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Time indep schrodinger equation

  • Thread starter jaredmt
  • Start date
  • #1
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Homework Statement


Show by direct substitution into the time-independent Schrodinger equation that such a particle could be described by the state function
Ψ(x) = Ψ_0exp(-ax^2) with a, Ψ_0 constants



Homework Equations


V(x) = (1/2)kx^2

time independant equation:
6cf16270fbb803cb3cfdff9b9e65a9aa.png




The Attempt at a Solution



i found d2Ψ/dx2 = Ψ(4a^2x^2-2a)

then i plugged that and V(x) into the formula and get:
(-h^2/2m)[4a^2x^2 - 2a]Ψ + Ψ(1/2)kx^2 = EΨ

then divided Ψ out and multiplied out the left side and got:
(-2h^2a^2x^2/m) +ah^2/m + (1/2)kx^2 = E

now im not sure where to go from here. does anybody know what to do or a website that tells how to do this proof?
 

Answers and Replies

  • #2
49
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Can u use tex, pretty please?

You have a,x for the wavefunction and you want the perturbation independent "answer" provided, so what do the a and x represent, in respect of time and time/perturbation independence?
 
  • #3
diazona
Homework Helper
2,175
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then divided Ψ out and multiplied out the left side and got:
(-2h^2a^2x^2/m) +ah^2/m + (1/2)kx^2 = E

now im not sure where to go from here. does anybody know what to do or a website that tells how to do this proof?
Is this state supposed to be an energy eigenstate? If so, remember that you have to find out what the energy is that corresponds to the wavefunction, and it should be independent of x.
 

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