Finding a constant within a wavefuntion for a harmonic oscillator

[Chronos000]

Homework Statement

The question states for a harmonic oscillator the wavefunction is:

$$\mu$$ = C*x*exp(-$$\alpha$$x²/2)

it then wants you to find $$\alpha$$.

using the standard hamiltonian:

H = -$$\hbar$$/2m d²/dx² + 1/2 mw²x²

I have differentiated $$\mu$$ twice and put it into the TISE.

for the left hand side of the TISE I have

3$$\alpha$$$$\hbar$$²/2m $$\mu$$ + $$\mu$$x² [ mw²/2 - $$\hbar$$²$$\alpha$$²/2m]

I have been given a hint that the 2nd term goes equals zero but I'm not entirely sure why.
Could it be something to do with the eigenvalues having no dependence on x so this term must cancel?

 

[ideasrule]

I have been given a hint that the 2nd term goes equals zero but I'm not entirely sure why.
Could it be something to do with the eigenvalues having no dependence on x so this term must cancel?

Yes, that's exactly the reason. Eigenvalues are, by definition, always constants. Eigenvalues of observables are always real constants. They can never depend on x or any other variable.

 

[Chronos000]

thanks for confirming