Having trouble with time (in)dependant solutions

[ChaosCon343]

Hey all -

I'm taking my first quantum class this year, and I'm still on really shaky ground about time-dependent and time-independent solutions of the Schrödinger equation. I understand that the time independent Schrödinger equation comes from separating your space and time variables, but I have trouble interpreting the meanings of both types of solutions. Particularly, why can you reconstruct any arbitrary $$\Psi(x,t)$$ out of the time-independent solutions? Rather, why can an electron in, say, an infinite square well potential have different $$\Psi$$'s?

 

[zenith8]

Solve the time-dependent Schroedinger equation $$i\hbar\frac{\partial}{\partial t}\Psi(x,t) = H\Psi(x,t)$$ by separation of variables to give the following particular solutions (which have the counterintuitive property of predicting time-independent observables):

$$\Psi(x,t) = \phi_E(x)e^{-\frac{i}{\hbar}Et}\;\;\;\;\;\;\;\;|\Psi(x,t)|^2 = |\phi_E(x)|^2$$

Where has the time gone? It is restored to us by a general solution to the TDSE - an arbitrary superposition of the particular solutions:

$$\Psi(x,t) = \sum_{n=1}^{\infty} a_n\phi_n(x) e^{-\frac{i}{\hbar}E_nt} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\text{\mathrm{(discrete\;spectrum)}}$$

$$\Psi(x,t) = \int_0^{\infty} a(E)\phi_E(x) e^{-\frac{i}{\hbar}Et}\;dE \;\;\;\;\;\;\;\;\text{\mathrm{(continuous \; spectrum)}}$$

Quite generally, a wave packet - a superposition of states having different energies - is required in order to have a time-dependence in the probability density and in other observable quantities, such as the average position or momentum of a particle.
Simplest example: a linear combination of just two particular solutions

$$\Psi(x,t) = a\phi_E(x) e^{-\frac{i}{\hbar}Et} + b\phi_{E'}(x) e^{-\frac{i}{\hbar}E't}$$.

The probability density is given by:

$$|\Psi(x,t)|^2 = |a|^2|\phi_E(x)|^2 + |b|^2|\Psi_{E'}(x)|^2 + 2\mathrm{Re}\left\{a^*b\phi_E^*(x)\phi_{E'}(x)\mathrm{e}^{-i\frac{(E'-E)t}{\hbar}}\right\}$$

All the time-dependence is contained in the interference term.