Verify Ψ is solution of quantum oscillator using H operator

[ElectricEel1]

Homework Statement

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.

Homework Equations

HΨ=EΨ
E_n = (n+½)hbar*w

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 .
Then when I apply the Hamiltonian operator to the wave function I can't see a way this would cancel down to (n+½)hbar*w. Have I gone about this the right way at all?

Thanks

 

[Dr Transport]

Think about what [itex] n [/itex] is for that wave function

 

[ElectricEel1]

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

 

[Dr Transport]

if that is the [itex] n [/itex] for that wave function, then that is the equation you must solve...

 

[ElectricEel1]

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

 

[Ray Vickson]

Homework Statement

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.

Homework Equations

HΨ=EΨ
E_n = (n+½)hbar*w

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 .
Then when I apply the Hamiltonian operator to the wave function I can't see a way this would cancel down to (n+½)hbar*w. Have I gone about this the right way at all?

Thanks

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.

 

[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

 

[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

 

[Ray Vickson]

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

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