Obtaining non-negative quantum mechanical distribution function

The distribution function obtained by smoothing the original Wigner function with a weighting function, is shown to be positive everywhere, if the weighting function itself is a Wigner transform of any normalised function. The marginal distribution is recovered with a special choice of this function.     

Foundations of phase-space quantum mechanics

In the present paper a general concept of a phase-space representation of the ordinary Hilbert-space quantum theory is formulated, and then, by using some elementary facts of functional analysis, several equivalent forms of that concept are analyzed. Several important physical examples are presented in Section 3 of the paper.Supported by the NSERC research grant No. A5206.On leave of absence from the Theoretical Physics Institute, University of Gdansk, Gdansk, Poland.     

Amplitude phase-space model for quantum mechanics

We show that there is a close relationship between quantum mechanics and ordinary probability theory. The main difference is that in quantum mechanics the probability is computed in terms of an amplitude function, while in probability theory a probability distribution is used. Applying this idea, we then construct an amplitude model for quantum mechanics on phase space. In this model, states are represented by amplitude functions and observables are represented by functions on phase space. If we now postulate a conjugation condition, the model provides the same predictions as conventional quantum mechanics. In particular, we obtain the usual quantum marginal probabilities, conditional probabilities and expectations. The commutation relations and uncertainty principle also follow. Moreover Schrödinger's equation is shown to be an averaged version of Hamilton's equation in classical mechanics.     

On constructing the wave function of a quantum particle from its Wigner phase-space distribution function

For a Schrödinger wave function, the part of the phase that depends only on time disappears in the construction of the corresponding Wigner phase-space (quasi)distribution function. Despite this, it can be recovered from the Wigner function using the quantum Hamilton–Jacobi equation. This is demonstrated for three simple cases.     

Trajectories and causal phase-space approach to relativistic quantum mechanics

We analyze phase-space approaches to relativistic quantum mechanics from the viewpoint of the causal interpretation. In particular, we discuss the canonical phase space associated with stochastic quantization, its relation to Hilbert space, and the Wigner-Moyal formalism. We then consider the nature of Feynman paths, and the problem of nonlocality, and conclude that a perfectly consistent relativistically covariant interpretation of quantum mechanics which retains the notion of particle trajectory is possible.     

Liouville dynamics for optimal stochastic phase-space representations of quantum mechanics

It is shown that the space P(Γ_s) of Γ_s-distribution functions ?(q, p) (Husimi transforms) can be described without reference to any conventional representation of the density operator ?. A Liouville-type differential equation governing the free time-evolution of ?_t(q, p) is derived and solved explicitly; the time dependence of this solution supports the thesis that ?(q, p) is a bona fide probability density observable with optimally accurate apparatus for the simultaneous measurement of position and momentum. Liouville-type equations are derived also for the case when local interactions described by analytic potentials are present. Probability currents corresponding to ?(q, p) are defined and it is shown that they obey a continuity equation at space-time points. Reduced Γ_s-distribution functions are defined and it is shown that they obey a BBGKY hierarchy of equations. A Brownian-motion experimental test of the underlying theory of measurement is suggested.     

On phase-space representations of quantum mechanics using Glauber coherent states

A phase-space formulation of quantum mechanics is proposed by constructing two representations (identified as pq and qp) in terms of the Glauber coherent states, in which phase-space wave functions (probability amplitudes) play the central role, and position q and momentum p are treated on equal footing. After finding some basic properties of the pq and qp wave functions, the quantum operators in phase-space are represented by differential operators, and the Schrödinger equation is formulated in both pictures. Afterwards, the method is generalized to work with the density operator by converting the quantum Liouville equation into pq and qp equations of motion for two-point functions in phase-space. A coordinate transformation between those points allows one to construct a cell in phase-space, whose central point can be treated as a parameter. In this way, one gets equations of motion describing the evolution of one-point functions in phase-space. Finally, it is shown that some quantities obtained in this paper are related in a natural way with cross-Wigner functions, which are constructed with either the position or the momentum wave functions.     

