Phase-space wavefunctions

Phase space quantum mechanics or symplectic quantum mechanics is a formulation of quantum mechanics that associates the phase-space formulation with a Hilbert space. Thus, it is possible to attribute a meaning for 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 $W(x,p)=\psi (x,p)\star \psi ^{\dagger }(x,p)$.

The Hilbert space aproach of quantum mechanics in phase space was first introduced by Oliveira et. al. in 2014 and it was followed by many authors .

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 $\phi (x,p)=\langle x,p|\phi \rangle$, 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 $\psi ^{\dagger }(x,p,t)$

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

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

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

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