Phase-space formulation

The phase space formulation of quantum mechanics is a physical theory, equivalent to the usual Schrödinger picture, that places position and momentum on equal footing in phase space. The two key features of the phase space formulation are that the quantum state is described by a quasi-probability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully detailed by Hip Groenewold in 1946 in his PhD thesis, with significant parallel contributions from Joe Moyal, each building off earlier ideas by Hermann Weyl and Eugene Wigner.

The chief advantage of the phase space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space." Quantum mechanics in phase space is often favored in certain quantum optics applications or in the study of decoherence though otherwise the formalism is less commonly employed in practical situations.

The foundational ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as deformation theory and noncommutative geometry.

Phase space distribution

Main article: Quasi-probability distribution

Main article: Wigner quasi-probability distribution

Main article: Weyl quantization

The phase space distribution $f(x,p)$ of a quantum state is a quasi-proability distribution. There are several different ways to represent the distribution. The most noteworthy is the Wigner representation, $W(x,p)$. Other representations (in approximately descending order of prevalence in the literature) include the Glauber-Sudarshan P, Husimi Q, Kirkwood-Rihaczek, Mehta, Rivier, and Born-Jordan representations. These alternatives are most useful when the Hamiltonian takes a particular form, such as normal order for the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it unless otherwise specified.

By construction, the phase space distribution is intended to have properties akin to the probability density in a 2n-dimensional phase space. For example, it is real-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta:

$\operatorname {P} [a\leq X\leq b]=\int _{a}^{b}\int _{-\infty }^{\infty }W(x,p)\,dp\,dx.$

If ${\hat {A}}(x,p)$ is an operator representing an observable, we may map it into the phase space as $A(x,p)$ using the Wigner transformation. Conversely, the operator may be restored via the Weyl quantization. We may take the expectation value of the observable with respect to the phase space distribution:

$\langle A\rangle =\int A(x,p)W(x,p)\,dp\,dx.$

The reader should be cautioned that, despite the similarity of appearance, $W(x,p)$ is not a true joint probability distribution because in general it can be negative, with exception only for (optionally squeezed) coherent states. For more details on the properties and interpretation of the Wigner function, see its main article.

Star product

Main article: Moyal product

The fundamental binary operator in the phase space formulation that replaces the standard operator multiplication is the star product, represented by the symbol ★. Each representation of the phase space distribution has a different star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner representation.

For notational convenience, we introduce the notion of left and right derivatives. For a pair of functions f and g, the left and right derivatives are defined as

$f{\stackrel {\leftarrow }{\partial }}_{x}g={\frac {\partial f}{\partial x}}\cdot g$

$f{\stackrel {\rightarrow }{\partial }}_{x}g=f\cdot {\frac {\partial g}{\partial x}}.$

The differential definition of the star product is

$f\star g=f\,\exp {\left({\frac {i\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})\right)}\,g$

where the argument of the exponential function can be interpreted as a power series. Additional differential relations allow this to be written in terms of a change in the arguments of f and g:

${\begin{aligned}f(x,p)\star g(x,p)&=f\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g(x,p)\\&=f(x,p)\cdot g\left(x-{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p+{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right)\\&=f\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p},p\right)\cdot g\left(x-{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{p},p\right)\\&=f\left(x,p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot g\left(x,p+{\frac {i\hbar }{2}}{\stackrel {\leftarrow }{\partial }}_{x}\right)\end{aligned}}.$

It is also possible to define the star product in an integral form, essentially through the Fourier transform:

$f(x,p)\star g(x,p)={\frac {1}{\pi ^{2}\hbar ^{2}}}\,\int f(x+x',p+p')\,g(x+x'',p+p'')\,\exp {\left({\frac {2i}{\hbar }}(x'p''-x''p')\right)}\,dx'dp'dx''dp''$

The energy eigenstates are known as stargenstates or ★genstates, and the associated energies are known as stargenvalues or ★genvalues. These are solved, analogously to the time-independent Schrödinger equation, by the ★genvalue equation

$H\star f=E\cdot f,$

where H is the Hamiltonian.

