You are here: start » courses » lecture » wvlec » wvlectimeevo

## 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.