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=====Lecture (xx minutes)=====
Slides: {{courses:lecture:wvlec:wvtimeevo_wiki.ppt|Time-dependent Schroedinger equation}}

This discussion goes over the solution to the TDSE,  
$\hat{H}\psi \left( x,t \right)=i\hbar \frac{\partial \psi \left( x,t \right)}{\partial t}$, as discussed in the //Spins// paradigm, but now in wave function language.  The students generally have much less recall of this particular topic than they do of others encountered in //Spins//, for example, the idea of projection (which they know well).

The important point is that the Hamiltonian operator is special, so it makes sense to write the general solution as a superposition of the (time independent) eigenfunctions of the Hamiltonian, and allow the time dependence to be in the expansion coefficients:
$\psi \left( {x,t} \right) = \sum\limits_n {c_n \left( t \right)\varphi _n \left( x \right)} $.

The mathematics falls out easily, the only mildly subtle point being to remind students that when an equation that is a sum of independent quantities is zero, the coefficients must be zero term by term.  They have encountered this idea before, but it's an important reminder.


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