Thanks everyone, that's explained the error quite well.
I think the intent of the "proof" in the context of that lecture was to demonstrate that stationary states are not physically realizable, because in that state nothing ever changes (all observables have constant expectation values). It...
The following lecture from University of Oxford contains an explanation of the constancy of probability distributions for all observables when a system is in a stationary state: http://www.youtube.com/watch?v=0yvX4jhzblY#t=15m35s. However, the derivation of the vanishing amplitude does not...