Phase-space wavefunctions

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The wave-function approach of quantum mechanics in phase space is a formulation of quantum mechanics elaborating the phase-space formulation with a Hilbert space. Thus, it is possible to assign a meaning to the wave function in phase space, $\psi (x,p,t)$. The wave function, $\psi (x,p,t)$, is associated to the Wigner distribution by means of the star product as $W(x,p,t)=\psi (x,p,t)\star \psi ^{\dagger }(x,p,t)$.

The wave-function approach of quantum mechanics in phase space was introduced by Torres-Vega and Frederick in 1990. (also see ) and it was followed by many authors

An advantage might be that off-diagonal Wigner functions used in superpositions are treated in an intuitive way, $\psi _{1}\star \psi _{2}$.

Wigner quasiprobability distribution

Main articles: Wigner quasiprobability distribution and Wigner–Weyl transform

There exists a family of different representations of phase space distribution, and all are interrelated. The most most popular is the Wigner representation, W(x, p), and historically discovered first. Other common representations are Glauber–Sudarshan P, and Husimi Q. Such alternatives are particularly useful where a particular form is taken by the Hamiltonian, such as normal order of Glauber – Sudarshan P-representation.

In the Wigner formalism the probability of a quantum system be in a interval of positions can be attributed to the Wigner functionintegrated over all momenta and the position interval

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

If Â(x, p) is an operator representing an observable, it may be mapped to phase space as A(x, p) through the Wigner transform. Conversely, this operator may be recovered by the Weyl transform.

The expectation value of the observable with respect to the phase-space distribution is

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

It should be noted that despite the similarity in appearance, W(x, p) is not a genuine joint probability distribution, because regions under it do not represent mutually exclusive states, as required in the third axiom of probability theory. Moreover, it can, in general, take negative values even for pure states, with the unique exception of (optionally squeezed) coherent states, in violation of the first axiom.

Star product

Main article: Moyal product

In the phase space formulation the standard operator multiplication is replaced by the star product, represented by the symbol $\bigstar$, ${\widehat {A}}\cdot {\widehat {B}}\rightarrow a\star b$, where $a$ and $b$ are functions in phase space. Each representation of the phase space distribution has its own characteristic star product, the symplectic quantum mechanics is constructed with the star product relevant to the Wigner-Weyl representation, also called Moyal product.

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

${\begin{aligned}f{\overleftarrow {\partial }}_{x}g&={\frac {\partial f}{\partial x}}\cdot g\\[5pt]f{\overrightarrow {\partial }}_{x}g&=f\cdot {\frac {\partial g}{\partial x}}.\end{aligned}}$

The definition of the star product in its differential form is given by

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

where the argument in the exponential function can be interpreted as a power series.

It is also possible to define the ★-product in a integral form, essentially through the Fourier transform:

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

The star product is non-commutative,

$f\star g\neq g\star f$.

It is associative

$f\star (g\star h)=(f\star g)\star h$.

The complex conjugate is an antilinear antiautomorphism

$(f\star g)^{\dagger }=g^{\dagger }\star f^{\dagger }$.

The scalar function $1$ still the identity in the new algebra

$1\star f=f\star 1=f$.

Star Operators

Instead of thinking in terms multiplication of function using the star product, we can shift to think in terms of operators acting in functions in phase space

$f(x,p)\star g(x,p)=f\left(x+{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{p},p-{\tfrac {i\hbar }{2}}{\overrightarrow {\partial }}_{x}\right)g(x,p)={\widehat {F}}g$.

Thus, we can define the following operators

${\widehat {p\,}}=p\star =p-i\hbar \partial _{x}$,

${\widehat {x}}=x\star =x+i\hbar \partial _{p}$.

These operators satisfy the uncertainty principle,

$[{\widehat {x}},{\widehat {p\,}}]=i\hbar$.

Symplectic Hilbert Space

To associate the Hilbert space, ${\mathcal {H}}$, with the phase space $\Gamma$, we will consider the set of complex functions of integrable square, $\psi (x,p)$ in $\Gamma$, such that

$\int dp\,dx\,\psi ^{\ast }(x,p)\psi (x,p)<\infty .$

Then we can write $\psi (x,p)=\langle x,p|\psi \rangle$, with

$\int dp\,dx\;|x,p\rangle \langle x,p|=1,$

where $\langle \psi |$ is the dual vector of $|\psi \rangle$. This symplectic Hilbert space is denoted by ${\mathcal {H}}(\Gamma ).$

Thus, it is now, with aid of the star product possible to construct a Schrödinger picture in phase space for $\psi (x,p)$

$\psi (x,p,t)=e^{-{\frac {i}{\hbar }}H\star \,t}\psi (x,p)$,

deriving both side by $t$, we have

$i\hbar {\frac {\partial }{\partial t}}\psi (x,p,t)=H\star \psi (x,p,t)$,

therefore, the above has the same role of Schrödinger equation in usual quantum mechanics.

