For a set of energy eigenstates [itex]|n\rangle[/itex] then we have the energy eigenvalue equation [itex]\hat{H}|n\rangle = E_{n}|n\rangle[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

We also have a commutator equation [itex] [\hat{H}, \hat{a^\dagger}] = \hbar\omega\hat{a}^{\dagger}[/itex]

From this we have [itex]\hat{a}^{\dagger}\hat{H}|n\rangle = (\hat{H}\hat{a}^{\dagger} - \hbar\omega\hat{a}^{\dagger})|n\rangle = E_{n}\hat{a}^{\dagger}|n\rangle[/itex]

and thus [itex] \hat{H}\hat{a}^{\dagger}|n\rangle = (E_{n} + \hbar\omega)\hat{a}^{\dagger} |n\rangle [/itex]. So, comparing this to [itex]\hat{H}|n\rangle = E_{n}|n\rangle[/itex], we can see that the state [itex]\hat{a}^{\dagger}|n\rangle[/itex] has more energy by an amount [itex]\hbar\omega[/itex]. Fair enough.

My question is to do with why this operator [itex]a^{\dagger}[/itex] is interpreted as a "creation" operator. If I'm not mistaken, it's interpreted as increasing the number of particles in a given state by 1, ie. creating a particle. Why is it not interpreted as just one particle existing before and after, but that particle now has an extra [itex]\hbar\omega[/itex] of energy, and just oscillates more violently in a higher excited state?

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# Quantum harmonic oscillator, creation & annihilation operators?

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