# Verify Ψ is solution of quantum oscillator using H operator

1. Feb 20, 2016

### ElectricEel1

1. The problem statement, all variables and given/known data
verify that Ψ(x) = ( 1/a√π)½ exp(-(x2/2a2))
is a solution to the TISE for linear harmonic oscillator. Where a = √(hbar/mw). and V(x) = ½ mw2x2.

2. Relevant equations
HΨ=EΨ
E_n = (n+½)hbar*w

3. The attempt at a solution

I've started by differentiating the wave function twice to find http://www5a.wolframalpha.com/Calculate/MSP/MSP2111e793aff6db9408500002d959h09e9180hfg?MSPStoreType=image/gif&s=28 [Broken].
Then when I apply the Hamiltonian operator to the wave function I cant see a way this would cancel down to (n+½)hbar*w. Have I gone about this the right way at all?

Thanks

Last edited by a moderator: May 7, 2017
2. Feb 20, 2016

### Dr Transport

Think about what $n$ is for that wave function

3. Feb 20, 2016

### ElectricEel1

pretty sure n=0, ive seen the full version of the wave function i have before and n=0 would make it cancel to what I have now but im still a little lost

4. Feb 20, 2016

### Dr Transport

if that is the $n$ for that wave function, then that is the equation you must solve....

5. Feb 20, 2016

### ElectricEel1

so E should equal 1/2 hbar*w but I cant find a way to make the Schrodinger equation I've written to cancel to that. Maybe I'm not understanding

6. Feb 20, 2016

### Ray Vickson

Show your complete, final, expression for $H \psi$, so we can tell if you have made an algebraic error or not. Right now, we cannot say where your problem lies.

Last edited by a moderator: May 7, 2017
7. Feb 20, 2016

### ElectricEel1

hey. since I last posted I think i got a solution.
my expression was
H = -hbar^2/2m d^2/dx^2 + 1/2 mw^2x^2

I took the second derivative of psi and wrote it as (x^2/a^4 - 1/a^2)*psi and then moved everything onto the left hand side with

hbar^2/2m (x^2/a^4 - 1/a^2)psi + (E-1/2mw^2x^2)psi=0

then collected x^2 terms and constant terms and found

E - hbar^2/2ma^2 = 0
so substituting a back into the equation it reduced down to (hbar*w)/2

8. Feb 21, 2016

### ElectricEel1

So I had to do the same problem but this time with

2xe*e^((-x^2)/(2a^2))

This time when I collected terms I ended with only one x^3,x^2,x,constant term each. Am I right in thinking this wave function is not a solution then? To the schroedinger equation and linear harmonic oscillator

9. Feb 21, 2016

### Ray Vickson

What, exactly, is your final expression for $H \psi$? Write out all the details!