.

To construct the Lie algebra of SO(4,2) we require that the set of operators {Γ*A*, *SAB*; *A*, *B* = 0, 1, 2, 3, 4} must close under the operations of commutation. By applying Jacobi's identity to Γ*A*, Γ*B*, and *SAB*, and requiring that Γ*<sup>A</sup>* and Γ*<sup>B</sup>* do not commute, we find

$$\left[ \mathbb{S}\_{AB\prime} \left[ \Gamma\_{A\prime} \Gamma\_B \right] \right] = 0 \quad A\prime \\ B = 0\prime 1\prime 2\prime 3\prime 4\ldots$$

Because we require that our Lie algebra closes, the commutator of Γ*<sup>A</sup>* and Γ*<sup>B</sup>* must be proportional to *SAB*. We normalize Γ, so

$$\text{tr}\left[\Gamma\_A \Gamma\_B\right] = -i S\_{AB} \quad A, B = 0, 1, 2, 3, 4. \tag{213}$$

If we define

$$S\_{A5} = \Gamma\_A = -S\_{5A} \quad A = 0,1,2,3,4. \tag{214}$$

and recall

$$A\_i = S\_{i4} \quad B\_i = S\_{io} \quad L\_i = \mathfrak{e}\_{ijk} S\_{jk} \quad S = S\_{40}$$

then we may unite all the commutations relations of Γ*<sup>A</sup>* and *SAB* in the single equation :

$$i\left[\mathcal{S}\_{\mathcal{A}\mathcal{B}\prime}\mathcal{S}\_{\mathcal{C}\mathcal{D}}\right] = i\left(\mathcal{g}\_{\mathcal{A}\mathcal{C}}\mathcal{S}\_{\mathcal{B}\mathcal{D}} + \mathcal{g}\_{\mathcal{B}\mathcal{D}}\mathcal{S}\_{\mathcal{A}\mathcal{C}} - \mathcal{g}\_{\mathcal{A}\mathcal{D}}\mathcal{S}\_{\mathcal{B}\mathcal{C}} - \mathcal{g}\_{\mathcal{B}\mathcal{C}}\mathcal{S}\_{\mathcal{A}\mathcal{D}}\right) \tag{215}$$

where A, B, .. = 0, 1, 2, 3, 4, 5 and *g*<sup>00</sup> = *g*<sup>55</sup> = −1; *gaa* = 1, *a* = 1, 2, 3, 4.

These are the commutation relation for the Lie algebra of SO(4,2). In terms of *A*, *B*, *L*, *S* and Γ*<sup>A</sup>* the additional commutation relations for the noncommuting generators are [53]:

$$\begin{array}{ll} \left[B\_{i\prime}, \Gamma\_{j}\right] = i\Gamma\_{0}\delta\_{ij} & \left[\Gamma\_{i\prime}, \Gamma\_{j}\right] = -i\epsilon\_{ijk}L\_{k} \\ \left[A\_{i\prime}, \Gamma\_{j}\right] = i\Gamma\_{4}\delta\_{ij} & \left[\Gamma\_{i\prime}, \Gamma\_{0}\right] = -iB\_{i} \\ \left[L\_{i\prime}, \Gamma\_{j}\right] = i\epsilon\_{ijk}\Gamma\_{k} & \left[\Gamma\_{i\prime}, \Gamma\_{4}\right] = -iA\_{i} \\ & \left[\Gamma\_{4\prime}, \Gamma\_{0}\right] = -iS \\ \left[B\_{i\prime}, \Gamma\_{0}\right] = i\Gamma\_{i} & \left[A\_{i\prime}, \Gamma\_{4}\right] = -i\Gamma\_{i} \\ \left[S, \Gamma\_{0}\right] = i\Gamma\_{4} & \left[S, \Gamma\_{4}\right] = \left.i\Gamma\_{0}\right] \end{array} \tag{216}$$

#### *7.2. Casimir Operators*

The Lie algebra of SO(4,2) is rank three, so it has three Casimir operators *W*2, *W*3, and *W*<sup>4</sup> [43]:

$$\mathcal{W}\_2 = -\frac{1}{2} \mathcal{S}\_{\mathcal{AB}} \mathcal{S}^{\mathcal{AB}} = Q\_2 + \Gamma\_A \Gamma^A \tag{217}$$

where *Q*<sup>2</sup> is the nonvanishing SO(4,1) Casimir operator Equation (190) and

$$W\_3 = \epsilon^{ABCD\mathcal{E}\mathcal{E}\mathcal{F}} S\_{\mathcal{A}\mathcal{B}} S\_{\mathcal{C}\mathcal{D}} S\_{\mathcal{E}\mathcal{F}} \tag{218}$$

$$W\_4 = S\_{\mathcal{A}\mathcal{B}} S^{\mathcal{B}\mathcal{C}} S\_{\mathcal{C}\mathcal{D}} S^{\mathcal{D}\mathcal{A}}.\tag{219}$$

Computation of *W*<sup>3</sup>

We can show that *W*<sup>3</sup> = 0 from dynamical considerations similar to those used in the discussion of SO(4,1) Casimir operators. The only terms that can be included in *W*<sup>3</sup> are scalars that are formed from products of three generators with different indices

$$\begin{aligned} \mathbf{B} \cdot \mathbf{A} &\times \Gamma, \quad \mathbf{A} \cdot \Gamma \times \mathbf{B}, \quad \Gamma \cdot \mathbf{B} \times \mathbf{A} \\ \Gamma\_4 \mathbf{L} \cdot \mathbf{B}, \quad \Gamma\_0 \mathbf{L} \cdot \mathbf{A}, \quad \mathbf{S} \Gamma \cdot \mathbf{L} \end{aligned} \tag{220}$$

It is interesting that these terms are actually all pseudoscalars. Terms like *<sup>B</sup>* · *<sup>A</sup>* <sup>×</sup> *<sup>L</sup>* are simply not possible because of the structure of *W*3. We know that **Γ** = (Γ1, Γ2, Γ3) must not be pseudovector, otherwise it would be proportional to *L*. Since it is a vector, it must equal a linear combination of *r* and *p*. Therefore, we conclude

$$
\Gamma \cdot L = L \cdot \Gamma = 0.\tag{221}
$$

Because **<sup>Γ</sup>** and *<sup>B</sup>* are both vectors and *<sup>L</sup>* is the only pseudovector we have **<sup>Γ</sup>** <sup>×</sup> *<sup>B</sup>* <sup>=</sup> <sup>Λ</sup>*<sup>L</sup>* . In order to determine the scalar Λ we evaluate the commutators

$$[B\_{k\prime} \left(\Gamma \times \mathcal{B}\right)\_k], \qquad \text{and } [\Gamma\_k \left(\Gamma \times \mathcal{B}\right)\_k] \tag{222}$$

and find

$$
\Gamma \times \mathcal{B} = \Gamma\_0 \mathcal{L} = -\mathcal{B} \times \mathcal{L}.\tag{223}
$$

The analogous equations for *A* and **Γ**, and for *A* and *B*, are

$$\begin{aligned} \Gamma \times A &= -\Gamma\_4 L = -A \times \Gamma \\ A \times \mathcal{B} &= \quad SL \ = -\mathcal{B} \times A. \end{aligned} \tag{224}$$

