The quantum-mechanical joint (quasi-) density in phase space (the Wigner distribution) for thermodynamic equilibrium is studied in the semiclassical limit h(cross) to 0. The approximation is of an asymptotic (WKB) type and nonperturbative in character. The equilibrium phase-space distribution can be expressed in terms of classical paths satisfying certain well defined conditions. These paths are complex-valued, i.e. classically forbidden. The resulting semiclassical expression for the equilibrium Wigner phase-space density agrees with the exact quantum result for the analytical solvable case of harmonic potentials for all temperatures.
The one-dimensional Wigner quasiprobability function is considered rigorously in the classical limit ($\ensuremath{\hbar}\ensuremath{\rightarrow}0$). With the implicit $\ensuremath{\hbar}$-dependence of the wave function $\ensuremath{\psi}$ ignored in the integral representation, this limit differs from ${|\ensuremath{\psi}(q)|}^{2}\ensuremath{\delta}(p)$ by terms $\frac{o(\ensuremath{\hbar})}{\ensuremath{\hbar}}$. With the implicit $\ensuremath{\hbar}$ dependence of $\ensuremath{\psi}$ accounted for (as it should be), rigorous asymptotic methods are applied, with the results that: (1) in the "classically allowed" region, the limit is an asymptotic exponential function of
q, multiplied by an energy-conserving $\ensuremath{\delta}$ function; (2) in the "nonclassically allowed" region, this limit is zero. The stationary-phase approach leads nowhere in this regard. A WKBJ-type solution for the radial Schr\"odinger equation is developed, with rigorous error bounds in the different situations. The Wigner function is computed for an ensemble of one-dimensional harmonic oscillators, and an associated phase-space "density matrix" is constructed therefrom. This is compared with the Slater sum for the ensemble---in general, and in the particular cases of high temperature and of $\ensuremath{\hbar}\ensuremath{\rightarrow}0$; where $\ensuremath{\hbar}\ensuremath{\rightarrow}0$, the asymptotic behavior of the Wigner function is explicitly manifest.
The Wigner function
WN(x,p) is a useful quantity to characterize the quantum fluctuations of an
N-body system in its phase space. Here we study
WN(x,p) for
N noninteracting spinless fermions in a
d-dimensional spherical hard box of radius
R at temperature
T=0. In the large-
N limit, the local-density approximation predicts that ${W}_{N}(\mathbf{x},\mathbf{p})\ensuremath{\approx}1/{(2\ensuremath{\pi}\ensuremath{\hbar})}^{d}$ inside a finite region of the
(x,p) plane, namely, for
∣x∣<R and
∣p∣<kF, where
kF is the Fermi momentum, while
WN(x,p) vanishes outside this region, or droplet, on a scale determined by quantum fluctuations. In this paper we investigate systematically, in this quantum region, the structure of the Wigner function along the edge of this droplet, called the Fermi surf. In one dimension, we find that there are three distinct edge regions along the Fermi surf and we compute exactly the associated nontrivial scaling functions in each regime. We also study the momentum distribution ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)$ and find a striking algebraic tail for very large momenta ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{\ensuremath{\rho}}}_{N}(p)\ensuremath{\propto}1/{p}^{4}$, well beyond
kF, reminiscent of a similar tail found in interacting quantum systems (discussed in the context of Tan's relation). We then generalize these results to higher
d and find, remarkably, that the scaling function close to the edge of the box is universal, i.e., independent of the dimension
d.
In this work we study the Wigner functions, which are the quantum analogues of the classical phase space density, and show how a full rigorous semiclassical scheme for all orders of ℏ can be constructed for them without referring to the actual coordinate space wavefunctions from which the Wigner functions are typically calculated. We find such a picture by a careful analysis around the stationary points of the main quantization equation, and apply this approach to the harmonic oscillator solving it for all orders of ℏ.
When Fermi's golden rule (FGR) is studied in the Wigner representation, the transition rate from an initial pure state or from an initial thermal distribution into a quasicontinuum manifold of degenerate states is given by an overlap integral of Wigner functions in phase space. In the semiclassical limit the transition rate is obtained by integrating over the regions in phase space where the energy difference between the initial and final potential surfaces is equal to the available energy. The integral is weighted by the initial probability density to be at that phase-space region. The classical limit of FGR is thus both simple and intuitive. In one dimension a relation to the Landau–Zener–Stuckelberg formula is established. The multi-dimensional case is considered by induction, proving that for separable multi-dimensional systems deviations of the logarithm of the transition rate from its classical limit scale at worst linearly with the dimension.