I'm trying to understand the relation between the time dependent and time independent Schrödinger equations. In particular, we know that the TDSE is
$$H\Psi=i\hbar \frac{\partial \Psi}{\partial t}$$
whereas the independent equation is the eigenvalue problem $$H\psi=E\psi$$
My main question is this: if we allow $\Psi$ to be independent of time (which is my interpretation of a 'time independent equation') then why don't we just get $H\Psi=0$? I can see where the eigenvalue problem comes from: suppose we had a separable solution to the TDSE $\Psi(x,t)=\psi(x)T(t)$. Then, $$TH\psi=i\hbar \dot{T}\psi \implies i\hbar \frac{\dot{T}}{T}=\frac{H\psi}{\psi}=E$$
For some constant $E$, so we get $T(t)=Ae^{-iEt/\hbar}$ and $H\psi=E\psi$.
This is interesting, but doesn't quite answer my question: why does the argument that $H\psi=0$ not work, and what about solutions which are not separable?
Best Answer
The "independent" in "time-independent Schrödinger equation" doesn't mean that the wavefunction $\psi(x,t)$ is independent of time, but that the quantum state it defines doesn't change with time.
Since $\psi(x)$ and $\mathrm{e}^{\mathrm{i}\phi}\psi(x)$ for any $\phi\in\mathbb{R}$ define the same quantum state, this does not imply $\partial_t\psi(x,t) = 0$. Indeed, as the solution shows, the time dependence $\mathrm{e}^{\mathrm{i}Et}$ is precisely the kind of dependence that is allowed.