From Equations (223) and (224), we see that because of the dynamical structure of the generators each of the quantities in the first line of Equation (220) is proportional to the quantity directly below in the second line. We also have shown that (Equations (194) and (221))

$$L \cdot B = L \cdot A = \Gamma \cdot L = 0.\tag{225}$$

Accordingly, each scalar in our list vanishes and

$$\mathcal{W}\_{\mathfrak{I}} = 0.\tag{226}$$

Computation of *W*<sup>2</sup>

In order to compute *W*<sup>2</sup> we need to evaluate

$$
\Gamma^2 \equiv \Gamma\_A \Gamma^A = \Gamma\_4^2 + \Gamma\_i \Gamma^i - \Gamma\_0^2. \tag{227}
$$

From the structure of *W*<sup>2</sup> as shown in Equation (217), we see that Γ<sup>2</sup> must be a number since *W*<sup>2</sup> and *Q*<sup>2</sup> are both Casimir operators and therefore equal numbers for a particular representation. Accordingly, we have

$$[\Gamma^2, \Gamma\_A] = 0.\tag{228}$$

From this equation, we can deduce a lemma allowing us to easily evaluate *W*<sup>2</sup> and *W*<sup>4</sup> in terms of the number Γ2. Using Equation (227) and the definition of *SAB* Equation (213) we find

$$
\Gamma^A \mathcal{S}\_{AB} + \mathcal{S}\_{AB} \Gamma^A = 0.
$$

Contracting Equation (212) with *gAC* gives

$$S\_{AB}\Gamma^A - \Gamma^A S\_{AB} = 4i\Gamma\_B.$$

Consequently, it must follow that

$$S\_{AB}\Gamma^A = 2i\Gamma\_B = -\Gamma^A S\_{AB}.\tag{229}$$

We are now able to evaluate the quantity

$$i S\_{AB} S^B\_{\ C} = i S\_{AB} [\Gamma^B, \Gamma\_C] = i (\mathcal{S}\_{AB} \Gamma^B \Gamma\_C - \mathcal{S}\_{AB} \Gamma\_C \Gamma^B).$$

Using Equation (212) for the commutator of *SAB* with Γ*<sup>C</sup>* and Equation (229) for the contraction *SAB*Γ*<sup>B</sup>* we prove the lemma

$$
\Gamma\_{AB} S^{B}\_{\ \subset} = 2i S\_{CA} - \Gamma\_A \Gamma\_{\ \subset} + \Gamma^2 \mathfrak{g}\_{AC}.\tag{230}
$$

The value of the SO(4,1) Casimir operator *Q*<sup>2</sup> = <sup>1</sup> <sup>2</sup> *<sup>g</sup>ACSABSB <sup>C</sup>* follows directly from the lemma:

$$Q\_2 = 2\Gamma^2. \tag{231}$$

Accordingly, we have from Equation (217)

$$\mathcal{W}\_2 = 3\Gamma^2. \tag{232}$$

Computation of *W*<sup>4</sup>

The Casimir operator *W*<sup>4</sup> can be written as

$$\mathcal{W}\_{4} = S\_{\rm AB} S^{\rm BC} S\_{\rm CD} S^{\rm DA} + S\_{\rm AB} S^{\rm B5} S\_{\rm FD} S^{\rm DA} + S\_{\rm 5B} S^{\rm BC} S\_{\rm CD} S^{\rm D5} + S\_{\rm 5B} S^{\rm B5} S\_{\rm FD} S^{\rm D5}.\tag{233}$$

where B, D = 0, 1, 2, 3, 4, 5 and A, C = 0, 1, 2, 3, 4.

In order to evaluate *<sup>W</sup>*<sup>4</sup> in terms of <sup>Γ</sup><sup>2</sup> we compute *<sup>S</sup>*AB*S*B*C*. Recalling <sup>Γ</sup><sup>A</sup> <sup>=</sup> *<sup>S</sup>*A5 we see

$$\mathcal{S}\_{\rm AB}\mathcal{S}^{\rm BC} = \Gamma\_{\rm A}\Gamma^{\rm C} + \mathcal{S}\_{\rm AB}\mathcal{S}^{\rm BC}.\tag{234}$$

Substituting the lemma Equation (230), we find

$$S\_{\rm AB} S^{\rm BC} = 2i S^{\mathbb{C}}\_{\rm A} - \Gamma^2 g\_{\mathcal{A}}^{\mathbb{C}}.\tag{235}$$

From Equation (229), it follows that

$$\mathbf{S}\_{\mathsf{SE}}\mathbf{S}^{\mathsf{BC}} = 2i\Gamma^{\mathsf{C}}.\tag{236}$$

Substituting Equations (231), (232), (235), and (236) into Equation (233) for *W*<sup>4</sup> we find

$$\mathcal{W}\_4 = 6(\Gamma^2)^2 - 24\Gamma^2. \tag{237}$$

The fact that the nonvanishing Casimir operators (*Q*2, *W*2, and *W*4) for SO(4,1) and SO(4,2) are given in terms of Γ<sup>2</sup> implies that the representation of SO(4,2) determines the particular representation of SO(4,l) appropriate to the hydrogenlike atom. In turn the value of Γ<sup>2</sup> is determined by the structure of the Γs in terms of the canonical variables. In Section 7.4, we derive these structures and find that

$$
\Gamma^2 = 1.
$$

Therefore, the quadratic SO(4,1) Casimir operator *Q*<sup>2</sup> has the value

$$\mathbf{Q}\_2 = 2$$

and the SO(4,2) Casimir operators have the values:

$$\mathcal{W}\_2 = \mathfrak{Z} \qquad \mathcal{W}\_3 = 0 \qquad \mathcal{W}\_4 = -18.5$$

The researchers that have published different representations of SO(4,2) based on the hydrogen atom that give their Casimir operators all have *W*<sup>2</sup> = 3 (or its equivalent) and *W*3=0 [34,79,82], however, two authors have representations with *W*<sup>4</sup> = 0 [34,82] and one [79] has *W*<sup>4</sup> = −12, as compared to our value of -18.

From the mathematical theory of representations, it follows that our representations of SO(4,1) and SO(4,2) are both unitary and irreducible. This means there is no subset of basis vectors that transform among themselves as either SO(4,1) or as SO(4,2).

#### *7.3. Some Group Theoretical Results*

In this section, we derive the transformation properties of the generators of SO(4,2) and then a novel contraction formula that will prove useful for situations in which we want to employ perturbation theory, for example, in our calculation of the radiative shift for the hydrogen atom in Section 8. We will work primarily with the SO(4,2) generators expressed as the combination of the SO(4,1) generators *SAB* and the five-vector Γ, with *gAB* = (−1, 1, 1, 1, 1) where *A*, *B* = 0, 1, 2, 3, 4.

