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In Sakruari's Modern Physics, it's written that


$$H(t)|n;t\rangle =E_n(t)|n;t\rangle $$ simply noting that at any particular time $t$, the states, and eigenvalues may change. If we now look for general solutions to SE of the form $$i\hbar \partial_t|\alpha;t\rangle =H(t)|\alpha;t\rangle $$

then we can write $$|\alpha;t\rangle =\sum_nc_n(t)e^{i\theta_n t}|n;t\rangle $$ where $$\theta_n(t)=-\frac{1}{\hbar}\int_0^tE_n(t')dt'$$


But consider what I have done. $$i\hbar\partial_t|n;t\rangle =H(t)|n;t\rangle =E_n(t)|n;t\rangle \rightarrow |n;t\rangle =e^{i\theta_nt}|n;0\rangle $$ $$|\alpha ;t\rangle =\sum_nc_n(t)|n;t\rangle =\sum_n c_n(t)e^{i\theta_nt}|n;0\rangle $$ This is different from what is done above. What's wrong with my thinking?

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2 Answers 2

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$$H(t)|n;t\rangle =E_n(t)|n;t\rangle \rightarrow |n;t\rangle =e^{i\theta_nt}|n;0\rangle$$

This states that the time dependence of eigenvalues of the time-dependent Hamiltonian is trivial: only phase factor changes. But this can't be right: we have no such constraints on time evolution of the Hamiltonian.

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The "evolution" of the basis states $|n;t\rangle$ is given by the requirement that it should be the $n$th eigenvalue of $H$ at time $t$. There is no requirement that this should match the evolution given by the Schrodinger equation. That is (if you will excuse the slightly butchered notation) $$ | n;0\;(t)\rangle := U(t) | n;0\rangle \ne |n;t\rangle $$ where $U(t)$ is the time evolution operator.

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