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Degenerate Perturbation Theory Question

  1. Mar 8, 2008 #1

    T-7

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    Hello,

    This is a question on perturbation theory - which I am trying to apply to the following example.

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

    The two-dimensional infinitely deep square well (with sides at x=0,a; y=0,a) is perturbed by the potential V(x)=[tex]\alpha(x^{2}+y^{2})[/tex]. What is the first-order correction to the energy of the first excited states (that is, those with quantum numbers (1,2), (2,1))?

    3. The attempt at a solution

    I have calculated [tex]\langle\left\right 1,2^{(0)}|\widehat{V}|2,1^{(0)} \rangle[/tex]. In Maple this is:

    > value( Doubleint( (2/a)*sin(Pi*x/a)*sin(2*Pi*y/a)*(x^2+y^2)*(2/a)*sin(2*Pi*x/a)*sin(Pi*y/a) ,x=0...a,y=0...a) );

    which is zero.

    I have calculated

    [tex]\langle\left\right 1,2^{(0)}|\widehat{V}|1,2^{(0)} \rangle[/tex]. In Maple this is:

    value( alpha*Doubleint(4*sin(Pi*x/a)*sin(2*Pi*y/a)*(x^2+y^2)*sin(Pi*x/a)*sin(2*Pi*y/a),x=0..a,y=0..a) )

    which is

    [tex]\alpha a^{2}\left( \frac{16\pi^{2}-15}{24 \pi^{2}} \right)[/tex]

    Am I to conclude from this that the perturbation, in *this* case, does not lift the degeneracy, and that the first order correction to the energy is the same for both states, and is given above?

    I had expected to form a matrix, work out its eigenvalues and eigenvectors, use that to construct a new basis etc. But the matrix I would form from the above would already be diagonalised, with a repeated eigenvalue of 1.

    I'd appreciate some speedy words of wisdom :-)

    Thanks folks
     
  2. jcsd
  3. Mar 8, 2008 #2

    pam

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    I don't see how your <12|V|21> integral could be zero.
     
  4. Mar 8, 2008 #3

    kdv

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    Why not?

    The integral of sin(x) sin(2x) x^2 is zero since it's odd with respect to the center of the well and the integral sin(x) sin(2x) y^2 is obviously zero.

    EDIT: Oops, I was wrong on this since I was thinking about a well going from -a/2 to a/2,
     
    Last edited: Mar 9, 2008
  5. Mar 8, 2008 #4

    kdv

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    It does seem to me that you are right. Here the degeneracy is not lifted.
     
  6. Mar 9, 2008 #5

    pam

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    Did you leave out the x^2+y^2 in the integrals from 0 to a?
    If you are integrating from -a/2 to +a/2, then change your question.
     
  7. Mar 9, 2008 #6

    T-7

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    Hi,

    I have done the integral on paper and on Maple. On Maple I put:

    > value( Doubleint( (2/a)*sin(Pi*x/a)*sin(2*Pi*y/a)*(x^2+y^2)*(2/a)*sin(2*Pi*x/a)*sin(Pi*y/a) ,x=0...a,y=0...a) );

    The (x^2+y^2) is there, as you can see. The result is zero (which is what I also get using paper and pen).

    Since the well is defined from 0 to a, I have used those as the limits of the integrals.
     
  8. Mar 9, 2008 #7

    olgranpappy

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    Homework Helper

    |1> and |2> are orthogonal. So,

    <1,2|x^2 + y^2 |2,1>

    =<1|x^2|2><2|1> + <1|2><2|y^2|1>

    = 0 + 0

    = 0

    OP: It's quite reasonable that the potential doesn't lift the degeneracy since the unperturbed potential (the length of the box) is the same in x and y and the perturbation also treats x and y on equal footing.
     
  9. Mar 9, 2008 #8

    pam

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    That's right! Sorry to have said the opposite. I didn't notice the x^2 did not enter the y integral.
     
  10. Mar 9, 2008 #9

    kdv

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    Ah yes..Ok, my mistake too. Of course, the x integral does not give zero but the y part will give zero so the whole thing is zero. My mistake too...
     
  11. Mar 10, 2008 #10

    T-7

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    Thanks for your thoughts folks. Much appreciated. :-)
     
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