Transformation Properties of the Generators

We can evaluate quantities like

$$\epsilon^{AB}\Gamma\_B(\theta) \equiv \epsilon^{iS\_{AB}\theta}\Gamma\_B\epsilon^{-iS\_{AB}\theta} \quad \text{no sum over A or B} \tag{238}$$

by expanding the exponentials in an infinite series and then using the commutation relations Equations (212) and (213) of the generators *SAB* and Γ*A*, Γ*<sup>B</sup>* repeatedly. However, it is easier to solve the differential equations satisfied by *AB*Γ*<sup>B</sup>* and to use the appropriate boundary conditions. Differentiating Equation (238) and using the commutation relations, we obtain the equations

$$\frac{d}{d\theta}\,^{AB}\Gamma\_B = -\mathcal{g}\_{BB}\,^{AB}\Gamma\_A \quad \frac{d^2}{d\theta^2}\,^{AB}\Gamma\_B = -\mathcal{g}\_{AA}\mathcal{g}\_{BB}\,^{AB}\Gamma\_B\tag{239}$$

which have the solution

$${}^{AB}\Gamma\_B = \Gamma\_B \cos\sqrt{\mathcal{g}\_{AA}\mathcal{g}\_{BB}}\theta + \frac{\mathcal{g}\_{BB}}{\sqrt{\mathcal{g}\_{AA}\mathcal{g}\_{BB}}}\Gamma\_A \sin\sqrt{\mathcal{g}\_{AA}\mathcal{g}\_{BB}}\theta. \tag{240}$$

Using a similar procedure we find

$$\epsilon^{i\Gamma\_A\theta} S\_{AB} \epsilon^{-i\Gamma\_A\theta} = S\_{AB} \cosh\sqrt{\mathcal{g}\_{AA}} \theta + \sqrt{\mathcal{g}\_{AA}} \Gamma\_B \sinh\sqrt{\mathcal{g}\_{AA}} \theta \tag{241}$$

$$e^{i\Gamma\_A \theta} \Gamma\_B e^{-i\Gamma\_A \theta} = \Gamma\_B \cosh\sqrt{\mathcal{g}\_{AA}} \theta + \frac{1}{\sqrt{\mathcal{g}\_{AA}}} S\_{AB} \sinh\sqrt{\mathcal{g}\_{AA}} \theta \tag{242}$$

where no summation over *A* or *B* is implied.

These formulae, Equation (240)–(242), give the SO(4,2) transformation properties of the SO(4,2) generators.

#### The Contraction Formula

If we multiply Equation (241) from the right by *ei*Γ*A<sup>θ</sup>* and then contract from the left with Γ*B*, we obtain

$$\sum\_{B} \Gamma^{B} e^{i\Gamma\_{A}\theta} \Gamma\_{B} = \left[ \left( 1 - \underline{\text{g}}\_{AA} \Gamma\_{A}^{2} \right) \cosh \sqrt{\underline{\text{g}}\_{AA}} \theta + \frac{2i\Gamma\_{A}}{\sqrt{\underline{\text{g}}\_{AA}}} \sinh \sqrt{\underline{\text{g}}\_{AA}} \theta \right] e^{i\Gamma\_{A}\theta} + \underline{\text{g}}\_{AA} \Gamma\_{A}^{2} e^{i\Gamma\_{A}\theta} \tag{243}$$

where we have used Γ<sup>2</sup> = 1 and Equation (229). Expanding the hyperbolic functions in terms of exponentials and collecting terms gives

$$\sum\_{B} \Gamma^{B} e^{i\Gamma\_{A}\theta} \Gamma\_{B} = \frac{1}{2} (1 + \frac{i\Gamma\_{A}}{\sqrt{\mathcal{S}AA}})^2 e^{i(\Gamma\_{A} - i\sqrt{\mathcal{S}AA})\theta} + \frac{1}{2} (1 - \frac{i\Gamma\_{A}}{\sqrt{\mathcal{S}AA}})^2 e^{i(\Gamma\_{A} + i\sqrt{\mathcal{S}AA})\theta} + g\_{AA} \Gamma\_{A}^{2} e^{i\Gamma\_{A}\theta}.\tag{244}$$

A Fourier decomposition of a function Γ*<sup>A</sup>* may be written

$$f(\Gamma\_A) = \frac{1}{2\pi} \int d\theta \, h(\theta) e^{i\Gamma\_A \theta}. \tag{245}$$

Consequently we have

$$\sum\_{B} \Gamma^{B} f(\Gamma\_{A}) \Gamma\_{B} = \frac{1}{2} (1 + \frac{i\Gamma\_{A}}{\sqrt{\mathcal{g}\_{AA}}})^{2} f(\Gamma\_{A} - i\sqrt{\mathcal{g}\_{AA}}) + \frac{1}{2} (1 - \frac{i\Gamma\_{A}}{\sqrt{\mathcal{g}\_{AA}}})^{2} f(\Gamma\_{A} + i\sqrt{\mathcal{g}\_{AA}}) + \mathcal{g}\_{AA} \Gamma\_{A}^{2} f(\Gamma\_{A}). \tag{246}$$

By performing a suitable rotation we can generalize this formula from functions of Γ*<sup>A</sup>* to functions of <sup>Γ</sup>*An<sup>A</sup>* where *nAn<sup>A</sup>* <sup>=</sup> <sup>±</sup>1. For *<sup>n</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>1 we start with <sup>Γ</sup>*<sup>A</sup>* <sup>=</sup> <sup>Γ</sup><sup>0</sup> and rotate to obtain a very general result

$$\sum\_{B} \Gamma\_B f(n\Gamma) \Gamma^B = \frac{1}{2} (n\Gamma + 1)^2 f(n\Gamma + 1) + \frac{1}{2} (n\Gamma - 1)^2 f(n\Gamma - 1) - (n\Gamma)^2 f(n\Gamma). \tag{247}$$

We will have occasion to apply this formula for the special case

$$f(n\Gamma) = \frac{1}{\Gamma n - \nu}.\tag{248}$$

Using the representation

$$\frac{1}{\Gamma n - \nu} = \int\_0^\infty ds e^{\nu s} e^{-\Gamma n s} \tag{249}$$

we obtain the result

$$
\Gamma\_A \frac{1}{\Gamma n - \nu} \Gamma^A = -2\nu \int\_0^\infty ds \, e^{\nu s} \frac{d}{ds} (\sinh^2 \frac{s}{2} \, e^{-\Gamma n \cdot s}) \tag{250}
$$

which is in a form convenient for perturbation calculations.

Derivation of the Γ*<sup>A</sup>* in terms of the Canonical Variables

For our basis states, we shall use eigenstates of (*Zα*)−<sup>1</sup> convenient for configuration space calculations (*ρ* = *na*/*r*). We choose these states rather than those convenient for momentum space calculations because they lead to simpler expressions for the Γ*<sup>A</sup>* in terms of the canonical variables, although the expression for *a* is slightly more complicated. Thus, our states obey the equation

$$
\left[\frac{1}{K\_1(a)} - n\right]|nlm\rangle = 0\tag{251}
$$

where *K*1(*a*) is given by Equation (116). We know that *K*−<sup>1</sup> <sup>1</sup> must commute with the generators of the SO(4) symmetry group **(***a*)*<sup>i</sup>* = *Si*<sup>4</sup> and *Sij* = *ijkLk*. This suggests that we choose

$$
\Gamma\_0 = [K\_1(a)]^{-1} = \sqrt{ar} \frac{p^2 + a^2}{2a^2} \sqrt{ar} = \frac{1}{2} \left( \frac{\sqrt{r} p^2 \sqrt{r}}{a} + ar \right) \tag{252}
$$

so that

$$(\Gamma\_0 - n) \, |nlm\rangle = 0.\tag{253}$$

This last equation is the Schrodinger equation expressed in our language of SO(4,2): our states |*nlm*) are eigenstates of Γ<sup>0</sup> with eigenvalue *n*.

