The Schrödinger equation is generally formulated in position space $$ i \hbar \frac{\partial}{\partial t}\psi(x,t) = \hat H_x \psi(x,t) = \left [ \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ]\psi(x,t) $$ or in momentum space $$ i \hbar \frac{\partial}{\partial t}\psi(p,t) = \hat H_p \psi(p,t) = \left[ \frac{p^2}{2m} + V(\frac{\hbar}{i}\frac{\partial}{\partial p},t)\right ]\psi(p,t). $$

But is it also possible to derive a Schrödinger equation for position $\times$ momentum $=$ phase space? I assume in that case it should be an equation acting on a wave function $\psi = \psi(x,p,t)$ . So I guess simply multiplying the two Schrödinger equations above is not enough. How could one then derive a Schrödinger equation for phase space, if it is possible at all?

The Schrödinger equation is generally formulated in position space $$ i \hbar \frac{\partial}{\partial t}\psi(x,t) = \hat H_x \psi(x,t) = \left [ \frac{-\hbar^2}{2m}\frac{\partial^2}{\partial x^2} + V(x,t)\right ]\psi(x,t) $$ or in momentum space $$ i \hbar \frac{\partial}{\partial t}\psi(p,t) = \hat H_p \psi(p,t) = \left[ \frac{p^2}{2m} + V\left(\frac{\hbar}{i}\frac{\partial}{\partial p},t\right)\right ]\psi(p,t). $$

But is it also possible to derive a Schrödinger equation for position $\times$ momentum $=$ phase space? I assume in that case it should be an equation acting on a wave function $\psi = \psi(x,p,t)$ . So I guess simply multiplying the two Schrödinger equations above is not enough. How could one then derive a Schrödinger equation for phase space, if it is possible at all?