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Journal of Mathematical Physics

Some Examples of Canonical Transformations in Quantum Mechanics

Francis R. Halpern

Published Mar 1, 1962• 0 citations
Publisher

Abstract

The set of all pairs of polynomials X(x,p) and P(x,p) in the position x and momentum p are found such that X and P have the same commutator as x and p. Several other types of representations of the commutation relations are studied. The reducibility, time, and space inversion properties of these are discussed. The unitary transformations connecting different representations are exhibited. The use of different representations to solve scattering and bound state problems is indicated. The application of one of the transformations to a partial diagonalization of a perturbed harmonic oscillator Hamiltonian is made.

DOI
10.1063/1.1703801

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Journal of Physics A Mathematical and General•Apr 1, 1984•0 citations
The set of all pairs of polynomials X(x,p) and P(x,p) in the position x and momentum p are found such that X and P have the same commutator as x and p. Several other types of representations of the commutation relations are studied. The reducibility, time, and space inversion properties of these are discussed. The unitary transformations connecting different representations are exhibited. The use of different representations to solve scattering and bound state problems is indicated. The application of one of the transformations to a partial diagonalization of a perturbed harmonic oscillator Hamiltonian is made.
While groups of point transformations such as rotations, translations and permutations have been extensively used in quantum mechanics, we cannot say the same of the more general groups of canonical transformations. To begin with, for the latter groups we have to determine their unitary representation in appropriate spaces. But once this is achieved canonical transformations can play as important a role in quantum mechanics as point transformations. We illustrate this in relation with the harmonic oscillator in one and many dimensions and for one and many particles. We show how radial matrix elements with respect to harmonic oscillator states can be determined group theoretically both in the Schroedinger and Heisenberg pictures. We discuss the application of canonical transformations to the three body nuclear problem. We briefly outline in the concluding section applications to Coulomb problems, many body systems, accidental degeneracy, and generalized integral transforms.
In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.
In the present paper we analyze the representations in quantum mechanics of classical canonical transformations that are nonbijective, i.e., not one-to-one onto. We take as the central example the canonical transformation that changes the Hamiltonian of a one-dimensional oscillator of frequency κ−1 into one of frequency k−1 where κ, k are relatively prime integers. For the particular case k=1, the mapping of the original phase space (x,p) onto the new one (x̄, p̄) is κ to 1 and the equivalent points in (x,p) are related by a cyclic group 𝒞κ of linear canonical transformations. When formulating this problem in Bargmann Hilbert space, the canonical transformation can be related to the conformal transformation w=zκ, which again is κ to 1 and where a group 𝒞κ also appears. This cyclic group proves fundamental for the determination of representations of the conformal transformation in Bargmann Hilbert space. To begin with, it suggests that while we can take in the original Bargmann Hilbert space a single component function, in the new Bargmann Hilbert space we must take a κ component one. In this way we can map in a one-to-one fashion the states and operators in the old and new Bargmann Hilbert spaces. When translating these results to ordinary Hilbert space, we get in an unambiguous way the quantization of the observables appearing in the equations that determine the representation of the classical canonical transformation relating oscillators of frequencies κ−1 and k−1. Furthermore, we also get the solutions of these equations, and the resulting representation is unitary. While our discussion is restricted to the problem mentioned above, in the concluding section we indicate our surmise for deriving systematically the unitary representation in quantum mechanics of arbitrary canonical transformations.
A canonical transformation changes variables such as coordinates and momenta to new variables preserving either the Poisson bracket or the commutation relations depending on whether the problem is classical or quantal respectively. Classically canonical transformations are well established as a powerful tool for solving differential equations. Quantum canonical transformations have been defined and used relatively recently because of the non-commutativeness of the quantum variables. Three elementary canonical transformations and their composite transformations have quantum implementations. Quantum canonical transformations have been mostly used in time-independent Schrodinger equations and a harmonic oscillator with time-dependent angular frequency is probably the only time-dependent problem solved by these transformations. In this work, we apply quantum canonical transformations to a harmonic oscillator in which both angular frequency and equilibrium position are time-dependent.
We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.
For pt.II see ibid., vol.16, no.7, p.1397-408 (1983). It is shown how quantum mechanical wavefunctions can be obtained from a sequence of simple canonical transformations, which map the given system onto a simple reference system. The resulting wavefunctions are at least uniformly valid up to order h(cross). Under some more restrictive conditions for the individual transformation steps the authors even find the exact wavefunctions. The essential point of the paper is to enlarge the conventional coordinate-momentum phase space by taking time and energy as an additional conjugate pair. In this extended space they exploit the possibility of using transformations which intermix energy and time with position coordinates and momenta. Compared with transformations in the conventional position-momentum phase space, they gain the advantage that scattering states and bound states can be treated in a unified way. Therefore this method is appropriate for systems with mixed spectra. In addition it allows for more flexibility in choosing the individual transformation steps. The practicability of the method is demonstrated by several examples.