To find Γ4, we calculate Γ<sup>4</sup> = −*i*[*S*, Γ0], using Equation (209) for *S*,

$$
\Gamma\_4 = \sqrt{ar} \frac{p^2 - a^2}{2a^2} \sqrt{ar} = \frac{1}{2} \left( \frac{\sqrt{r} p^2 \sqrt{r}}{a} - ar \right). \tag{254}
$$

Sometimes it is convenient to use the linear combinations

$$
\Gamma\_0 - \Gamma\_4 = ar \qquad \Gamma\_0 + \Gamma\_4 = \frac{\sqrt{r}p^2\sqrt{r}}{a} \tag{255}
$$

which can be used to express the dipole transition operator [33]. We can find Γ*<sup>i</sup>* from Equation (216), Γ*<sup>i</sup>* = −*i*[*Bi*, Γ0]

$$
\Gamma\_i = \sqrt{r} p\_i \sqrt{r} \tag{256}
$$

which we might have guessed initially since [*rpi*,*rpj*] ∼ *Lk*. Every component of Γ*<sup>A</sup>* is Hermitean, consequently the generators *SAB* given by the commutators Equation (213) are also Hermitian. We may explicitly verify that these expressions for Γ*<sup>A</sup>* lead to a consistent representation of all generators in the SO (4,2) Lie algebra.

Under a scale change generated by *S*, Γ*<sup>i</sup>* is invariant and Γ<sup>4</sup> and Γ<sup>0</sup> transform in the same manner as *a* and *B* (Equation (210)): they retain their form but *a* is transformed into *eλa*:

$$e^{i\lambda S} \left\{ \frac{\Gamma\_0}{\Gamma\_4} \right\} e^{-i\lambda S} = \frac{1}{2} \left( \frac{\sqrt{r} p^2 \sqrt{r}}{e^{\lambda} a} \pm e^{\lambda} ar \right) \tag{257}$$

The scale change generates an inner automorphism of SO(4,2) characterized by a different value of the parameter *a*.

### *7.4. Subgroups of SO(4,2)*

The two most significant subgroups are [53]:

1. *Li*, *ai* or *Sjk*, *Si*4, forming an SO(4) subgroup. These generators commute with Γ<sup>0</sup> and therefore constitute the degeneracy group for states of energy <sup>−</sup>*a*2/(2*m*) and fixed principal quantum number *<sup>n</sup>* (or fixed coupling constant *na*/*m*). The Casimir operator for this subgroup is

$$
\sigma^2 + L^2 = n^2 - 1 = \Gamma\_0^2 - 1.\tag{258}
$$

We discussed this subgroup in Section 4.2 in terms of *<sup>L</sup>* and *<sup>A</sup>* and the states <sup>|</sup>*nlm*. The same results are obtained with the generators *<sup>L</sup>* and *<sup>a</sup>* with the states <sup>|</sup>*nlm*). For example, we have the raising and lowering operators for *m* and *l* (Equations (93) and (94)). With the definition

$$L\_{\pm} = L\_1 \pm iL\_2 \tag{259}$$

it follows that

$$\left[L\_{3\nu}L\_{\pm}\right] = \pm L\_{\pm} \tag{260}$$

which gives

$$L\_{\pm}|nlm\rangle = \sqrt{(l(l+1) - m(m\pm 1))} \, |nl\,\, m \pm 1\rangle \qquad L\_{3}|nlm\rangle = m|nlm\rangle. \tag{261}$$

for *l* ≥ 1. In analogy to *L*<sup>±</sup> one can define

$$a\_{\pm} = a\_1 \pm ia\_2 \tag{262}$$

which obey the relations

$$[a\_3, a\_\pm] = \pm L\_3 \qquad [L\_3, a\_\pm] = \pm a\_\pm \tag{263}$$

and

$$\begin{split} a\_{\pm}|nlm\rangle &= \mp \left( \frac{(n^2 - (l+1)^2)(l+2 \pm m)(l+1 \pm m)}{4(l+1)^2 - 1} \right)^{\frac{1}{2}} |n|l+1 \ m \pm 1\rangle \\ &\pm \left( \frac{(n^2 - l^2)(l \mp m)(l-1 \mp m)}{4l^2 - 1} \right)^{\frac{1}{2}} |n|l-1 \ m \pm 1\rangle \end{split} \tag{264}$$

for *l* ≥ 1. The action of *a*<sup>±</sup> is not directly analogous to that of *L*±, because we are using |*nlm*) as basis states. If we used |*na*3*l*<sup>3</sup> = *m*) as basis states, the action would be similar. An operator that only changes the angular momentum is *a*<sup>3</sup>

$$|a\_3|nlm\rangle = \left(\frac{(n^2 - (l+1)^2)((l+1)^2 - m^2)}{4(l+1)^2 - 1}\right)^{\frac{1}{2}} |n|l + 1 |m\rangle + \left(\frac{(n^2 - l^2)(l^2 - m^2)}{4l^2 - 1}\right)^{\frac{1}{2}} |n|l - 1 |m\rangle. \tag{265}$$

for *l* ≥ 1. Since *a*<sup>3</sup> commutes with *L*<sup>3</sup> and Γ0, it does not change *n* or *m*.

2. Γ4, *S* = *S*40, Γ0, forming a SO(2,1) subgroup. These operators commute with *L* but not with Γ0, hence then can change *n* but not *L* or *m*. The Casimir operator for this subgroup is

$$
\Gamma\_0^2 - \Gamma\_4^2 - S^2 = L^2 = l(l+1). \tag{266}
$$

We can define the operators [53]

$$j\_1 = \Gamma\_4 \qquad j\_2 = S \qquad j\_3 = \Gamma\_0 \tag{267}$$

with commutators

$$\begin{bmatrix} j\_1 \ j\_2 \end{bmatrix} = -i\mathbf{j}\_3 \quad \begin{bmatrix} j\_2 \ j\_3 \end{bmatrix} = i\mathbf{j}\_1 \quad \begin{bmatrix} j\_3 \ j\_1 \end{bmatrix} = i\mathbf{j}\_2 \tag{268}$$

We can define the raising and lowering operators

$$j\_{\pm} = j\_1 \pm i j\_2 = \Gamma\_4 \pm i \text{S} \tag{269}$$

which obey the commutation relations

$$[j\_{\pm \prime}, j\_3] = \mp j\_{\pm}.\tag{270}$$

We find (in analogy to Equation (261))

$$
\Gamma\_0|nlm\rangle = n|nlm\rangle \quad (\Gamma\_4 \pm i\mathcal{S})|nlm\rangle \\
= \sqrt{n(n \pm 1) - l(l+1)}|n \pm 1\, lm\rangle \tag{271}
$$

We can express the action of Γ<sup>0</sup> − Γ<sup>4</sup> = *ar* on our states

$$\begin{array}{rcl} \text{Var}|nlm\rangle = \frac{1}{2}\left( (n)(n-l) - l(l+1) \right)^{\frac{1}{2}}|n-1\ lm m\rangle & + n|n\ lm m\rangle & + \frac{1}{2}\left( (n)(n+l) - l(l+1) \right)^{\frac{1}{2}}|n+1\ lm m\rangle & \end{array} \tag{272}$$

