1. Introduction
The study of exact solutions to the Schrödinger equation for a general harmonic oscillator has attracted considerable interest in the literature thanks to the pivotal role of the oscillator in physics. Constructing exact solutions for a harmonic oscillator based on various ideas and methods and finding connections between them expands the knowledge about this fundamental system.
The well-known stationary states of a quantum harmonic oscillator in the coordinate representation are obtained by the separation of variables in the Schrödinger equation with the harmonic oscillator potential. Glauber proposed standard coherent states for a harmonic oscillator, which is the prototype for most coherent states [
1,
2]. The coherent states form a very convenient representation for problems in quantum mechanics. They can be created from the ground state by the displacement operator and can be expanded in terms of the harmonic oscillator Hamiltonian eigenstates. Coherent states are described in a wealth of superb reference books and papers, e.g., [
3,
4,
5].
An alternative to the separation-of-variables method is the noncommutative integration method (NIM) proposed in [
6] for linear partial differential equations and developed in [
7,
8,
9,
10] (see Ref. [
11] for details). This method essentially uses the symmetry of a differential equation and its algebra of symmetry operators and allows one to construct a basis of solutions that, in general, differ from solutions constructed by the separation of variables and from coherent states. The NIM was effectively used to construct exact solutions to the Schrödinger, Klein–Gordon [
6,
8], and Dirac [
9,
10,
12] equations, and also for classification of external fields in equations with symmetries in the Riemannian spaces of general relativity in [
13,
14,
15,
16].
This paper describes the development and application of the NIM for solving the Schrödinger equation for a quantum harmonic oscillator, using symmetry in this problem. We will be looking for the NIM solutions, which can be regarded as analogs of coherent states in the sense of [
4,
17] in view of the close relation of NIM with the group symmetry of the quantum harmonic oscillator.
The paper is organized as follows.
Section 2 introduces the basic notation and a special
-representation of Lie algebras necessary for applying the method of noncommutative integration.
Section 3 considers the Schrödinger equation for a harmonic oscillator and shows that its symmetry algebra in the class of first-order linear differential operators forms the oscillatory Lie algebra
. The next
Section 4 considers the
-representation of the oscillatory Lie algebra
and the generalized Fourier transform on the Lie group
of the Lie algebra
.
Section 5 shows that the Schrödinger equation for an oscillator is equivalent to some right-invariant system of equations on the group
G. Integrating this system by noncommutative integration, we obtain a basis of solutions and compare it with a system of coherent states.
Section 6 contains some concluding remarks.
2. -Representation of Lie Groups
Using the orbit method [
18,
19], we define an irreducible
representation of the Lie algebra
, which is a key element of the NIM.
First, we recall some necessary definitions related to the orbit method on Lie groups, which we will need in what follows. The degenerate Poisson–Lie bracket,
defines a Poisson structure in the space
[
18]. Here,
are the coordinates of a linear functional
relative to the dual basis
,
is a commutator in the Lie algebra
, and the natural pairing between the spaces
and
is denoted by
. The number
of functionally independent Casimir functions
,
, with respect to the bracket (
1) is called the
index of the Lie algebra.
The Lie group
G acts on the coalgebra
by the coadjoint representation
:
and splits
into the coadjoint orbits (K-orbits). Orbits of maximum dimension
are called
non-degenerate [
18].
Let
be a non-degenerate coajoint orbit passing through a general covector
. Locally, one can always introduce the Darboux coordinates
on the orbit
in which the Kirillov form
defining a symplectic structure on the coajoint orbits has the canonical form
,
, and
are called the canonical coordinates. We assume that the transition from the local coordinates
f on the orbit
to the canonical coordinates
is given if the set of functions
,
is defined in such a way that
Consider the functions
, which are linear in the variables
:
Denote by
a complex extension of the Lie algebra
. It was shown in Ref. [
7] that canonical functions (
2) can be constructed if for the functional
there exists a subalgebra
in the complex extension
of the Lie algebra
satisfying the conditions
The subalgebra
is called the
polarization of the functional
. In this case, the vector fields
are infinitesimal generators of a local transformation group
of a partially holomorphic manifold
Q. Equation (
3) assumes that the functionals from
can be prolonged to
by linearity. Note that for non-degenerate coajoint orbits, there always exist the canonical functions with the form (
2).
Let
be a space of complex functions on the manifold
Q with a measure
and inner product given by
where
denotes the complex conjugate of
. Functions of the space
are square-integrable on the manifold
Q.
