For a Wigner-Ville quasi-probability distribution $W(q,p,t)$ and Hamiltonian $H(q,p)$, we can write a quantum Liouville equation:

$\frac{\partial W(q,p,t)}{\partial t} = -\{\{W(q,p,t) , H(q,p )\}\}$

where $\{\{a, b\}\}$ is the Moyal bracket. This comes from applying a Wigner transformation to the Von Neumann equation for density matrices. Are there similar, clean-looking, quantum Liouville equations for the evolution of the Glauber-Sudarshan P, the Husimi Q, or other representations?

up vote 1 down vote accepted

I got an answer to this offline. The answer is that the form looks the same for all representations with a suitable redefinition of the star product. For example, in the Husimi Q representation, we can define a new kind of star product as follows:

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

If $T$ is the map from Wigner to Husimi, then this product has the following property:

$ T(f \star g) = T(f) \circledast T(g) $

Therefore the quantum Liouville equation in the Husimi representation is

$ \frac{\partial Q}{\partial t} = - \frac{1}{i\hbar} (Q \circledast H - H \circledast Q) $

See e.g. this reference.

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.