As previously mentioned, the operator *S* generates scale changes as shown in Equation (257), where the value of *a* is changed. We can also express the action of *S* equivalently as transforming Γ<sup>0</sup> into Γ<sup>4</sup>

$$e^{iS\lambda}\Gamma\_0 e^{-iS\lambda} = \Gamma\_0 \cosh\lambda - \Gamma\_4 \sinh\lambda \qquad e^{iS\lambda}\Gamma\_4 e^{-iS\lambda} = \Gamma\_4 \cosh\lambda - \Gamma\_0 \sinh\lambda. \tag{273}$$

#### *7.5. Time Dependence of SO(4,2) Generators*

For a generator to be a constant it must commute with the Hamiltonian as discussed in Section 2.1. Because the SO(4,2) group is the non-invariance or spectrum generating group, the additional generators do not all commute with the Hamiltonian and may have a harmonic time dependence as discussed in Section 2.2. It is notable that as far as we know only one paper considers the time dependence of the generators of non-invariance groups in general and one considers SO(4,2) specifically [97,136]. Our results certainly clarify and make explicit the time dependence, and show that it is just a particular aspect of the SO(4,2) transformations. In our representation with basis states |*nlm*; *a*), the Hamiltonian, which is the generator of translations in time, has been transformed into Γ<sup>0</sup> and the Schrodinger energy eigenvalue equation has become Γ0|*nlm*) = *n*|*nlm*). Accordingly, all of the generators that commute with Γ<sup>0</sup> are constants of the motion, which includes *a*, *L*. The other operators, *B*, **Γ**, *S*, Γ<sup>4</sup> have a time dependence given by Equations (241) and (242), for example

$$S(t) = e^{iHt} S(0) E^{-iHt} = e^{i\Gamma\_0 t} S e^{-i\Gamma\_0 t} = S \cos t + \Gamma\_4 \sin t. \tag{274}$$

$$
\Gamma\_4(t) = \varepsilon^{iHt} \Gamma\_4(0) E^{-iHt} = \varepsilon^{i\Gamma\_0 t} \Gamma\_4 e^{-i\Gamma\_0 t} = \Gamma\_4 \cos t - S \sin t. \tag{275}
$$

Consequently, terms like *j*<sup>±</sup> have a simple exponential time dependence

$$j\_{\pm}(t) = j\_{\pm}(0)e^{\pm it}.\tag{276}$$

Similarly **<sup>Γ</sup>** <sup>±</sup> *<sup>i</sup><sup>B</sup>* has an exponential time dependence.

#### *7.6. Expressing the Schrodinger Equation in Terms of the Generators of SO(4,2)*

We can write the Schrodinger equation for the energy eigenstate *En* <sup>=</sup> <sup>−</sup>*a*<sup>2</sup> *<sup>n</sup>*/2*m* of a particle in a Coulomb potential in terms of the SO(4,2) generators, which are expressed in terms of the energy <sup>−</sup>*a*2/2*m*, by making a scale change. From Section 4.3, Equation (114), the relationship between the Schrodinger energy eigenstate <sup>|</sup>*nlm* and the eigenstate of (*Zα*)−<sup>1</sup> is:

$$|nlm;a\rangle = e^{-iS\lambda\_n} \sqrt{\rho(a\_n)}|nlm\rangle\tag{277}$$

where

$$
\epsilon^{\lambda\_n} = \frac{a\_n}{a} \quad \rho(a\_n) = \frac{n}{a\_n r}. \tag{278}
$$

Substituting Equation (277) into the eigenvalue equation Equation (253) for |*nlm*; *a*) and employing the transformation Equation (273), we find the usual Schrodinger equation can be expressed in SO(4,2) terms as

$$
\langle (\Gamma n - n) \sqrt{\rho(a\_n)} | nlm \rangle = 0 \tag{279}
$$

where

$$
\Gamma n \equiv \Gamma\_A n^A = \Gamma\_0 n^0 + \Gamma\_i n^i + \Gamma\_4 n^4 \tag{280}
$$

$$n^o = \cosh \lambda\_{\mathbb{H}} = \frac{a^2 + a\_n^2}{2a a\_n}, \quad n^i = 0, \quad n^4 = -\sinh \lambda\_{\mathbb{H}} = \frac{a^2 - a\_n^2}{2a a\_n} \tag{281}$$

and *nAn<sup>A</sup>* = *n*<sup>2</sup> <sup>4</sup> <sup>−</sup> *<sup>n</sup>*<sup>2</sup> <sup>0</sup> = −1.

Equation (279) expresses Schrodinger's equation for an ordinary energy eigenstate |*nlm* with energy *EN* <sup>=</sup> <sup>−</sup>*a*<sup>2</sup> *<sup>n</sup>*/2*m* in the language of SO(4,2). It shows the relationship between these energy eigenstates and the basis states of (*Zα*)−<sup>1</sup> used for the SO(4,2) representation,

#### **8. SO(4,2) Calculation of the Radiative Shift for the Schrodinger Hydrogen Atom**

In the 1930's, it was generally believed that the Dirac equation predicted the energy levels of the hydrogen atom with excellent accuracy, but there were some questions about the prediction that the energy levels for a given principal quantum number and given total angular momentum were independent of the orbital angular momentum. To finally resolve this issue, in 1947, Willis Lamb and his student Robert Retherford at Columbia University in New York City employed rf spectroscopy and exploited the metastability of the hydrogen 2*s*1/2 level in a beautiful experiment and determined that the 2*s*1/2 and 2*p*1/2 levels were not degenerate and that the energy difference between the levels was about 1050 MHz, or 1 part in 106 of the 2*s*1/2 level [5,148]. Shortly thereafter Hans Bethe [6] published a ground breaking nonrelativistic quantum theoretical calculation of the shift assuming it was due to the interaction of the electron with the ground state electromagnetic field of the quantum vacuum field. This radiative shift accounted for about 96% of the measured shift. The insight that one needed to include the interaction of the atom with the vacuum fluctuations and how one could actually do it ushered in the modern world of quantum electrodynamics [7]. Here, we compute in the non-relativistic dipole approximation and to first order in the radiation field, as did Bethe, the radiative shift, but we use group theoretical methods based on the SO(4,2) symmetry of the non-relativistic hydrogen atom as developed in this paper. Bethe's calculation required the numerical sum over intermediate states to obtain the average value of the energy of the states contributing to the shift. In our calculation, we do not use intermediate states, and we derive an integral equivalent to Bethe's log, and more generally derive the shift for all levels in terms of a double integral.

An expression for the radiative shift Δ*NL* for energy level *EN* of a hydrogen atom in a state |*NL* can be readily obtained using second order perturbation theory (to first order in *α* the radiation field) [6,149–151]

$$
\Delta\_{NL} = \frac{2ac}{3\pi m^2} \sum\_{n}^{s} \int\_{0}^{\omega\_c} d\omega \frac{(E\_n - E\_N) \left< NL | p\_i | n \right> \langle n | p\_i | NL \rangle}{E\_n - E\_N + \omega - i\varepsilon},\tag{282}
$$

where *ω<sup>C</sup>* is a cutoff frequency for the integration that we will take as *ω<sup>c</sup>* = *m*.

