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Revista Mexicana de Física

Representation of canonical transformations in quantum mechanics

G. F. Torres del Castillo, H. Martı́nez, Ramiro Julián Sánchez, J.M. Zárate Paz

Published Jan 1, 2009• 1 citation
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Abstract

The transformation of the wave functions induced by a given canonical transformation in the classical phase space, (q i,p i) - (Q i, Pi), is considered. In the examples presented here, the kernel of the integral transform turns out to be essentially exp (q i, Qi) is defined by Pi dQi In the case of the time evolution, which is a canonical transformation, the kernel of the transform is the propagator, and is obtained directly by making use of the solution to the classical equations of motion.

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The transformation of the wave functions induced by a given canonical transformation in the classical phase space, (q i,p i) - (Q i, Pi), is considered. In the examples presented here, the kernel of the integral transform turns out to be essentially exp (q i, Qi) is defined by Pi dQi In the case of the time evolution, which is a canonical transformation, the kernel of the transform is the propagator, and is obtained directly by making use of the solution to the classical equations of motion.
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.
For pt.I see ibid., vol.16, no.11, p.3509-23 (1982). Sufficient conditions are given for constructing quantum mechanical wavefunctions by the sole knowledge of an appropriate sequence of classical canonical transformations which map a given Hamiltonian onto the new position variable. The transformation kernel for each individual step of this sequence is given by the semiclassical limit expression; it is a function of the generator of this transformation step only. The transformation kernel for the total transformation and the time-dependent wavefunction are obtained as a multiple integral over the product of the transformation kernels of the various intermediate steps. The practicability of this procedure is demonstrated by several examples. The authors consider explicitly time-dependent systems with one degree of freedom.
For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space-time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrödinger equation, the transformation function K(x,t′;y,t′) in the space representation on the basis of the composition rule K(x, t″; y, t′) = ∫K(x, t″; z, t)dzK(z, t; y, t′) (1) coupled with the supposition that it is approximated to zeroth order in the quantum of action h by the so-called semi-classical kernel Ko(x, t″; y, t′) = [(i/h)∂2S/∂x∂y]1/2 exp [(i/ℏ) S(x,t″; y, t′] (2) written in terms of the classical action S(x, t″; y, t′) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t″ − t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.
We consider general time-dependent quadratic Hamiltonian systems which are connected by canonical transformations and give the same classical equations of motion. In those systems, we demonstrate that canonical transformations in classical mechanics correspond to unitary transformations in quantum mechanics. The wave functions and the propagators are evaluated using the invariant operator method. However, the values of some functions of the canonical variables q and p are not equal to the values of the same functions of the other canonical variables Q and P, but the values of the functions of q and the kinetic momentum pk{\mathrm{p}}_{\mathrm{k}}pk​ are equal to those of the other Q and Pk{\mathrm{P}}_{\mathrm{k}}Pk​ in classical mechanics. We prove that these also hold in the quantum treatment. The uncertainty relations of momentum and position are evaluated for the two Hamiltonians.