Transformations In Quantum Mechanics Research Articles

What is a Gauge Transformation in Quantum Mechanics?

In classical theory, a physical state is an equivalence class under gauge transformations. Is the same true in quantum theory? The physical quantum states are the solutions of Dirac's quantum constraint equation. They cannot be constructed as equivalence classes under the ``simple'' gauge transformations generated by the Dirac constraints. However, we show here that they can be constructed as equivalence classes under suitably defined ``complete'' gauge transformations. The complete gauge transformations are generated by the action of the quantum constraints on arbitrary individual components of the state.

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.

Energy spectrum for two-dimensional potentials in very high magnetic fields

A method, analogous to supersymmetry transformation in quantum mechanics, is developed for a particle in the lowest Landau level moving in an arbitrary potential. The method is applied to two-dimensional potentials formed by Dirac \ensuremath{\delta} scattering centers. In the periodic case, the problem is solved exactly for rational values of the magnetic flux (in units of flux quantum) per unit cell. The spectrum is found to be self-similar, resembling the Hofstadter butterfly [Phys. Rev. B 14, 2239 (1976)].

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.

Anti-Isospectral Transformations in Quantum Mechanics

In this letter we demonstrate that there exists a remarkable class of quantum models admitting Z2 anti-isospectral transformations which change the form of the potential and invert the sign of a certain finite set of energy levels. These transformations help one to construct interesting classes of non-monic polynomials which are mutually orthogonal on two different intervals with two different weight functions.

Relations of canonical and unitary transformations for a general time-dependent quadratic Hamiltonian system

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 ${\mathrm{p}}_{\mathrm{k}}$ are equal to those of the other Q and ${\mathrm{P}}_{\mathrm{k}}$ 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.

Introducing nonlinear gauge transformations in a family of nonlinear Schrödinger equations.

In earlier work we proposed a family of nonlinear time-evolution equations for quantum mechanics associated with certain unitary group representations [Doebner and Goldin, Phys. Lett. A 162, 397 (1992); J. Phys. A 27, 1771 (1994)]. Such nonlinear Schr\"odinger equations are expected to describe irreversible and dissipative quantum systems. Here we introduce and justify physically the group of nonlinear gauge transformations necessary to interpret our equations. We determine the parameters that are actually gauge invariant and describe some of their properties. Our conclusions contradict, at least in part, the view that any nonlinearity in quantum mechanics leads to unphysical predictions. We also show how time-dependent nonlinear gauge transformations connect our equations to those proposed by Kostin [J. Chem. Phys. 57, 3589 (1972)] and by Bialynicki-Birula and Mycielski [Ann. Phys. 100, 62 (1976)]. We believe our approach to be a fundamental generalization of the usual notions about gauge transformations in quantum mechanics. \textcopyright{} 1996 The American Physical Society.

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.

O(3,3)-like symmetries of coupled harmonic oscillators

In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) or SL(4,r) in the four-dimensional phase space. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r), respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics.

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.

Correspondence between the classical and quantum canonical transformation groups from an operator formulation of the wigner function

An explicit expression of the “Wigner operator” is derived, such that the Wigner function of a quantum state is equal to the expectation value of this operator with respect to the same state. This Wigner operator leads to a representation-independent procedure for establishing the correspondence between the inhomogeneous symplectic group applicable to linear canonical transformations in classical mechanics and the Weyl-metaplectic group governing the symmetry of unitary transformations in quantum mechanics.

Quantum probabilities, Kolmogorov probabilities, and informational probabilities

The relations between quantum probabilities, Kolmogorov probabilities, and informational probabilities are studied against the background offered by the concept of a quantum mechanical probability tree built in previous work. It is shown that the quantum mechanical transformation theory goes beyond the Kolmogorov concept of probabilities. It is furthermore shown that the quantum mechanical concept of probability is of the same essence as the informational one. The analyses that produce these conclusions bring forth the first lines of a general mathematical representation of the emergence and circulation of patterns of any kind.

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.

DOUBLE WAVE DESCRIPTION OF THE MOTION OF A CHARGED PARTICLE IN A UNIFORM MAGNETIC FIELD

Double wave quantum theory is applied to describe the motion of charged particle in a uniform magnetic field. This description of the motion of the particle is complete. One can tell the value of an arbitrary physical quantity at any time. The probability and the formula to calculate mean values in ordinary quantum mechanics comes from the average of double wave description over a certain ensemble. The rule of gauge transformation in quantum mechanics can be seen from the average over an ensemble.

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.

Wigner distribution functions and the representation of a non-bijective canonical transformation in quantum mechanics

In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1, where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.

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.

Symmetry transformations in quantum mechanics

We introduce a new definition of symmetry transformations in quantum mechanics which contains the textbook definition as a special case. We findseveral conserved quantities associated with one (nonunitary) symmetry transformation. We present two examples.

Niels Bohr's discussions with Albert Einstein, Werner Heisenberg, and Erwin Schrödinger: The origins of the principles of uncertainty and complementarity

In this paper, the main outlines of the discussions between Niels Bohr with Albert Einstein, Werner Heisenberg, and Erwin Schrodinger during 1920–1927 are treated. From the formulation of quantum mechanics in 1925–1926 and wave mechanics in 1926, there emerged Born's statistical interpretation of the wave function in summer 1926, and on the basis of the quantum mechanical transformation theory—formulated in fall 1926 by Dirac, London, and Jordan—Heisenberg formulated the uncertainty principle in early 1927. At the Volta Conference in Como in September 1927 and at the fifth Solvay Conference in Brussels the following month, Bohr publicly enunciated his complementarity principle, which had been developing in his mind for several years. The Bohr-Einstein discussions about the consistency and completeness of qnautum mechanics and of physical theory as such—formally begun in October 1927 at the fifth Solvay Conference and carried on at the sixth Solvay Conference in October 1930—were continued during the next decades. All these aspects are briefly summarized.

On the classification of the homogeneous 16-vertex models on a square lattice: I. Connection with the boson Bogoliubov transformation

We consider the classification of all homogeneous 16-vertex models on a square lattice which was proposed by Gaaff and Hijmans. We show that this classification can be translated into (that is, reformulated in terms of) a classification of matrices which closely resembles one already investigated with some success in the literature, viz. the classification connected with the boson Bogoliubov transformation in quantum mechanics. The main difference between our approach and the one of Gaaff and Hijmans who also made a reformulation, is that we can keep our considerations, in conformity with the actual physical problem, within the field of real rather than complex numbers. In the accompanying paper this leads to results with features not present in the analysis over C , which point to a possible connection with a new kind of phase transitions.