If you have a wave function $\Psi$ of a system consisting of an electron and the vibrational modes of the crystal, THEN we represent the wavefunction $\Psi%$ to be in the Hilbert Space formed by the tensor product of the Hilbert spaces corresponding to the electron with the Hilbert Space corresponding to the vibrational modes if and only if there isn't an instantaneous interaction between the electrons and the vibrational modes; First of all, this is true right?
The Born-Oppenheimer Approximation technique tells us that we can write the wavefunction $\Psi$ as a product wavefunction- as a product of the electronic ($\phi$) and vibrational modes' ($\zeta$) wavefunctions. We write $\Psi$= $\phi \zeta$ ?
Now my main question:
Is the Born-Oppenheimer Approximation technique equivalent to saying that the Hilbert Space representation of the space that $\Psi$ is situated in is the tensor product space of the electronic and vibrational mode wavefunctions?