Consider the usual 1D quantum harmonic oscillator with the typical hamiltonian in P and X and with the usual ladder operators defined.(adsbygoogle = window.adsbygoogle || []).push({});

i) I have to prove that given a generic wave function [itex] \psi [/itex], [itex] \partial_t < \psi (t) |a| \psi (t)> [/itex] is proportional to [itex]< \psi (t) | a | \psi (t) >[/itex] and determine their ratio.

Here I tried to express [itex] \psi(t) [/itex] as an infinite sum of the eigenstates with time evolution operator applied to it but I think there must definitely be some less clumsy way.

ii) Construct an operator [itex] Q_k [/itex] such that it only allows transitions between the states [itex] |n> [/itex] and [itex] |n \pm k > [/itex] ([itex] |n> [/itex] being the nth eigenstate of the N operator).

In this question I really did not get why the answer couldn't just be [itex] a [/itex] or [itex] a^_\dagger [/itex].

Thanks in advance

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# Quantum Harmonic oscillator

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