Energy is a cornerstone concept in physics, embodying both conservation principles and the dynamics of systems. In quantum mechanics, energy manifests as the expectation value of the Hamiltonian operator, yet its intuitive understanding often remains elusive to students. This paper adopts a pedagogical approach to demystify quantum energy by employing three key phase-space representations: the Glauber-Sudarshan P-function, the Husimi Q-function, and the Wigner function. We show that, despite their distinct mathematical frameworks and interpretations, all three representations yield equivalent expressions for the mean energy of thermal states. This result provides an engaging platform for introducing concepts such as coherent states, operator ordering, and quasiprobability distributions. By bridging classical and quantum perspectives in an accessible format, we present a teaching model that not only conveys technical skills but also cultivates deeper conceptual insights, making it feasible to implement in a single lecture. This approach facilitates a richer understanding of the interplay between quantum mechanics and statistical physics, preparing students for advanced topics in quantum theory.
The problem of quantization in the phase-space formulation of quantum mechanics is considered. An integral equation for the phase-space eigenfunctions is derived which is equivalent to the standard eigenvalue equation for a quantum mechanical operator. A differential form is also given. A variational principle is derived for quasiprobability distributions. It is shown that the expected value of the classical Hamiltonian calculated with a trial quasiprobability distribution will be greater than the ground state energy only if the distribution is chosen from a certain class of functions. The notion of ψ-representability is introduced to classify these functions. They represent distributions which correspond to possible quantum mechanical states. Also, a general relation is given between different distribution functions.
Various phase-space distributions, from the celebrated Wigner function, to the Husimi
Q function and the Glauber–Sudarshan
P distribution, have played an interesting and important role in the phase-space formulation of quantum mechanics in general, and quantum optics in particular. A unified approach to all these distributions based on the notion of the
s -ordered phase-space distribution was introduced by Cahill and Glauber. With the intention of illuminating the physical meaning of the parameter
s , we interpret the
s -ordered phase-space distribution as the Wigner function of a state under the Gaussian noise channel, and thus reveal an intrinsic connection between the
s -ordered phase-space distribution and the Gaussian noise channel, which yields a physical insight into the
s -ordered phase-space distribution. In this connection, the parameter
−s/2 (rather than the original
s ) acquires the role of the noise occurring in the Gaussian noise channel. An alternative representation of the Gaussian noise channel as the scaling-measurement preparation in a coherent states is illuminated. Furthermore, by exploiting the freedom in the parameter
s , we introduce a computable and experimentally testable quantifier for optical nonclassicality, reveal its basic properties, and illustrate it by typical examples. A simple and convenient criterion for optical nonclassicality in terms of the
s -ordered phase-space distribution is derived.
This article provides a concise review of quantum phase space theory, beginning with its foundational principles and the properties of standard quantum quasi-probability distributions, specifically the Wigner, Husimi Q, and Glauber--Sudarshan P functions. We discuss the intrinsic limitations of these distributions, such as the appearance of negative values and phase-space blurring. A significant portion of this review highlights recent theoretical developments, particularly the quantum Wannier basis. This approach establishes a unitary mapping between the Hilbert space and a discretized phase space, yielding a genuine probability distribution in phase space and thereby providing a basis-dependent entropy for pure quantum states. Furthermore, we examine Bourgain's nonperiodic basis as a theoretical framework to circumvent the constraints imposed by the Balian--Low theorem. These developments provide practical tools for numerical studies based on the quantum Wannier basis, as well as conceptual benchmarks for understanding the localization limits of orthonormal phase-space representations.
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.
This thesis describes quantum mechanics in the phase space formulation. We introduce quantization and in particular the Weyl quantization. We study a general class of phase space distribution functions on phase space. The Wigner distribution function is one such distribution function. The Wigner distribution function in general attains negative values and thus can not be interpreted as a real probability density, as opposed to for example the Husimi distribution function. The Husimi distribution however does not yield the correct marginal distribution functions known from quantum mechanics. Properties of the Wigner and Husimi distribution function are studied to more extent. We calculate the Wigner and Husimi distribution function for the energy eigenstates of a particle trapped in a box. We then look at the semi classical limit for this example. The time evolution of Wigner functions are studied by making use of the Moyal bracket. The Moyal bracket and the Poisson bracket are compared in the classical limit. The phase space formulation of quantum mechanics has as advantage that classical concepts can be studied and compared to quantum mechanics. For certain quantum mechanical systems the time evolution of Wigner distribution functions becomes equivalent to the classical time evolution stated in the exact Egerov theorem. Another advantage of using Wigner functions is when one is interested in systems involving mixed states. A disadvantage of the phase space formulation is that for most problems it quickly loses its simplicity and becomes hard to calculate.