On the nonclassical character of the phase-space representations of quantum mechanics

The quasiclassical representations of quantum theory, generalizing the concept of a phase-space representation of quantum mechanics, are studied with particular emphasis on some questions connected with the Jordan structure of the classical and quantum algebras of observables. A generalized version of the theorem of Gleason, Kahane, and Zelazko is used to establish some nonclassical features of these representations.Supported by the NSERC research grant No. A5206.On leave of absence from the Theoretical Physics Institute, University of Gdask, Gdask, Poland.     

A precaution needed in using the phase-space formulation of quantum mechanics

We explain the origin of the apparent discrepancy recently reported between results obtained by the phase-space formulation of quantum mechanics and conventional (Schrödinger) quantum mechanics. We show how to arrive at a complete agreement.     

A curiosity concerning non-negative quantum distribution functions

Approaching the problem of non-negative quantum distribution functions from the viewpoint of the correspondence rule, it is shown that the class of such functions proposed earlier by Kuryshkin is the only admissible class.     

On Wigner functions and a damped star product in dissipative phase-space quantum mechanics

Dito and Turrubiates recently introduced an interesting model of the dissipative quantum mechanics of a damped harmonic oscillator in phase space. Its key ingredient is a non-Hermitian deformation of the Moyal star product with the damping constant as deformation parameter. We compare the Dito-Turrubiates scheme with phase-space quantum mechanics (or deformation quantization) based on other star products, and extend it to incorporate Wigner functions. The deformed (or damped) star product is related to a complex Hamiltonian, and so necessitates a modified equation of motion involving complex conjugation. We find that with this change the Wigner function satisfies the classical equation of motion. This seems appropriate since non-dissipative systems with quadratic Hamiltonians share this property.     

Supersymmetry in problems of quantum mechanics

A connection is discussed between the group SU(2) and supersymmetry for a series of quantum mechanical problems. It is pointed out that the impossibility of factorizing Hamiltonians obtained based on representations of the group SU(2) indicates that the supersymmetry of the system is broken.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 19–23, July, 1988.In conclusion, we thank A. V. Borisov, V. Ch. Zhukovskii, I. V. Tyutin, and S. M. Khoroshkin for discussing results of this work.     

Some aspects of gravitational quantum mechanics

String theory, quantum geometry, loop quantum gravity and black hole physics all indicate the existence of a minimal observable length on the order of Planck length. This feature leads to a modification of Heisenberg uncertainty principle. Such a modified Heisenberg uncertainty principle is referred as gravitational uncertainty principle(GUP) in literatures. This proposal has some novel implications on various domains of theoretical physics. Here, we study some consequences of GUP in the spirit of Quantum mechanics. We consider two problem: a particle in an one-dimensional box and free particle wave function. In each case we will solve corresponding perturbational equations and compare the results with ordinary solutions.     

Some remarks on a model of supersymmetric quantum mechanics

We develop a recently proposed model within supersymmetric quantum mechanics that puts a group structure on the creation and annihilation operators. We apply the scheme to a variety of quantum mechanical problems and work out a two-term energy recursion equation when the overall group structure isU(1, 1).     

教学研究:介绍量子力学几个基本概念——兼答《关于量子几何相位的评注》中的几个主要问题

介绍了量子绝热定理的物理含义及成立的条件，认为有关主要献（Aharonov-Anandan,Bohm，孙昌璞等）的表述是正确的，而《关于量子几何相位的评注》^[1]（以下简称《评注》）相应的表述不完全正确。在此基础上，认为这些献和教材（R.Shankar)得出的涉及Berry绝热相位的一些论述（不含Berry绝热相因子的瞬时能量本征态不满足含时Schroedinger方程等）也是正确的，而《评注》的论述与此相反。《评注》认为只有γn(C)才是Berry相位。本作则倾向于把γn（t）叫做Berry绝热相位，而把γn（C）=γn（T）-γn（0）叫做几何相位（geometric phase)^[2]。     

Controversial problems of measurements within quantum mechanics

It is explained why quantum mechanics applies principally to single systems and not to ensembles. A thorough analysis of thought-experiments shows clearly that irreversibility is connected with the storing of information rather than with the act of measurement. In order to avoid paradoxes one has to admit that the wave function does not represent the state of the system in itself, but information acquired in consequence of a complete measurement. The meaning of the time-energy uncertainty relation for stable systems is discussed.