2.1. GUP Algebra Depending Linearly on
For the first example of this dependence, consider the commutation relations
for which
. An operator representation of the above commutation relations is
The momentum eigenstates are in this case
so
Consider now the usual non-relativistic classical Hamiltonian
The quantisation of this Hamiltonian by the rule (
8) implies the following Schrödinger equation:
or
The associated time-independent Schrödinger equation is
By performing the change in variables
which is well-defined for the interest region
and
, we have that
and
Replacing Equations (
19) and (
20) in (
17), we obtain the following Schrödinger equations for the new variable
y:
Due to its simplicity, we will analyse the case of infinite square-well potential and explore the consequences of the modification of the Heisenberg commutations relations for its eigenstates and energy spectra. For this purpose, consider the potential
given by (see
Figure 1)
In terms of
y,
U is given by
So for
the Schrödinger equation is
For
, the above equation can be written as
with
The general solution of (
25) is
or in terms of
x,
Since the potential is infinite at
and
, the wave function must be zero at these two points so that
The first condition implies
and the second condition gives
Since
A cannot be zero (we are looking for non-trivial solutions), the above equation gives
so
The eigenfunctions are thus
whose energies are by virtue of (
26)
The energies
of the usual unmodified case (
) can be obtained by taking the limit
of the Equation (
34)
Thus, the quotient between the modified and unmodified energies is
Thus, all energy eigenvalues are uniformly re-scaled in this case.
Figure 2 shows how
changes as a function of
.
The constant
in Equation (
33) can be determined as usual, by normalising the wave function to unity:
i.e., for our case
which gives
After evaluating the integral, we obtain
from which follows
and the eigenfunctions are explicitly
Let us consider the case
so that
Figure 3 and
Figure 4 show
as a function of
x, for different values of
and
n, when
.
Let us now consider the associated probability densities. In this case
Figure 5 and
Figure 6 show the graph of
as a function of
x for different values of
and
n, when
.
Note, that in all cases, for small values of , the probability density of the original solution (i.e., for the unmodified system) is recovered.
2.2. GUP Algebra Depending Non-Linearly on
If, instead of the commutation relation (
8), one considers the generalisation
for
, the momentum operator has the representation
The momentum eigenfunctions are, in this case, the solutions of the equation
that is
or
where
whose solution is
Explicitly, the first functions
are
Let us now consider the eigenfunctions in the presence of the potential well. The Hamiltonian is in the region
,
whose eigenfunctions are the momentum states
with energies
. To satisfy the boundary conditions (
29) one must construct a superposition of two states: one of positive momentum
p and another of negative momentum
of the form
The boundary conditions (
29) imply that
and
From the first of these equations, we have that
so, the second equation gives
Since
we have
or
which implies the quantisation condition for momentum
that is
The energy eigenvalues are
The quotient of the energies
of the modified system over the original energies
is
so that the energies are uniformly rescaled for each value of
n.
The corresponding eigenfunctions are
that is
The corresponding probability densities are
If we define the non-normalised densities by
we have explicitly for the first
n values that
Figure 7 and
Figure 8 show the unnormalised probability density
for various values of
,
k and
n, when
.
From the results shown in
Figure 3,
Figure 4,
Figure 5,
Figure 6,
Figure 7 and
Figure 8, one can ask: why the wave functions and their density probabilities are deformed in that way, and why the density probabilities are shirking to the left side of the potential well.
To obtain an answer, one can consider the Hamiltonian operator (for the free particle) associated with the modified Heisenberg commutation relation (
46), which is given by
By taking the the classical limit
[
21], the corresponding classical Hamiltonian function is
then, the Hamilton equations of motion give
and
Because the classical Hamiltonian is time-independent, the energy is conserved, so
is a constant, thus
so
By replacing the above equation in (
76), obtains
The right side of (
78) is, by the Newton equation, the force
that acts over the particle for each
n value, so
The above two relations have interesting implications because these equations establish that the modified Heisenberg commutation relation (
46) generates a force over the initial free particle at both classical and quantum levels. In the first case, by generating the force
and, in the second one, by deforming the probability densities. Thus, gravity, by modifying the Heisenberg commutation relations, can generate forces on the particles, and in this way, gravitation could generate the other three forces of nature.
Note, that the classical force
is negative for
; thus, the particle accelerates to the wall’s left side. That is consistent with the shift of probability densities to the left side of the potential well.
Figure 9 shows the classical force (
79) for
. Note, that the force has a maximum. From (
79), the position
for which the force reaches its maximum value can be computed as
Note, that for the case , the maximum is at the origin, but for , the maximum is at the origin’s right side. When increases, the maximum is shifted towards .
2.3. GUP Depending Linearly on the Momentum
Consider now the following GUP commutation relations depending on the momentum
These commutation relations can be represented by the differential operators
where
is the standard momentum operator for when
.
The eigenfunctions of
are
that is
For the square well potential (
22), the Hamiltonian in the region
is
The eigenfunctions (
84) of the momentum operator
are also eigenfunctions of the Hamiltonian:
with energy eigenvalues
Figure 10 shows the spectrum of
for various values of
.
Figure 11 presents a more detailed version of the spectrum for
and
in the momentum region
.
