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It says in Griffith's chapter 2.1, that:

$$\tag{2.14} \Psi(x,t)~=~\sum_{n=1}^{\infty}c_n\,\psi_n(x) e^{(-iE_n t/\hbar)}$$ It so happens that every solution to the (time-dependent) Schrodinger equation can be written in this form [...].

By "solution" does he mean solutions on separable form

$\tag{2.1} \Psi(x,t)~=~\psi(x)f(t),$

which he stated to begin with?

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  • $\begingroup$ 2.14 is pretty much equivalent to saying that every vector can be written as a linear combination of a set of basis vectors. $\endgroup$ – Joren Jun 7 '13 at 13:46
  • $\begingroup$ @Joren: It looks similar, but it is not the same. When you represent an exact eigenfunction $\psi_k$ of the total Hamiltonian $\hat{H}$ as a linear combination of the eigenstates of a non perturbed Hamiltonian $\hat{H}_0$ like $\psi_k=\sum c_{kn}\psi_n^{(0)}$, in experiment you will never find any state with energy $E_n^{(0)}$. $\endgroup$ – Vladimir Kalitvianski Jun 7 '13 at 21:42
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No, by a solution he means any solution, i.e. a function $\Psi(x,t)$ that satisfies Schrödinger's equation and that will typically not be writable in the $a(x)b(t)$ separated form. The claim that any solution may be written in the summation form you quoted is the claim that the solutions in the separated form are "sufficient" because the most general solution may be written as their superposition.

There isn't really any ambiguity in the text.

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  • $\begingroup$ @user25504: A general solution of this type does not have a certain energy. In a measurement process one observes different eigenstates with probabilities $|c_n|^2$, not one eigenstate. Certain and conserved is the mean energy of the state $\bar{E}=\langle \Psi|\hat{H}|\Psi\rangle$ (if the Hamiltonian is time-independent, of course). $\endgroup$ – Vladimir Kalitvianski Jun 7 '13 at 12:30
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That wave function is a series

$$\Psi(x,t)=\sum_{n}c_{n}\psi_{n}(x)e^{-iE_{n}t/\hbar} $$

every term (of index, say $j$) looks like a separable solution

$$\Psi_{j}(x,t)=\psi_{j}(x)e^{-iE_{j}t/\hbar}=\psi(x)f(t)$$

and because the equation is linear, if $\Psi_{j}$ is a solution of the time dependent Schrödinger equation, then

$$\Psi (x,t)=\sum_{n}c_{n}\Psi_{n}(x,t) $$

is the most general solution.

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