I am given the following Hamiltonian, H=H1=p22m+12mω21x2
Now, my question is, at time t1, we have a whole list of operators which can act on the state like a1 and a†1 i.e the annihilation and creation operators and similarly at time t2 as a2 and a†2. Now, if I act a1 on |02⟩, or perform similar operations, what do I get? Will the eigenvectors of a1 also be the eigenvectors of a2?
Answer
a1 is not hermitian so be careful about its eigenvectors. Most importantly, the definition of the creation and destruction operators involves the frequency ω so the creation operator on both sides of t=0 will not be the same so the harmonic oscillator eigenstates, i.e. the number states, will not be the same. This is evident because the length scale for the number states depends also on ω.
In particular, the change of scale is equivalent to a squeezing transformation so the vacuum state for t2 will not be the vacuum state for t1. You will need to expand |02⟩ in terms of t1-states in order compute the action of a1, i.e. given λ1=√mω1ℏ ψn(x)=1√2nn!(λ21π)1/4e−λ21x2/2Hn(λ1x)
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