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The quantum mechanical relationships between time-dependent oscillators, Hamilton-Jacobi theory and an invariant operator are clarified by making reference to a system with a generalized oscillator. We introduce a linear transformation in position and momentum, and show that the correspondence between classical and quantum transformations is exactly one-to-one. We found that classical canonical transformations are constructed from quantum unitary transformations as long as we are concerned with linear transformations. We also show the relationship between the invariant operator and a linear transformation.

Canonical transformations are a highlight in classical mechanics. They give not only solutions to classical mechanical systems, but also an insight into the quantization of them. However, the idea of canonical transformations has so far not been fully utilized in quantum systems. This issue was raised by Dirac [

These various methods have been investigated for various purposes. However, there has been no unification of these various methods. The purpose of the present paper is to provide a unified description of these methods in terms of a linear transformation for position and momentum by referring to the system of a generalized oscillator. Since we are concerned with a linear canonical transformation, the classical and quantum correspondence is one-to-one. We also show that the invariant operator can be considered as part of a linear canonical transformation.

The organization of the paper is as follows. In Section 2, we define a linear canonical transformation in position and momentum and apply this to a genelarized oscillator. Moreover, we show two special cases of linear canonical transformations. One is the transformation to a time-dependent oscillator, and the other is to construct a Hamilton- Jacobi theory. In Section 3, we introduce a unitary operator that generates a linear transformation in position and momentum in quantum mechanics. We apply this unitary operator to a genelarized oscillator and obtain the same results as the classical cases mentioned in Section 2. In Section 4, we introduce an invariant operator. We show that this is also derived from a unitary operator that generates a linear transformation. We give two special cases for the coefficients of the linear transformation. The foregoing research is just one special case of a linear canonical transformation. Section 5 is devoted to a summary.

A linear canonical transformation is defined by a transformation from old position and momentum variables

where A, B, C and D are real functions of time t.

with

which is known from classical mechanics [

Now the Hamiltonian which we consider in this paper is

where

where

The dot above the variables denotes the time derivatives of the variables. For later use, we derive the equation of motion for p from the Hamiltonian (4),

where

We will see these equations in later sections.

The transformed Hamiltonian K is derived from the classical mechanics [

From the linear canonical transformation (1), we obtain

and

Collecting these equations together, we obtain the transformed Hamiltonian K as follows

Up till now, we have placed no constraint on the coefficients A, B, C and D, except

We assign the coefficients of (1) [

Substituting these coefficients and their time derivatives

into (12), we obtain the transformed Hamiltonian

where

and

Next let us consider another constraint. We impose the additional condition on the coefficients.

These constraints show that the coefficients are the solution to the equation of motion (5) and (7), that is,

where

We substitute (18) into the transformed Hamiltonian (12). For the time-derivative parts in (12), we obtain

so that the transformed Hamiltonian becomes zero;

which means that the new variables

Since we are concerned with linear canonical transformations, the classical and quantum correspondence is exactly one-to-one [

where

Next we show a “normal-ordering” of the unitary operator (24). For this we introduce the operators [

which form the SU(1, 1) Lie algebra

We rewrite the unitary operator (24) in the “normal-ordered” form as

where

and

Corresponding to the classical linear transformation (1), the new quantum variables

we obtain

where from (28),

To recognize this statement, let us consider the generalized oscillator. The quantum counterpart of the Hamiltonian (4) is

and the quantum counterpart of the transformed Hamiltonian

Substituting (27b) and (31) into (32), we obtain

and using (A.18),

Collecting these equations together, we obtain the transformed Hamiltonian

The time derivative of the condition

Then we recover the same form of the transformed Hamiltonian

with (12) in the classical transformed Hamiltonian. It is realized that the linear transformation in position and momentum gives the same transformed Hamiltonian (12) and (39). So, the same constraints (13) and (18) give the same results for the time- dependent oscillator and the Hamilton-Jacobi relations, as mentioned in section 2. As long as we are concerned with linear canonical transformations, the correspondence between the canonical transformation in classical mechanics and the unitary transformation in quantum mechanics is one-to-one. Referring to the generalized oscillator, the quantum unitary transformation is constructed in parallel with the classical canonical transformation.

An invariant operator

This was first investigated for the time-dependent harmonic oscillator [

where x, y and z are real functions of time t. The time derivative of

In order to satisfy (40), we demand for the coefficients x, y and z, that

On the other hand, the invariant operator

whose eigenvalues and eigenfunctions are well known in elementary quantum mechanics.

We see that this unitary operator is also a linear canonical transformation [

To satisfy (41), we assign

for x, y and z. In other words, the coefficients A, B, C and D in the unitary operator

The Hamiltonian (44) can be written down in terms of annihilation and creation operators (

whose eigenvalues and eigenstates are given by [

where

This implies that we define the time-dependent operators by

and the invariant operator

and its eigenstates are

These are the same eigenvalues as for the time-independent case

When we choose the squeezing coefficients

then we construct the same results as in [

The invariant operator is classified according to an auxiliary equation. We will see two cases below.

There are some kinds of invariant function that are classified as auxiliary equations. One example is [

where

Equation (47a) means

and from the derivative of

which fulfills the condition (47b). Equation (43b) gives

where

and

With the initial condition (

This is the auxiliary equation [

which is identical with (47c).

Another kind of invariant function [

From (43a) and (47a), we obtain

then,

which fulfills the condition (47b). Eq.(43b) gives

where

and

With the initial condition (

This is the auxiliary equation [

This is an inhomogeneous differential equation of (5). From (61) and (62), z becomes

which is identical with (47c).

We investigated a linear transformation in position and momentum by referring to a generalized oscillator. We found that the correspondence between the classical canonical transformation and the quantum unitary transformation is one-to-one, that is, as long as we are concerned with linear transformations, all classical transformations can be constructed as quantum ones. As examples of this, the transformation to a time- dependent oscillator and the construction of Hamilton-Jacobi theory are derived both for the classical and quantum cases. The notion of linear transformations is also applicable to the invariant operator. On choosing the coefficients for the linear transformation, we were able to repeat the results obtained in previous work.

Ogura, A. (2016) Classical and Quantum Behavior of Generalized Oscillators in Terms of Linear Canonical Transformations. Journal of Modern Physics, 7, 2205-2218. http://dx.doi.org/10.4236/jmp.2016.715191

In this Appendix, we derive the “normal-ordering” of (27a). To accomplish this program, we apply the idea of Truax [

where

subject to the constraint

where primes indicate differentiation with respect to

we obtain

From the theorem (29) and the commutation relations (26), we obtain

We identify the coefficients of the respective basis elements of the Lie algebra and obtain a system of coupled nonlinear equations [

with initial conditions

Together, Equations ((A.7a), (A.7c), and (A.8)) imply

a Riccati equation for

with constant coefficients. Subject to the initial conditions, this equation has the solution

where

Substituting (A.12) into (A.8) for

We can integrate the differential Equation (A.7a) to get the following

To obtain the final result, choose

where A, B, C and D are defined by (28). This is the desired expression for the “normal ordering” of the unitary operator. We decompose this unitary operator in three parts and assign

The following equations

and

are helpful.