Phase space formulation

Builds on Phase space Wigner quasiprobability distribution Weyl quantization Moyal product Density matrix

Definition

Let $n \ge 1$ , $\hbar > 0$ , and let $H$ be a quantum Hamiltonian on $L^2(\R^n)$ with Weyl symbol $h(x, p)$ on the phase space $\R^{2n}$ . The phase space formulation of the quantum system is the following description of it by functions on $\R^{2n}$ :

a state, given by a density operator $\rho$ , is represented by its Wigner function $W_\rho(x, p)$ ;
an observable $A$ is represented by its Weyl symbol $a$ , so that $A = \mathrm{Op}^w(a)$ ;
the expectation value of $A$ in the state $\rho$ is $\langle A \rangle_\rho = \int_{\R^{2n}} a\, W_\rho\, dx\, dp$ ;
the product of observables $\mathrm{Op}^w(a)\mathrm{Op}^w(b)$ is represented by the Moyal product $a \star b$ ;
the time evolution of the state is given by the Moyal equation $\frac{\partial W}{\partial t} = \frac{1}{i\hbar}\left(h \star W - W \star h\right).$

Remarks

Clause 5 is the translation of the von Neumann equation $i\hbar\, \partial_t \rho = [H, \rho]$ , because $W_\rho$ is $(2\pi\hbar)^{-n}$ times the Weyl symbol of $\rho$ . Since $h \star W - W \star h = i\hbar\{h, W\} + O(\hbar^3)$ , the equation reduces to the classical Liouville equation $\partial_t W = \{h, W\}$ up to terms of order $\hbar^2$ .

Example: for the harmonic oscillator $h = \frac{p^2}{2m} + \frac{m\omega^2 x^2}{2}$ , all terms of order $\hbar^2$ and higher vanish, because $h$ is quadratic. So $\partial_t W = \{h, W\}$ exactly, and $W$ is transported along the classical elliptical orbits.

Non-example: a general probability density $f(x, p)$ on $\R^{2n}$ need not be the Wigner function of any state. For instance, a Gaussian $f = \frac{1}{\pi\varepsilon} e^{-(x^2 + p^2)/\varepsilon}$ with $n = 1$ and $\varepsilon < \hbar$ violates the bound $\abs{W} \le 1/(\pi\hbar)$ at the origin.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Density matrix concept
Moyal product concept
Phase space concept
Weyl quantization concept
Wigner quasiprobability distribution concept
Phase space formulation definition

Tag	`B9R2E2188P`
URI	`m2://B9R2E2188P`
Kind	definition
Writing	W0 stub
MSC	81S30, 53D55
Defines	Phase space formulation
Uses	Phase space, Wigner quasiprobability distribution, Weyl quantization, Moyal product, Density matrix
Changed	3 November 2026

Phase space formulation

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here