Quantum mechanical annihilation/creation operators?
In the lecture, you learned about the operators a † and a . Determine the time-dependent expectation values ⟨x⟩(t), ⟨p⟩(t), and ⟨x 2 ⟩(t) using these creation and annihilation operators.
The solution for ⟨x⟩(t) appears to be:
The factor 2/3 results from the pre-factor alpha.
I don't quite understand the step from the second line to the third line of the solution. Why does <Psi1|A|Psi0> equal 1 and <Psi2|A|Psi1> equal square root, and why don't the other factors from the multiplication contribute anything?
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because
a^dagger |n> = root(n+1)|n+1>
and
a|n>=root(s)|n-1>
and the energy properties form an orthonormal basis.
but a and a^dagger should be exchanged in the solution