This expression, which is the same as Bethe's, has been derived by inserting a complete set of states |*nn*|, a step that we eliminate with our group theoretical approach:

$$
\Delta\_{\rm NL} = \frac{2a}{3\pi m^2} \int\_0^{\omega\_c} d\omega \langle \mathcal{N}L | p\_i \frac{H - E}{H - (E - \omega) - i\epsilon} p\_i | \mathcal{N}L \rangle \tag{283}
$$

If we add and subtract *ω* from the numerator, we find the real part of the shift is

$$\text{Re}\Delta\_{NL} = \frac{2a}{3\pi m^2} \text{Re} \int\_0^{\omega\_c} d\omega \left[ \langle NL \vert p^2 \vert NL \rangle - \omega \Omega\_{NL} \right] \tag{284}$$

where

$$
\Omega\_{\rm NL} = \langle NL \vert p\_i \frac{1}{H - E\_N + \omega - i\epsilon} p\_i \vert NL \rangle \tag{285}
$$

and

$$H = \frac{p^2}{2m} - \frac{Za}{r}.\tag{286}$$

The imaginary part of the shift gives the width of the level [7].

The matrix element Ω*NL* can be converted to a matrix element of a function of the generators <sup>Γ</sup>*<sup>A</sup>* taken between eigenstates <sup>|</sup>*nlm*) of (*Zα*)−1. To do this we insert factors of 1 <sup>=</sup> <sup>√</sup>*<sup>r</sup>* <sup>√</sup><sup>1</sup> *<sup>r</sup>* and use the definitions of the Γ*<sup>A</sup>* in terms of the canonical variables, Equations (254)–(256). Letting the parameter *a* take the value *aN*, we obtain the result

$$
\Omega\_{NL} = \frac{m\nu}{N^2} (NL)\Gamma\_i \frac{\Gamma\_i}{\Gamma n(\frac{\pi}{\nu}) - \nu} \Gamma\_i |NL\rangle \tag{287}
$$

where

$$n^0(\underline{\boldsymbol{\xi}}) = \frac{2 + \underline{\boldsymbol{\xi}}}{2\sqrt{1 + \underline{\boldsymbol{\xi}}^2}} = \cosh\phi \qquad n^i = 0 \qquad n^4(\underline{\boldsymbol{\xi}}) = -\frac{\underline{\boldsymbol{\xi}}}{2\sqrt{1 + \underline{\boldsymbol{\xi}}^2}} = -\sinh\phi \tag{288}$$

and

$$\zeta = \frac{\omega}{|E\_N|} \qquad \nu = \frac{N}{\sqrt{1+\zeta^2}} = \text{Ne}^{-\phi}. \tag{289}$$

From the definitions we see *φ* = <sup>1</sup> <sup>2</sup> *ln*(<sup>1</sup> <sup>+</sup> *<sup>ξ</sup>*) <sup>&</sup>gt; 0 and *nA*(*ξ*)*nA*(*ξ*) = <sup>−</sup>1. The quantity

$$\nu = \frac{mZ\kappa}{\sqrt{-2m(E\_N - \omega)}}$$

may be considered the effective principal quantum number for a state of energy *EN* − *ω*. The contraction over *i* in Ω*NL* may be evaluated using the group theoretical formula Equation (250):

$$\begin{split} \Omega\_{NL} &= -2\frac{m\nu^2}{N^2} \int\_0^\infty ds e^{\nu s} \frac{d}{ds} \left( \sinh^2 \frac{s}{2} M\_{NL}(s) \right) \\ &- m \frac{\nu}{N^2} (NL) \Gamma\_4 \frac{1}{\Gamma n(\frac{x}{\nu}) - \nu} \Gamma\_4 |NL \rangle + m \frac{\nu}{N^2} (NL) \Gamma\_0 \frac{1}{\Gamma n(\frac{x}{\nu}) - \nu} \Gamma\_0 |NL \rangle \end{split} \tag{290}$$

where

$$M\_{NL}(s) = (NL|e^{-\Gamma n(\frac{\sigma}{\delta})s}|NL). \tag{291}$$

In order to evaluate the last-two terms in Ω*NL*, we can express the action of Γ<sup>4</sup> on our states in terms of Γ*n*(*ξ*) − *ν*. Substituting the equation

$$
\Gamma\_0|NL\rangle = N|NL\rangle\tag{292}
$$

into the expression for Γ*n*(*ξ*) − *ν*, with *n*(*ξ*) given by Equation (288), gives

$$
\Gamma\_4 = N - \left(\frac{1}{\sinh \phi}\right) \left(\Gamma n(\xi) - \nu\right) \tag{293}
$$

when acting on the state |NL). If we substitute Equation (293) into the expression for the *Re*Δ*ENL* Equation (284) and simplify using the virial theorem

$$(NL|p^2|NL) = a\_N^2$$

we find that the term in *p*<sup>2</sup> exactly cancels the last two terms in Ω*NL*, yielding the result

$$Re\Delta E\_{NL} = \frac{4ma(Za)^4}{3\pi N^4} \int\_0^{\phi\_c} d\phi \sinh\phi e^{\phi} \int\_0^{\infty} ds \, e^{\nu s} \frac{d}{ds} \left(\sinh^2 \frac{s}{2} M\_{NL}(s)\right) \tag{294}$$

where

$$\phi\_{\varepsilon} = \frac{1}{2} \ln \left( 1 + \frac{\omega\_{\varepsilon}}{|E\_N|} \right) = \frac{1}{2} \ln \left( 1 + \frac{2N^2}{(Za)^2} \right) \tag{295}$$

and *ω<sup>c</sup>* = *m*.

Comparison to the Bethe Logarithm

The first order non-relativistic radiative shift is commonly given in terms of the Bethe logarithm *γ*(*N*, *L*), which is interpreted as the average over all states, including scattering states, of *ln* <sup>|</sup>*En*−*EN*<sup>|</sup> <sup>1</sup> <sup>2</sup> *<sup>m</sup>*(*Zα*)<sup>2</sup> [149] :

$$\begin{array}{l} \gamma(N,L)\sum\_{n}^{\mathbb{S}}(E\_{n}-E\_{N}) \left< N0 \left| p\_{i} \right| n \right> \left< n \left| p\_{i} \right| N0 \right>\\ = \sum\_{n}^{\mathbb{S}}(E\_{n}-E\_{N}) \left< NL \left| p\_{i} \right| n \right> \left< n \left| p\_{i} \right| NL \right> \ln \frac{\left| E\_{n}-E\_{N} \right|}{\frac{1}{2}m(Za)^{2}} \end{array} . \tag{296}$$

We use the dipole sum rule [150]

$$2\sum\_{n}^{s} \left(E\_{n} - E\_{N}\right) \left< N \left| p\_{i} \right| n \right> \left< n \left| p\_{i} \right| N \right> = - \left< N \left| \nabla^{2} V \right| N \right>\tag{297}$$

and apply it for the Coulomb potential <sup>∇</sup>2*V*(*r*) = <sup>4</sup>*πZαδ*(*r*). The use of the Bethe log allowed Bethe to take the logarithmic expression obtained from the frequency integration outside the summation over the states, and replace it with the average value. Only the S states contribute to the expectation value in Equation (297), giving, from Equation (282), an expression for the shift

$$\operatorname{Re}\Delta\mathcal{E}\_{NL} = \left[\frac{4m}{3\pi}a(Za)^4\right]\frac{1}{N^5}\left\{\delta\_{L0}\ln\frac{2}{(Za)^2} - \gamma(N,L)\right\}.\tag{298}$$