The linear operators
implement the irrep
-representation of a Lie algebra
in
and can be seen as the result of
-quantization on the coajoint orbit
[
6,
7].
We assume that with respect to the inner product (
4) the operators
are Hermitian.
Let
and
be left- and right-invariant vector fields on a Lie group
G, respectively. We will be interested in generalized solutions of the system of equations
The functions
provide the lift of the
-representation of the Lie algebra
to the local unitary representation
of its Lie group
G,
and satisfy the relations
where
. Note that by properly defining the measure
in the parameter space
J, we can check the completeness and orthogonality properties for the functions
,
where
is the generalized Dirac delta function with respect to the right Haar measure
on the Lie group
G.
Note that the functions
are defined globally on the Lie group
G iff the Kirillov condition of integerness of the orbit
holds [
18]:
Here,
is a one-dimensional homology group of the stationarity subgroup
.
Let
be the space of functions having the form
The function
belongs to the space
with respect to the variables
q and
. Within the framework of the NIM, the relation (
12) is a generalization of the Fourier transform with kernel
to the case of the Lie group
G. From the relations, (
10) follows the expression for the inverse Fourier transform in the form
The transformations (
12) and (
13) induce the corresponding transformation of the left- and right-invariant fields on the Lie group
G:
It follows from (
14) that functions from the space
that have the form (
12) are eigenfunctions of the Casimir operators
:
Since the left- and right-invariant vector fields on the Lie group
G are transformed into
-representation operators under the transformation (
12), the Casimir operators in the
-representation are constants.
3. Symmetry Algebra of a Quantum Harmonic Oscillator
The states of a one-dimensional quantum harmonic oscillator in the coordinate representation
,
are described by the wave function
, which satisfies the nonstationary Schrödinger equation
where
is the mass of the quantum particle,
is the frequency of the harmonic oscillator, and
ℏ is Planck’s constant.
The well-known wave functions of the harmonic oscillator in terms of the Hermite polynomials
are [
4]
The eigenstates
for the Hamiltonian
are called Fock’s or number states,
. The Fock states are orthonormal and form a complete basis such that any other state of the harmonic oscillator may be written in terms of them.
We can define the annihilation and creation operators by the formulas
respectively. The time dependent-coherent states
are eigenstates of the annihilation operator
,
where the eigenvalue of the operator
is a complex number
, which is a function of time
t. The coherent states may be written as
In the coordinate representation, we have
The real and imaginary parts of the quantum number
z characterize the mean values of position and momentum operators:
Equation (
15) admits four integrals of motion in the class of the first-order linear differential operators:
These operators form the Lie algebra
with non-zero commutation relations
The algebra
with the commutation relations (
20) is called the oscillatory Lie algebra. In the next section, we will construct a special irreducible
-representation of this Lie algebra, which is necessary for solving the Equation (
15) in terms of the noncommutative integration method.
4. λ-Representation of the Oscillatory Lie Algebra
Let
be some fixed basis of the Lie algebra
,
and let
be the commutator in
,
An arbitrary element
is determined by its components
with respect to the chosen basis,
. In turn, an arbitrary element
of the dual space
is determined by the components of
with respect to the basis
dual to the basis
,
.
The Lie algebra
admits two Casimir functions
Nondegenerate orbits of the coadjoint representation (K-orbits) pass through the parametrized covector
,
Denote by
the local Lie group of the Lie algebra
. Let us introduce canonical coordinates of the second kind,
, on the group
as
The group composition law in the coordinates (
21) has the form
The Lie group
acts on itself by the left
and right
shifts. The left-invariant vector fields
on the group
in local coordinates (
21) are
The right-invariant vector fields
are in turn defined by the expressions
The Lie group
is unimodular and the Haar measure coincides with the Lebesgue measure
. Suppose that the coordinates
values take values in
.
There exists a three-dimensional complex subalgebra
of the complex extension
of the algebra
subject to the functional
, so that
. This subalgebra is a complex polarization corresponding to the linear functional
. This polarization corresponds to the canonical transition
The
-representation operators are of the form
The function space
is invariant under the
-representation operators and is a Hilbert space with respect to the scalar product (
4) with the measure
The functions of the space
are entire analytic functions of the complex variable
q. The generalized Dirac function in the space
,
is defined by the expression
By integrating the system of Equation (
7), we obtain
where
is the action of the group
on the complex manifold
Q, which is given by the generators
so we have
The representation (
9) becomes an induced representation of the Lie group
and, according to (
24), has the form
It can be shown that any
-representation of the Lie algebra in the class of the first-order linear partial differential operators leads to the induced representation of the Lie group constructed in the framework of the Kirillov orbit method (see Refs. [
6,
7,
9]). The relations (
10) are satisfied with respect to the measure
The direct Fourier transform (
12) in the space
has the form
For an invariant second-order differential equation on the group
written as
where
,
, and
C are constants, the general solution is sought in the form (
25) in the framework of NIM. Then, we have the reduced equation for the function
,
which is an ordinary differential equation with respect to the independent variable
. Equation (
27) will be called the Equation (
26) in the
-
representation, and the transition from (
26) to (
27) will be called the
non-commutative reduction of Equation (
26).
