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) 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). An association with the Schrödinger wavefunction can be made by \psi(q,p)=e^{-ixp/2\hbar}\int g(x')\phi(x+x')e^{-(i/\hbar)px'}dx', letting g(x')=\phi^*(-\frac{z}{2}), we have \psi(q,p)=\int \phi(x-\frac{z}{2})\phi(x+\frac{z}{2})e^{-(i/\hbar)pz}dz. Then \psi(x,p)\propto W(q,p).
Torres-Vega–Frederick representation With the operators of position and momentum a
Schrödinger picture is developed in phase space :i\hbar\frac{\partial}{\partial t}\psi(x,p,t)=\widehat{H}_{{}_\text{TV}}\psi(x,p,t) . The Torres-Vega–Frederick distribution is :f_{{}_\text{TV}}=|\psi_{{}_\text{TV}}(q,p)|^2 .
Oliveira representation 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 equation 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, for this reason \psi is sometimes called quasiamplitude of probability. 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 static Wigner distribution function is a \star-genfunction of the \star-genvalue equation, a result well known in the usual phase-space formulation of quantum mechanics. In the case where \psi(q,p)\propto W(q,p), worked in the beginning of the section, the Oliveira approach and phase-space formulation are indistinguishable, at least for pure states. == Equivalence of representations ==