Comparing the shift in terms of *MNL* Equation (294) to the shift in terms of *γ*(*N*, *L*) we find that the Bethe log is

$$\gamma(N,L) = \int\_0^{\phi\_c} d\phi \sinh\phi \, e^{\phi} \int\_0^{\infty} ds \, e^{\upsilon s} \frac{d}{ds} \left(\sinh^2 \frac{s}{2} \, M\_{NL}(s)\right) - \delta\_{L0} \ln \frac{1}{(Za)^2} \tag{299}$$

#### *8.1. Generating Function for the Shifts*

We can derive a generating function for the shifts for any eigenstate characterized by *N* and *L* if we multiply Equation (291) by *<sup>N</sup>*4*eβ<sup>N</sup>* and sum over all *<sup>N</sup>*, *<sup>N</sup>* <sup>≥</sup> *<sup>L</sup>* <sup>+</sup> 1. To simplify the right side of the resulting equation, we use the fact that the O(2,1) algebra of Γ0, Γ4, and *S* closes. We can compute the sum on the right hand side:

$$\sum\_{N=L+1}^{\infty} e^{-\beta N} M\_{NL} = \sum\_{N=L+1}^{\infty} \left( NL \vert e^{-\hat{f} \cdot \Psi} \vert NL \right). \tag{300}$$

where

$$\varepsilon^{-j\cdot\Psi} \equiv \varepsilon^{-\beta\Gamma\_0} \varepsilon^{-s\Gamma u(\xi)}.\tag{301}$$

We perform a *j* transformation, such that

$$
\varepsilon^{-j\_1\psi} \to \varepsilon^{-j\_3\psi} = \varepsilon^{-\Gamma\_0\psi} \tag{302}
$$

$$\sum\_{N=L+1}^{\infty} e^{-\beta N} M\_{NL} = \sum\_{N=L+1}^{\infty} \left( NL \middle| e^{-j\_N \phi} \middle| NL \right) = \sum\_{N=L+1}^{\infty} e^{-N\phi} \tag{303}$$

$$=\frac{e^{-\psi(L+1)}}{1-e^{-\psi}}.\tag{304}$$

In order to find a particular *MNL*, we must expand the right hand side of the equation in powers of *e*−*<sup>β</sup>* and equate the coefficients to those on the left hand side. First, we need an equation for *e*−*ψ*. This can be obtained using the isomorphism between *j* and the Pauli *σ* matrices:

$$(\Gamma\_4, \mathcal{S}, \Gamma\_0) \to (j\_1, j\_2, j\_3) \to (\frac{i}{2}\sigma\_1, \frac{i}{2}\sigma\_2, \frac{1}{2}\sigma\_3) \tag{305}$$

Using the formula

$$e^{\frac{i}{2}sn \cdot \sigma} = \cos\frac{s}{2} + i\mathbf{n} \cdot \sigma \sin\frac{s}{2} \tag{306}$$

where <sup>|</sup>*n*<sup>|</sup> <sup>=</sup> 1, we find

$$\cosh\frac{\psi}{2} = \cosh\frac{\beta}{2}\cosh\frac{s}{2} + \sinh\frac{\beta}{2}\sinh\frac{s}{2}\cosh\phi. \tag{307}$$

We can rewrite this equation in a form easier for expansion

$$e^{+\frac{1}{2}\psi} = de^{\frac{1}{2}\beta} + be^{-\frac{1}{2}\beta} - e^{-\frac{1}{2}\psi} \tag{308}$$

where

$$\begin{cases} d = \cosh\frac{s}{2} + \sinh\frac{s}{2}\cosh\phi\\ b = \cosh\frac{s}{2} - \sinh\frac{s}{2}\cosh\phi \end{cases} \tag{309}$$

Let *β* become very large and iterate the equation for *e*<sup>−</sup> <sup>1</sup> <sup>2</sup> *<sup>ψ</sup>* to obtain the result

$$e^{-\psi} = Ae^{-\beta} \left[ 1 + A\_1 e^{-\beta} + A\_2 e^{-2\beta} + \dots \right] \tag{310}$$

where

$$\begin{aligned} A &= A\_0 = \frac{1}{d^2} \\ A\_1 &= -\left(\frac{2}{d}\right) \left(b - d^{-1}\right) \\ A\_2 &= 3d^{-2} \left(b - d^{-1}\right)^2 - 2^{-2} \left(b - d^{-1}\right) \\ &\vdots \end{aligned} \tag{311}$$

Note *<sup>b</sup>* <sup>−</sup> *<sup>d</sup>*−<sup>1</sup> <sup>=</sup> <sup>−</sup>*d*−<sup>1</sup> sinh<sup>2</sup> *<sup>s</sup>* <sup>2</sup> sinh<sup>2</sup> *<sup>φ</sup>*.

#### *8.2. The Shift between Degenerate Levels*

Expressions for the energy shift between degenerate levels with the same value of N may be obtained directly from the generating function using Equations (294) and (304). We find

$$\sum\_{N=L+1} \varepsilon^{-\beta N} N^4 \operatorname{Re} \Delta E\_{NL} - \sum\_{N=L'+1}^{\infty} \varepsilon^{-\beta N} N^4 \operatorname{Re} \Delta E\_{NL'} = 0$$

$$\frac{4ma(Za)^4}{3\pi} \int\_0^{\varphi\_c} d\phi e^{\phi} \sinh\phi \int\_0^{\infty} ds e^{\nu s} \frac{d}{ds} \left(\sinh^2\frac{s}{2} \frac{e^{-\psi(L+1)} - e^{-\psi(L'+1)}}{1 - e^{-\psi}}\right). \tag{312}$$

For an example, consider *L* = 1, *L* = 0. For the shifts between levels we obtain

$$\sum\_{N=2}^{\infty} e^{-\beta N} N^4 \operatorname{Re} \left(\Delta E\_{NO} - \Delta E\_{N1}\right) + \operatorname{Re} \Delta E\_{10} e^{-\beta} =$$

$$\frac{4ma(z\alpha)^4}{3\pi} \int\_0^{\phi\_c} d\phi \, e^{\phi} \sinh\phi \int\_0^{\infty} ds e^{\nu s} \frac{d}{ds} \left(\sinh^2 \frac{s}{2} e^{-\psi}\right) \tag{313}$$

Substituting Equation (310) for *<sup>e</sup>*−*ψ*, using the coefficient *AAN*−<sup>1</sup> of *<sup>e</sup>*−*Nβ*, gives

$$\operatorname{Re}(\Delta E\_{N0} - \Delta E\_{N1}) = \frac{4ma(Za)^4}{3\pi N^4} \int\_0^{\phi\_c} d\phi e^{\phi} \sinh\phi \int\_0^{\infty} ds e^{\nu s} \frac{d}{ds} \left(\sinh^2\frac{s}{2} A A\_{N-1}\right). \tag{314}$$

where *A* and *AN*−<sup>1</sup> are given in Equation (311) in terms of the variables of integration *s* and *φ*.

