Lecture (xx minutes)

Slides: 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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