5. The Schrödinger Equation on the Oscillatory Lie Group
Let us show that the Schrödinger Equation (
15) describing the quantum harmonic oscillator is equivalent to the following system of equations on the Lie group
Indeed, the general solution to the Equations (29) and (30) can be written as
Substituting
into the second Equation (
28), we obtain the Schrödinger equation for the function
in the form
Thus, we have reduced the Schrödinger equation to the system of Equations (29) and (30) on the Lie group
, for which the set of basic left-invariant vector fields (
22) is a set of non-commuting integrals of motion forming the Lie algebra
.
Let us integrate the system (29) and (30) using the NIM. We are looking for a solution to this system in the form (
25). Then, we obtain the non-commutative reduced system of equations for the function
as
The system (
32) says that the quantum harmonic oscillator corresponds to the orbit of the coadjoint representation
of the group
, which passes through the parameterized covector
, and the function
describing the quantum harmonic oscillator in terms of the
-representation does not depend on variable
. From (
32), we have
Substituting (
33) into (
25) yields the general solution
Equation (
25) gives the general solution to the system of Equations (29) and (30). According to Equation (
31), the general solution to the Schrödinger equation is obtained from (
34) by setting
,
,
. It is convenient to represent the general solution to the Schrödinger equation as follows. Let us introduce a set of functions
which satisfies the completeness and orthogonality conditions:
Then the general solution of the Schrödinger equation, according to (
34), is written as
Moreover, for the solution norm (
36), we have
As a result, using the NIM, we have found a general solution (
36) to the Schrödinger equation for the quantum harmonic oscillator. We say that this solution describes the
H-state of the harmonic oscillator. Let us show that for a given solution (
36), stationary solutions are obtained, which are determined from the equation
Substituting (
36) into (
38) by the function
, we obtain the equation
From here, we obtain
The function
belongs to the space
iff
, and
n is an integer. This condition results in the well-known spectrum of the quantum harmonic oscillator:
. The corresponding wave functions on the manifold
Q coincide with the basis functions
up to a normalization factor:
Then, (
36) provides the well-known expression for the wave functions of the harmonic oscillator in terms of the Hermite polynomials (
16) as
Thus, Fock’s states
of the harmonic oscillator in the
-representation (
39) generate the space
in which the
-representation of the oscillatory group acts.
Comparing (
18) and (
35), we obtain the relationship between the
H-states and the harmonic oscillator coherent states in the form
From here, we can see that the
H-solution (
36) is related to coherent states of the harmonic oscillator, but it differs from the latter by a constant factor. In bracket notation, the solution (
36) can be represented as
Here
is a coherent state with a wave function (
18), and the wave function (
36) corresponds to the state
. Accordingly, for mean values, one can obtain
From (
17), it is easy to write out the expansion of H-states
in terms of Fock’s states:
Thus, as the result of applying the NIM to the system of Equations (29) and (30), we have obtained the
H-states (
40) of the harmonic oscillator, which, up to a normalization factor, coincide with known coherent states
.
6. Conclusions
In this paper, we have shown that the oscillatory Lie algebra
naturally arises as the Lie algebra formed by the symmetry operators (
19) of the Schrödinger equation, (
15) and the Schrödinger equation itself for the harmonic oscillator is equivalent to a system of right-invariant equations on the corresponding Lie group
. As a result of the noncommutative integration of this system, a complete set of solutions (
36) (
H-solutions) is found. Moreover, the quantum harmonic oscillator corresponds to the only non-degenerate orbit
of the adjoint representation of the Lie group
. It is shown that the Fock states of the harmonic oscillator in the
-representation form a Hilbert space
, which is invariant under the operators of the
-representation (
23) constructed along the given orbit. It turns out that the
H-solutions are eigenvalues for the annihilation operator
, and therefore, they differ from the known coherent states of the harmonic oscillator by a factor that does not depend on
t and
x (see Equation (
40)) but depends on the complex quantum number
u.