General Expression for *MNL*

Once we have a general expression for *MNL*, we can use Equation (294) to calculate the shift for any level *ENL*. We can obtain expressions for the values of *MNL* by letting *β* become large, expanding the denominator in Equation (304) and equating coefficients of powers of *e*−*β*. For large *β*, we have large *ψ*. We have

$$\frac{e^{-\psi(L+1)}}{1 - e^{-\psi}} = \sum\_{m=1}^{\infty} e^{-\psi(m+L)}$$

and for large *β* it follows from Equation (310) that

$$\sum\_{N=L+1}^{\infty} e^{-\beta N} M\_{NL} = \sum\_{m=1}^{\infty} \left[ e^{-\beta} A \left( 1 + A\_1 e^{-\beta} + \dots \right) \right]^{m+L} \dots \tag{315}$$

Using the multinomial theorem [124], the right side of the equation becomes

$$\sum\_{m=1}^{\infty} A^{m+L} \sum\_{r,s,t,\dots} \frac{(m+L)!}{r!s!t!\dots} A\_1{}^s A\_2^t \dots a^{-\beta(m+L+s+2t+\dots)}.\tag{316}$$

where *r* + *s* + *t* + ... = *m* + *L*.

To obtain the expression for *MNL*, we note *N* is the coefficient of *β* so *N* = *m* + *L* + *s* + 2*t* + ... = *r* + 2*s* + 3*t* + ... Accordingly we find

$$M\_{NL} = \sum\_{r,s,t,\dots} A^{(r+s+t+\dots)} \frac{(r+s+t+\dots)!}{r!s!t!\dots} A\_1^s A\_2^t \dots \tag{317}$$

where *r* + 2*s* + 3*t* + ... = *N* and *r* + *s* + *t* + . . . >L.

By applying this formula, we obtain the results:

N = 1:

$$M\_{10} = A\tag{318}$$

N = 2:

$$\begin{aligned} M\_{20} &= A^2 + AA\_1\\ M\_{21} &= A^2 \end{aligned} \tag{319}$$

Shifts for N = 1 and N = 2

To illustrate these results, we can calculate the shift for a given energy level using Equation (294). For *N* = 1, we note from Equation (318) that *M*<sup>10</sup> = *A*, and from Equation (311) that *A* = 1/*d*2. We find that the real part of the radiative shift for the 1*S* ground state is

$$\operatorname{Re}\Delta E\_{10} = \frac{4ma(Za)^4}{3\pi} \int\_0^{\Phi\_0} d\phi e^{\phi} \sinh\phi \int\_0^{\infty} ds e^{s e^{-\phi}} \frac{d}{ds} \frac{1}{\left(\cosh\frac{s}{2} + \cosh\phi\right)^2} \tag{320}$$

For the shift between two states Equation (314) can be used. For the N = 2 Lamb shift between 2S-2P states, the radiative shift to first order in *α* is

$$\operatorname{Re}(\Delta E\_{20} - \Delta E\_{21}) = \frac{ma(Za)^4}{6\pi} \int\_0^{\phi\_c} d\phi e^{\phi} \sinh^3\phi \int\_0^{\infty} ds e^{2s e^{-\phi}} \frac{d}{ds} \frac{1}{\left(\coth\frac{s}{2} + \cosh\phi\right)^4} \tag{321}$$

The s integral can be computed in terms of a Jacobi function of the second kind [127].

As one check on our group theoretical methods, we can compare our matrix elements (10|*eiSφ*|*n*0) with those of Huff [45]. To go from Equation (301) to Equation (302), we did a rotation *R*(*φ*) = *eiφ<sup>S</sup>* generated by *S* that transformed Γ*n* into Γ0. For *N*, *L* = 1, 0 we have

$$M\_{10} = (10|e^{-\Gamma \text{ns}}|10) = (10|R(\phi)e^{-\Gamma\_0 \varepsilon}R^{-1}(\phi)|10) = \frac{1}{(\cosh\frac{s}{2} + \sinh\frac{s}{2}\cosh\phi)^2} \tag{322}$$

Expanding the hyperbolic functions, we get

$$M\_{10} = \frac{4e^{-s}}{(1 + \cosh \phi)^2} \left[1 - e^{-s} \tanh^2 \frac{s}{2}\right]^{-2}$$

$$\xi = \frac{4}{(1 + \cosh \phi)^2} \sum\_{n=1}^{\infty} n e^{-ns} \left(\tanh^2 \frac{\phi}{2}\right)^{n-1}. \tag{323}$$

We can also compute *M*<sup>10</sup> by inserting a complete set of states and using Γ0|*n*0) = *n*|*n*0) in Equation (322). Because the generator *S* is a scalar, only states with *L* = 0, *m* = 0 can contribute:

$$M\_{10} = \sum\_{n!m} \varepsilon^{-ns} |n\rangle\langle 10|R(\phi)|n0\rangle|^2. \tag{324}$$

Comparing this to Equation (323), we make the identification

$$|\langle 10|R(\phi)|n0\rangle|^2 = \frac{4n}{(1+\cosh\phi)^2} \left(\tanh^2\frac{\phi}{2}\right)^{n-1}.\tag{325}$$

Huff computes this matrix element by analytically continuing the known *O*(3) matrix element of *ei Jy<sup>φ</sup>* obtaining

$$\left| \langle 10 \vert R(\phi) \vert n0 \rangle \right|^2 = \frac{4n}{\cosh^2 \phi - 1} \left( \tanh^2 \frac{\phi}{2} \right)^n \cdot \left[ {}\_2F\_1(0, -1; n; \frac{1}{2}(1 - \cosh \phi)) \right]^2. \tag{326}$$

By algebraic manipulation and using <sup>2</sup>*F*<sup>1</sup> = 1 for the arguments here, we see that this result agrees with our much more simply expressed result from group theory.

#### **9. Conclusions and Future Research**

Measuring and explaining the properties of the hydrogen atom has been central to the development of modern physics over the last century. One of the most useful and profound ways to understand its properties is through its symmetries, which we have explored in this paper, beginning with the symmetry of the Hamiltonian, which reflects the symmetry of the degenerate levels, then the larger non-invariance and spectrum-generating groups, which include all of the states. The successes in using symmetry to explore the hydrogen atom led to use of symmetry to understand and model other physical systems, particularly elementary particles.

The hydrogen atom will doubtless continue to be one of testing grounds for fundamental physics. Researchers are exploring the relationship between the hydrogen atom and quantum information [152], the effect of non-commuting canonical variables [*xi*, *xj*] = 0 on energy levels [153–155], muonic hydrogen spectra [156], and new physics using Rydberg states [157–162]. The ultra high precision of the measurement of the energy levels has led to new understanding of low Z two body systems, including muonium, positronium, and tritium [151]. As mentioned in the introduction, measurements of levels shifts are currently being used to measure the radius of the proton [2]. We can expect that further investigations of the hydrogen atom and hydrogenlike atoms will continue to reveal new vistas of physics and that symmetry considerations will play an important part.

**Funding:** This research received no external funding.

**Acknowledgments:** I would especially like to thank Lowell S. Brown very much for the time and energy he has spent on my education as a physicist. I also thank Peter Milonni for his encouragement and his helpful comments and many insightful discussions and MDPI for the invitation to be the Guest Editor for this special issue of Symmetry on Symmetries in Quantum Mechanics.

**Conflicts of Interest:** The author declares no conflict of interest.

## **References and Notes**


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