Why Does Transition Amplitude Not Equate to Time Evolution in Quantum Mechanics?

[HomogenousCow]

I cannot see why the probaability ampltiude for an initial state to turn into another state is the inner product between the "another state" and the propogator acting on the initial state, since this is just equivalent to the inner product between the "another state" and the evolved initial state at the time of the evaluation.
For example, if I try it with the initial state as an energy eigenstate |1> and want to know the probability of it turning into a state |1>+|2> ( normalization constants implied), I would get a non zero result, but then that must be wrong because my initial state had zero probability of being in |2>. Thus my initial state should never be able to turn into |1>+|2>.
I can only see this working with eigenstates but not general ones, what am I missing?

 

[Sonderval]

When you do the measurement, of course you can get to a new state that is different from teh one you started with. When you start with a spin in +z direction and ask for the amplitude to measure a spin in +x-direction, the probability will be non-zero, eben though +x can be wirtten as a linear combination of +z and -z.

 

[HomogenousCow]

but then how does it reconcile my example, an energy eigen state cannot evolve into a superposition of other eigenstates no matter how much time has passed

 

[USeptim]

As far as I knwo, there is a transition amplitude when there is a perturbation hamiltonian H1 so that <2|H1|1> <> 0. There are other facts as the spontanous emision that can make a state to fall without need of H1 into another of less energy. Anyway the main idea is that an external perturbation can cause a change from a state to another.

 

[HomogenousCow]

not talking about that here, bump need exlplanation

 

[andrien]

there is a simple explanation.if it is in state|1> then (with normalization factor of 1/√2),it has 50% probability to turn into the other one where |1> shares of course 50% part.

 

[Sonderval]

The transition amplitude is not the time evolution. It gives you the probability of finding the system in the new state when actually doing a measurement. Since this will force the system to a new state, the measurement is a physical interaction with the system.