CRC Press eBooks

- Phase Space Picture and Canonical Transformations

S. Rajasekar, R. Velusamy

Published Dec 10, 2014• 0 citations

Abstract

Given the wave function ψ(x, t) we compute the probability density function ρ(x, t) = |ψ(x, t)|2 and easily visualize the distribution. Knowing ψ(x, t) the distribution in momentum px is given by σ(px, t) = |χ(px, t)|2 = ∣∣∣ 1√ 2π ψ(x, t)e−ikx dx ∣∣∣2 , (6.1) where χ(px, t) is the momentum wave function. The momentum distribution is diﬃcult to visualize from the given ψ(x, t). Thus, we wish to have a function that can display the probability distribution in the variables x and px simultaneously. Recall that, in classical mechanics phase space is used to visualize the solutions of systems. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture [1-8]. The idea is that every state of a quantum system is describable by a distribution (or function) on the classical phase space. In the phase space picture, starting from the Schrödinger wave function, we are able to construct the Wigner distribution function in terms of x and px. It does not represent a joint probability distribution for x and px because the uncertainty principle disallows the simultaneous determination of these variables with desired accuracy.

DOI: 10.1201/b17827-10

Related papers

Phase Space Picture and Canonical Transformations

S. Rajasekar, R. Velusamy

This chapter presents some of the features of the phase space picture. For quantum mechanical systems Eugene Paul Wigner introduced a quantum analogue of a phase space probability distribution called Wigner distribution function or phase space picture. Using the Wigner distribution function it is possible to perform a canonical transformation in quantum mechanics. The phase space picture is a useful candidate to illustrate the transition from classical to quantum mechanics. The Poisson brackets and canonical transformations are part of the effective formulation of classical mechanics. In the infinite height potential video sequences of the evolution of Wigner functions for a wave function of a wave packet and its mirror wave packet is constructed. The chapter also presents the application of the canonical transformations and phase space distribution function in the study of wave packet spread and coherent and squeezed states.

CRC Press eBooksOct 5, 20220 citations

Relationship between the Wigner function and the probability density function in quantum phase space representation

Qian Shu Li, Gong Min Wei, Li Qiang Lü

This paper discusses the relationship between the Wigner function, along with other related quasiprobability distribution functions, and the probability density distribution function constructed from the wave function of the Schr\"odinger equation in quantum phase space, as formulated by Torres-Vega and Frederick (TF). At the same time, a general approach in solving the wave function of the Schr\"odinger equation of TF quantum phase space theory is proposed. The relationship of the wave functions between the TF quantum phase space representation and the coordinate or momentum representation is thus revealed.

Physical Review AAug 20, 200426 citations

Probability Representation of Quantum States

Olga V. Man’ko, Vladimir I. Man’ko

The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.

EntropyApr 29, 202143 citations

- Momentum Representation

S. Rajasekar, R. Velusamy

In classical physics one needs to know the velocity or momentum of a particle in order to determine its evolution. In quantum mechanics the wave function ψ already contains information of possible momenta. What one has to do is to apply harmonic analysis to the function ψ. Sometimes it is helpful to describe quantum states in terms of wave function of momentum. This requires decomposition of ψ in terms of various momentum states and constructing a new function C(k). The value of C(k) of each k gives the strength of the contribution to the k−momentum state of ψ. The space of k is called momentum space.

CRC Press eBooksDec 11, 20140 citations

Quasiprobability distributions

Ulf Leonhardt

Wigner representation In classical optics (Born and Wolf, 1999), the state of the electromagnetic oscillator is perfectly described by the statistics of the classical amplitude α . The amplitude may be completely fixed (then the field is coherent), or α may fluctuate (then the field is partially coherent or incoherent). In classical optics as well as in classical mechanics, we can characterize the statistics of the complex amplitude?or, equivalently, the statistics of the components position q and momentum p introducing a phase space distribution W ( q , p ). (As explained in Section 3.1, the real and the imaginary part of the complex amplitude α can be regarded as the position and the momentum of the electromagnetic oscillator.) The distribution W ( q , p ) quantifies the probability of finding a particular pair of q and p values in their simultaneous measurement. Knowing the phase space probability distribution, all statistical quantities of the electromagnetic oscillator can be predicted by calculation. In this sense the phase space distribution describes the state in classical physics. All this is much more subtle in quantum mechanics. First of all, Heisenberg's uncertainty principle prevents us from observing position momentum simultaneously and precisely. So it seems there is no point in thinking about quantum phase space. But wait! In quantum mechanics we cannot directly observe quantum states either. Nevertheless, we are legitimately entitled to use the concept of states as if they were existing entities (whatever they are). We use their properties to predict the statistics of observations.

Cambridge University Press eBooksFeb 18, 20101 citation

Wigner function and the probability representation of quantum states

Margarita A. Man’ko, Vladimir I. Man’ko

The relation of theWigner function with the fair probability distribution called tomographic distribution or quantum tomogram associated with the quantum state is reviewed. The connection of the tomographic picture of quantum mechanics with the integral Radon transform of the Wigner quasidistribution is discussed. The Wigner–Moyal equation for the Wigner function is presented in the form of kinetic equation for the tomographic probability distribution both in quantum mechanics and in the classical limit of the Liouville equation. The calculation of moments of physical observables in terms of integrals with the state tomographic probability distributions is constructed having a standard form of averaging in the probability theory. New uncertainty relations for the position and momentum are written in terms of optical tomograms suitable for directexperimental check. Some recent experiments on checking the uncertainty relations including the entropic uncertainty relations are discussed.

arXiv:1403.3189EPJ Web of ConferencesJan 1, 20141 citation