# second quantization - time dependent basis

In the second quantization time-independent field operator can be expanded in the orthonormal basis: $$\hat{\Psi}(\mathbf{x}) = \sum\limits_{i}\hat{a}_{i}\ \phi_{i}(\mathbf{x})$$

Time evolution of the field operator is given by: $$\hat{\Psi}(\mathbf{r},t) = e^{it/\hbar \hat{H}}\hat{\Psi}(\mathbf{x})e^{-it/\hbar \hat{H}} = \sum\limits_{i}\hat{a}_{i}(t)\ \phi_{i}(\mathbf{x})$$ Time dependence is transferred to the operators. Can we have time-dependent basis states rather than operators: $$\hat{\Psi}(\mathbf{r},t) = \sum\limits_{i}\hat{a}_{i}\ \phi_{i}(\mathbf{x},t)$$ ? How would the Fock state look like ?

• you are using the Heisenberg picture, where the time dependence is absorbed into the the operators.. If you work in the Schrodinger picture then your basis states will have time dependence... – Bruce Lee Feb 22 '16 at 14:08
• I can't find second quantization in Schrodinger picture. Everyone is doing it in Heisenberg picture. – WoofDoggy Feb 22 '16 at 14:15
• actually it can be done...why the second quantisation is done in Heisenberg picture (or in interaction picture) is because it is more convenient to do so... – Bruce Lee Feb 22 '16 at 14:25
• So now the question is: where I can find a good reference about second quantization in Schrodinger picture? – WoofDoggy Feb 22 '16 at 14:29
• I'am afraid I don't know... :)...you can edit and re-ask it for a resource recomendation...or ask about the problems in second quantization in Schrodinger picture... – Bruce Lee Feb 22 '16 at 14:32

The answer is yes only if $\hat{H}$ is quadratic in the creation and annihilation operators (with suitable additional regularity assumptions). In other words, it is possible only if the map is a so-called Bogol'ubov transformation.
If $\hat{H}$ is more than quadratic, it is easy to see (heuristically at least) that it cannot preserve the linear form of the field operator, and so OP's last expression cannot be true.