It should be noted that when
p becomes very negative, the energy tends asymptotically to a constant value
. In fact
For this asymptotic value
of the energy, there exists a maximum positive momentum
that has the same energy, that is
or
so
with solutions
The above equations imply that (see
Figure 11 below)
On the other hand, if the energy of the particle is in the interval
then, as can be seen in
Figure 11, there are two values of the momentum
p, one positive
and the other negative
, which give the same value of
E. In fact, from
one can solve for the momentum
p in terms of the energy
E according to
that is
Note, that if
in such a way that
as can be seen in
Figure 11. Thus, for an energy
E in the interval
there exist two distinct momentum eigenstates
and
which have the same energy
E. Since the values of
and
in (
96) or (
97) depend on the
E, these momenta must be related. Using Equation (
96) we see that
Adding both equations gives
from which we can solve for
as
One must note that
is negative, since for
it follows that
which implies that
and therefore,
in Equation (
101) gives the logarithm of a number between zero and one, and therefore,
is negative.
Note, also that Equation (
97) implies that if
, i.e., the quotient
, then
is the logarithm of a negative number, so
becomes a complex number. In this situation, it is not possible to find stationary states because the boundary conditions can only be satisfied at one side of the potential well.
Let us now consider the problem of determining the eigenstates associated with the potential well. In the figure below, two momentum eigenstates are illustrated:
, which has momentum
(and Thus, represents a wave travelling to the right) and
, which has momentum
(which represents a wave travelling to the left). Because of the structure of the asymmetric energy spectrum in
Figure 11, the energy of
is greater than that of
. Thus, waves moving to the left have a lower energy than those moving to the right (see
Figure 12).
If we wish to generate a standing wave that vanishes at
and
, we must superimpose two states in opposite directions but with the same energy
E. For these states to exist, the following condition must be fulfilled (see
Figure 11):
For
E in the above range,
and
are two momentum eigenstates, one with positive (
) and the other with negative (
), but with the same energy
E. Thus, we can consider the state
The boundary conditions (
29) imply
The above conditions require that
, therefore,
and since
(otherwise the wave function is null), the following quantisation relation is obtained:
that is
implying that the momenta have discrete values
y
such that
Using Equations (
97) we obtain the quantisation condition for the energies
:
or
that is
By defining
and
, i.e.,
then, we have
or
By means of Equation (
97), we can obtain the momenta
and
as
We can now compare the energies of the modified systems to the original ones by means of the quotient
, which is given by
Here, the energy eigenvalues are not uniformly rescaled, but depend on each state
n.
Figure 13 shows the quotient
for various values of
n as a function of the parameter
for
. Note, that for larger values of
n, the quotient tends more rapidly to zero.
Let us now consider the probability densities of the different states. These are given by
Due to the quantisation relation (
104), we finally obtain
However, as
we have
that is
Note, that this density is proportional to the probability density of the original system (i.e., with
). The amplitude
is obtained by normalising the wave functions by means of Equation (
40), which gives
This suggests that the wave functions of the modified systems should be the same as the original ones. In fact, by Equation (116), the new eigenfunctions are
and due to the quantisation of momentum (
104) we have that
so that
Thus, we see that the new eigenfunctions differ from the original ones by a complex function
of unit norm, which does not change the value of the corresponding probability densities. Thus, the new probability densities are the same as the original ones, but the energy spectrum changes completely.
To conclude this subsection, we will mention a few words about the strange energy behaviour in terms of momentum given by Equation (
95). As indicated above, a wave travelling to the right with momentum
has an energy greater than a wave travelling to the left with momentum
as indicated in
Figure 12.
Because our Hamiltonian is time-independent, energy must be conserved. This fact implies that for a fixed energy, waves moving to the right have momentum magnitude lower than those moving to the left (see
Figure 11 above). One can understand this awkward behaviour by considering the classical limit of (
85). As shown in [
22], the classical Hamiltonian function is just given by (
95). Now, consider a simple classical non-relativistic collision between a mass
(with initial velocity
) with a second stationary mass
. If
and
are the velocities after the collision, conservation of momentum implies that
and using (
95), energy conservation gives
By solving this system, it can be shown that in the limit
, the velocity
with which mass
bounces back is given by
For
, the usual behaviour, in which a ball reverses its velocity when it hits a wall, is recovered. However, for non-zero
, the speed with which it bounces to the left is greater than the speed it had when it was moving to the right. Thus, changes generated by the deformed Heisenberg commutation relations could be detected by simply throwing balls over a wall and measuring velocities. All these questions and the connection of GUPs with classical mechanics will be analysed in a detailed way in our next article [
21].
It should be mentioned here that this awkward energy property, in terms of the momentum, can be related to what mathematicians call quasi metric spaces [
23,
24,
25,
26], i.e., spaces in which the distance measured when moving to the right differs from that measured when moving to the left. In the case at hand, energy or momentum can be taken as a measure analogous to distance. In this sense, the case of linear deformation would represent these quasimetric spaces in the physical world.
Finally, note that as is shown in [
3], for a deformed Heisenberg commutation relation of the form
for some function
, the above algebra can be represented by the differential operators
where the function
is solution to the differential equation [
3]
and
is the standard momentum operator. Thus, the Hamiltonian for the particle inside the square square-well is
In the classical limit
, the Hamiltonian (
110) goes to the classical Hamiltonian function
whose Hamiltonian equations of motion are
and
Note that (
111) implies that there are no forces acting classically on the particle and,
is constant, so (
112) implies that the velocity is also constant. Thus, for a deformed Heisenberg commutation relation whose right side depends only on the momentum as in (
107), the particle inside the well has no force acting on it, so the probability density must be the same as the non-deformed case, that is, the free-particle density as we are shown explicitly above for the case of linear deformation.