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2-D Harmonic Osc. with Perturbation

  1. Nov 16, 2007 #1
    2-D Harmonic Oscillator with Perturbation

    1. The problem statement, all variables and given/known data

    A 2-D harmonic oscillator has an energy Ensubxnsuby and wavefunctions [tex]\phi[/tex]nsubxnsuby

    The first excited states are 2-fold degenerate E[tex]_{01}[/tex]=E[tex]_{10}[/tex]=2[tex]\hbar[/tex][tex]\omega[/tex]

    What are the energies and wavefunctions if we add the perturbation

    H'=gwyp[tex]_{x}[/tex]?



    2. Relevant equations

    En=[tex]\hbar[/tex][tex]\omega[/tex](nx + ny + 1)

    [tex]\phi[/tex]subnxny=[tex]\phi[/tex]subnx(x)[tex]\phi[/tex]subny(y)

    H=g[tex]\omega[/tex]yPsubx

    3. The attempt at a solution

    I'm unsure exactly how to approach this problem, and I think it's a combination of not knowing exactly how to work with this perturbation and never having dealt with 2-D Harm. Osc. I suppose for the energies just a simple degenerate calculation of the three matrix elements of H' (where I come to trouble working with the ypsubx) and then plugging in for the two fold degeneracy equation. The wavefunctions would be a similar sum over H'mn m!=n. Also I know the wavefunctions (just nx and ny of 1-D 0 and 1 states multiplied together), just don't know where to go from here. Any help would be appreciated. THanks
     

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    Last edited: Nov 16, 2007
  2. jcsd
  3. Nov 16, 2007 #2

    CompuChip

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    Homework Helper

    Firstly, welcome to PF proto0.

    Secondly, a little hint to make your post more legible: I see that you use LaTeX to produce Greek letters, but the subscripts are terrible, if you don't mind me saying so. If you put the entire thing between the TeX tags (and replace the word "sub" with an underscore (_) everywhere, as you did in E[tex]_{01}[/tex] it will look much better, e.g.
    The first excited states are 2-fold degenerate [tex]E_{01} = E_{10} = 2 \hbar \omega[/tex] (note how that looks better, and if you click it you'll see how I made it).

    Having said that, I'll await the attachment to be approved and answer your question when I get back :smile:
    (Excuse me for not really answering your question right now, but I have to go now and don't want to rush into posting something.)
     
  4. Dec 24, 2007 #3
    Hi, i'm having the same question, but with a perturbation like K'xy

    Beeing H' the perturbated hamiltonian, [tex]H'=H_{0}+K'xy[/tex]

    I'm suposed to solve the secular equation right? By calculating the [tex]H_{ij}[/tex], is that the way?


    Thanks
     
    Last edited: Dec 24, 2007
  5. Dec 26, 2007 #4

    Dr Transport

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    Yes, the perturbation [itex]H'=H_{0}+K'xy [/itex] couples states of different levels, remember from c your class you should have seen [itex] <n'|x|n> ~ c_{1}\delta_{n+1,n} + c_{2}\delta_{n-1,n} [/itex]. Since your perturbation is separable, you should be able to get it quickly.
     
  6. Dec 26, 2007 #5
    Got it! Thanks a lot for your help! :tongue:
     
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