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I know base kets in the Heisenberg picture are $U^\dagger |{a}\rangle$ but if the base kets are the base of the hamiltonian, and the hamiltonian is independent of time, are all of the base kets stationary as well?

Specifically my question is: Consider a particle subject to the simple harmonic oscillator potential with state $\exp[\frac{-ipa}{\hbar}]|0\rangle$, what is the probability that this state is found in the ground state at time t? I think it is independent of time, but I don't know a completely solid line of reasoning to say this is so, because I am confused as to what is a base ket and what is a state ket here, and why I can just say that it is time independent.

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    $\begingroup$ In the Heisenberg picture, all states are time-independent. $\endgroup$ – Danu Sep 29 '13 at 22:20
  • $\begingroup$ Base states, or base kets, are not time-independent. $\endgroup$ – walczyk Sep 29 '13 at 23:13
  • $\begingroup$ To clarify: in $\exp(-ipa/\hbar)$ is $p$ supposed to be a c-number or the momentum operator? $\endgroup$ – Michael Brown Sep 30 '13 at 0:59
  • $\begingroup$ momentum operator, as a whole you can think of the exponential as a displacement operator. $\endgroup$ – walczyk Sep 30 '13 at 1:18

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