Time evolution

The time evolution of the phase space distribution is given by a quantum Liouville theorem. This formula can be obtained by applying the Wigner transformation to the density matrix version of the quantum Liouville equation. In any representation of the phase space distribution with its associated star product, this is

${\frac {\partial f}{\partial t}}=-{\frac {1}{i\hbar }}\left(f\star H-H\star f\right),$

or for the Wigner function in particular,

${\frac {\partial W}{\partial t}}=-\{\{W,H\}\}=-{\frac {2}{\hbar }}W\sin \left({{\frac {\hbar }{2}}({\stackrel {\leftarrow }{\partial }}_{x}{\stackrel {\rightarrow }{\partial }}_{p}-{\stackrel {\leftarrow }{\partial }}_{p}{\stackrel {\rightarrow }{\partial }}_{x})}\right)\ H=-\{W,H\}+O(\hbar ^{2}),$

where $\{\{\cdot ,\cdot \}\}$ is the Moyal bracket and $\{\cdot ,\cdot \}$ is the classical Poisson bracket. This gives a very concise interpretation of the correspondence principle in that this equation manifestly reduces to the classical Liouville theorem in the limit $\hbar \rightarrow 0$.

Examples

Simple harmonic oscillator

Main article: quantum harmonic oscillator

The Hamiltonian for the simple harmonic oscillator is

$H={\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}.$

We may write the ★genvalue equation as

${\begin{aligned}H\star f&=\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)\star f\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x+{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{p}\right)^{2}+{\frac {1}{2m}}\left(p-{\frac {i\hbar }{2}}{\stackrel {\rightarrow }{\partial }}_{x}\right)^{2}\right)\cdot f\\&=\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\stackrel {\rightarrow }{\partial }}_{x}^{2}\right)\right)\cdot f\\&\,\,\,\,\,+{\frac {i\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot f\\&=E\cdot f\end{aligned}}.$

Consider first the imaginary part of the ★genvalue equation.

${\frac {\hbar }{2}}\left(m\omega ^{2}x{\stackrel {\rightarrow }{\partial }}_{p}-{\frac {p}{m}}{\stackrel {\rightarrow }{\partial }}_{x}\right)\cdot f=0$

This implies that we may write the ★genstates as functions of a single argument,

$f(x,p)=F\left({\frac {1}{2}}m\omega ^{2}x^{2}+{\frac {p^{2}}{2m}}\right)=F(u).$

With this change of variables, it is possible to write the real part of the ★genvalue equation in the form of a modified Laguerre equation, the solution of which involves the Laguerre polynomials as

$F_{n}(u)={\frac {(-1)^{2}}{\pi \hbar }}L_{n}\left({\frac {u}{\hbar \omega }}\right)e^{u/\hbar \omega }$

with associated ★genvalues

$E_{n}=\hbar \omega \left(n+{\frac {1}{2}}\right).$

Free particle angular momentum

Suppose a particle is initially in a minimally uncertain Gaussian state with the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state is

$W(\mathbf {x} ,\mathbf {p} ,t)={\frac {1}{(\pi \hbar )^{3}}}\exp {\left(-\alpha ^{2}r^{2}-{\frac {p^{2}}{\alpha ^{2}\hbar ^{2}}}\left(1+\left({\frac {t}{\tau }}\right)^{2}\right)+{\frac {2t}{\hbar \tau }}\mathbf {x} \cdot \mathbf {p} \right)}$

where α is a parameter describing the initial width of the Gaussian and $\tau =m/\alpha ^{2}\hbar$. Initially the position and momenta are uncorrelated. Thus in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel. On the other hand, the position and momentum should become increasingly correlated as the state evolves because portions of the distribution far from the origin in position require a large amount of momentum to be reached. Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the usual quantum mechanical notion of the nonzero ground state angular momentum's orientation independence:

$K_{rad}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}\left({\frac {3}{2}}-{\frac {1}{1+(t/\tau )^{2}}}\right)$

$K_{ang}={\frac {\alpha ^{2}\hbar ^{2}}{2m}}{\frac {1}{1+(t/\tau )^{2}}}$