To show that $W(x,p,t)=\psi (x,p,t)\star \psi ^{\dagger }(x,p,t)$, we take the 'Schrödinger equation' in phase space and 'star multiply' by the right for $\psi ^{\dagger }(x,p,t)$

$i\hbar {\frac {\partial \psi }{\partial t}}\star \psi ^{\dagger }=H\star \psi \star \psi ^{\dagger }$,

where $H$ is the classical Hamiltonian of the system. And taking the complex conjugate

$-i\hbar \,\psi \star {\frac {\partial \psi ^{\dagger }}{\partial t}}=\psi \star \psi ^{\dagger }\star H$,

subtracting both equations we get

${\frac {\partial }{\partial t}}(\psi \star \psi ^{\dagger })=-{\frac {1}{i\hbar }}[(\psi \star \psi ^{\dagger })\star H-H\star (\psi \star \psi ^{\dagger })]$,

which is the time evolution of Wigner function. The $\star$-genvalue is given by the time independent equation

$H\star \psi =E\psi$.

Star multiplying for $\psi ^{\dagger }(x,p,t)$ on the right, we obtain

$H\star W=E\,W$.

Therefore the Wigner Distribution function is a $\star$-genfunction of the $\star$-genvalue equation.

Harmonic oscillator in phase space

Main article: quantum harmonic oscillator

The Wigner quasiprobability distribution of the quantum harmonic oscillator: a) n = 0, b) n = 1, c) n = 5.

The Hamiltonian of one dimensional simple harmonic oscillator is

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

The star operator is then given by

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

${}\;\;\;={\frac {1}{2m}}\left(p^{2}-i\hbar p\partial _{q}-{\frac {1}{4}}\partial _{q}^{2}\right)+{\frac {1}{2}}m\omega ^{2}\left(q^{2}+i\hbar q\partial _{p}-{\frac {1}{4}}\partial _{p}^{2}\,\right)$.

Thus the time evolution equation for the quasiamplitude of probability is

$H\star \Psi =i\hbar {\frac {\partial \Psi }{\partial t}}$.

We can use separation of variables to get

$H\star \psi =E\,\psi$,

with $\Psi =\exp {\Big (}{-{\frac {i}{\hbar }}Et}{\Big )}\psi$.

Choosing $\psi$ to be real, we can separate the real and imaginary part of the $\star$genvalue equation

$\left({\frac {1}{2}}m\omega ^{2}\left(x^{2}-{\frac {\hbar ^{2}}{4}}{\partial }_{p}^{2}\right)+{\frac {1}{2m}}\left(p^{2}-{\frac {\hbar ^{2}}{4}}{\partial }_{x}^{2}\right)-E\right)\psi =0$,

and

${\frac {\hbar }{2}}\left(m\omega ^{2}x{\partial _{p}}-{\frac {p}{m}}{\partial _{x}}\right)\psi =0$.

This equation system can be satisfied by

$\psi _{n}(x,p)=C_{n}e^{-(m\omega ^{2}x^{2}+{\frac {p^{2}}{m}})}L_{n}{\Big (}2{\big (}m\omega ^{2}x^{2}+{\frac {p^{2}}{m}}{\big )}{\Big )}$,

where $L_{n}(x)$ is the Laguerre polinomial.

The Wigner function $W(x,p)=\psi (x,p)\star \psi ^{\dagger }(x,p)$.

For the ground state we have

$\psi _{0}=C_{0}e^{-2H/\hbar \omega }$

Therefore,

$W_{0}=C_{0}e^{-2h/\hbar \omega }\star \psi _{0}=C_{0}e^{-2{\hat {h}}/\hbar \omega }\psi _{0}=C_{0}e^{-2E_{0}/\hbar \omega }\psi _{0}$

$={\frac {{C_{o}}^{2}}{e}}e^{(m\omega ^{2}x^{2}+{\frac {p^{2}}{m}})/\hbar \omega }$.

In General

$W_{n}(h)={\frac {(-1)^{n}}{\pi \hbar }}L_{n}\left(4{\frac {h}{\hbar \omega }}\right)e^{-2h/\hbar \omega }~.$

The $\star$genvalue is

$E=\hbar \omega {\bigg (}n+{\frac {1}{2}}{\bigg )}$.