Canonical Transformations In Quantum Mechanics Research Articles

Quantized normal matrices: some exact results and collective field formulation

We formulate and study a class of U ( N ) -invariant quantum mechanical models of large normal matrices with arbitrary rotation-invariant matrix potentials. We concentrate on the U ( N ) singlet sector of these models. In the particular case of quadratic matrix potential, the singlet sector can be mapped by a similarity transformation onto the two-dimensional Calogero–Marchioro–Sutherland model at specific couplings. For this quadratic case we were able to solve the N-body Schrödinger equation and obtain infinite sets of singlet eigenstates of the matrix model with given total angular momentum. Our main object in this paper is to study the singlet sector in the collective field formalism, in the large- N limit. We obtain in this framework the ground state eigenvalue distribution and ground state energy for an arbitrary potential, and outline briefly the way to compute bona-fide quantum phase transitions in this class of models. As explicit examples, we analyze the models with quadratic and quartic potentials. In the quartic case, we also touch upon the disk-annulus quantum phase transition. In order to make our presentation self-contained, we also discuss, in a manner which is somewhat complementary to standard expositions, the theory of point canonical transformations in quantum mechanics for systems whose configuration space is endowed with non-Euclidean metric, which is the basis for constructing the collective field theory.

Hamiltonian Formalism in Quantum Mechanics

Heisenberg motion equations in Quantum mechanics can be put into the Hamilton form. The difference between the commutator and its principal part, the Poisson bracket, can be accounted for exactly. Canonical transformations in Quantum mechanics are not, or at least not what they appear to be; their properties are formulated in a series of Conjectures. To Vladimir Igorevich Arnol’d with admiration, on occasion of his 60th birthday.

Linear canonical transformations in quantum mechanics

We find explicit unitary operators that implement linear canonical transformations of the quantum mechanical operators for a system with N degrees of freedom. We then relate the operators effecting this transformation to the previous formulation of quantum canonical transformations in terms of an effective generating function introduced by Ghandour.

Probability distributions for the phase difference.

In this work we analyze the quantum phase properties of pairs of electromagnetic field modes. Since phases differing by 2\ensuremath{\pi} are physically indistinguishable, we propose a general procedure to obtain the correct mod(2\ensuremath{\pi}) probability distributions for the phase difference. This allows us to investigate the properties of a number of phase approaches. This procedure provides deeper insight into the quantum nature of the phase difference. We relate this problem to the representation of nonbijective canonical transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.

Non-Hermitian techniques of canonical transformations in quantum mechanics.

The quantum-mechanical version of the four kinds of classical canonical transformations is investigated by using non-Hermitian operator techniques. To help understand the usefulness of this approach, the eigenvalue problem of a harmonic oscillator is solved in two different types of canonical transformations. The quantum form of the classical Hamilton-Jacobi theory is also employed to solve time-dependent Schr\"odinger wave equations, showing that when one uses the classical action as a generating function of the quantum canonical transformation of time evolutions of state vectors, the corresponding propagator can easily be obtained.

Unitary equivalence of the metric and holonomy formulations of (2+1)-dimensional quantum gravity on the torus.

Recent work on canonical transformations in quantum mechanics is applied to transform between the Moncrief metric formulation and the Witten-Carlip holonomy formulation of (2+1)-dimensional quantum gravity on the torus. A nonpolynomial factor ordering of the classical canonical transformation between the metric and holonomy variables is constructed which preserves their classical modular transformation properties. An extension of the definition of a unitary transformation is briefly discussed and is used to find the inner product in the holonomy variables which makes the canonical transformation unitary. This defines the Hilbert space in the Witten-Carlip formulation which is unitarily equivalent to the natural Hilbert space in the Moncrief formulation. In addition, gravitational $\ensuremath{\theta}$ states arising from "large" diffeomorphisms are found in the theory.

Nonclassical states of light and canonical transformations

Representations of nonlinear nonbijective canonical transformations in quantum mechanics are discussed. Due to the nonbijectivity the classical phase space has a Riemann sheet structure, and a family of partial isometries translating this structure into quantum mechanics is constructed. If a unitary representation is required, a new variable-the ambiguity spin-has to be introduced in order to recover bijectivity following the approach of Moshinsky (1981) and co-workers. The new degree of freedom is analysed in terms of multiboson operators. The application of this formalism to some non-classical states of light is discussed.

Canonical transformations in quantum mechanics: A canonically invariant path integral

A particular form of the path integral is presented that allows the implementation of general canonical transformations in quantum mechanics, as well as a consistent quantization procedure.

Comment on quantum representations of non-bijective canonical transformations

A recent letter by J. Deenen on 'non-bijective canonical transformations in quantum mechanics' (see ibid., vol.14, p.L273, 1981) is analysed and found to be incomplete. The general representation problem of non-bijective canonical transformations is shown to involve infinite sums of the quotient and tensor product spaces defined by Plebanski and Seligman (1982).

Non-bijective canonical transformations in quantum mechanics

Proposes a new formulation for the representation of non-bijective canonical transformations in quantum mechanics. These are represented by a unitary operator acting on a Hilbert space H(X)H similar to the one used for the description of a two-dimensional system.

Wigner distribution functions and the representation of canonical transformations in quantum mechanics

Shows how for classical canonical transformations the authors can pass, with the help of Wigner distribution functions, from their representation U in the configurational Hilbert space to a kernel K in phase space. The latter is a much more transparent way of looking at representations of canonical transformations, as the classical limit is reached when h(cross) to 0 and the successive quantum corrections are related with the power of h(cross)2n, n=1,2, et seq.

Canonical transforms. II. Complex radial transforms

Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n-dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut-Girardello, one-dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of SL(2,C) and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker-Campbell-Hausdorff and special function relations.

Canonical transforms. I. Complex linear transforms

Recent work by Moshinsky et al. on the role and applications of canonical transformations in quantum mechanics has focused attention on some complex extensions of linear transformations mapping the position and momentum operators x̂ and p̂ to a pair η̂ and ζ̂ of canonically conjugate, but not necessarily Hermitian, operators. In this paper we show that for a continuum of complex linear canonical transformations, a related Hilbert space of entire analytic functions exists with a scalar product over the complex plane such that the pair η̂, ζ̂ can be realized in the Schrödinger representation η and −id/dη. We provide a unitary mapping onto the ordinary Hilbert space of square-integrable functions over the real line through an integral transform. The transform kernels provide a representation of a subsemigroup of SL(2,C). The well-known Bargmann transform is the special case when η̂ and iζ̂ are the harmonic oscillator raising and lowering operators. The Moshinsky-Quesne transform is regained in the limit when the canonical transformation becomes real, a case which contains the ordinary Fourier transform. We present a realization of these transforms through hyperdifferential operators.

Canonical Transformations in Quantum Mechanics

A review of canonical transformations in quantum mechanics is outlined with a view toward their use in the solution of specific problems. Several typical problems are worked out by the method of canonical transformations to emphasize the utility of this approach. In this way one may obtain an insight into the nature and limitation of the connection that exists between the canonical transformations of classical mechanics and the unitary transformations of quantum mechanics. The purpose of this paper is to achieve certain expository and pedagogical objectives, and no original contribution to the theory is intended.

Some Examples of Canonical Transformations in Quantum Mechanics

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.