Adjusting the Zero Point Energy

As students of quantum mechanics should know, the solution to the quantum harmonic oscillator Hamiltonian gives an elegant result:

En=(n+12)ωE_n = (n+\frac{1}{2})\hbar\omega

There's a constant term in the energy when nn is zero. This is called the Zero Point Energy.

A student proposes that we get rid of this ridiculous term by simply redefining the Hamiltonian as

H=p22m+mω22x212ωH=\frac{p^2}{2m}+\frac{m\omega^2}{2}x^2-\frac{1}{2}\hbar\omega

After all, we do this all the time with gravity problems simply by adjusting the reference point.

Is this student successful in removing this offset?

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