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Quantum mechanics in DIRAC notation

  1. Nov 13, 2004 #1
    Consider a particle in a harmonic pscillator potential V (x) is given by
    [tex]V = \frac{1}{2}m\omega^2[/tex]

    Also [tex]\hat a = n^\frac{1}{2}|n-1>[/tex], and

    [tex]\hat a\dagger = (n-1)^\frac{1}{2}|n-1>[/tex]

    where

    [tex]
    \hat a = \frac{\beta}{\sqrt 2}(\hat x + \frac{i\hat p}{m\omega})
    [/tex]

    [tex]
    \hat a\dagger = \frac{\beta}{\sqrt 2}(\hat x - \frac{i\hat p}{m\omega})
    [/tex]

    Construct a matrix representation for
    [tex]\hat a[/tex]
    [tex]\hat a\dagger[/tex]
    [tex]\hat N = \hat a\hat \dagger[/tex]
    and
    [tex]\hat H[/tex]

    where[tex]\hat H[/tex] is the Hamiltonian.

    If someone can show me one and explain it to me, I will try the rest by myself before asking questions about them.

    I am really desparate :confused:

    Please help
    James
     
    Last edited: Nov 13, 2004
  2. jcsd
  3. Nov 13, 2004 #2
    Please someone....any help...just show me how to do one and I will try my best for the rest...please

    James
     
  4. Nov 13, 2004 #3
    This is the quantum harmonic oscillator where [tex]\hat a\dagger[/tex] and [tex]\hat a[/tex] are the step down and step up (SOMETIMES CALLED LADDER) operators according to your definitions.

    Correction: I think you meant to say [tex]\hat N[/tex] = [tex]\hat a\dagger \hat a[/tex]
     
    Last edited: Nov 13, 2004
  5. Nov 14, 2004 #4
    Anybody......please...something.

    james
     
  6. Nov 15, 2004 #5
    The matrix reprentation is just a matrix of a bunch of matrix elements (make sure you understand matrix elements before you do this problem

    but this matrix representation should be in a particular basis. By the way you also wrote the ladder operators incorrectly. You have an operator on the lhs and a ket vector on the right hand side, and the rasing and lowering are both minus. They should be:

    [tex]\hat{a}^{\dagger} = \sqrt{n+1}|n+1>[/tex]
    [tex]\hat{a} = \sqrt{n}|n-1>[/tex]

    But now that you have these relations you just need to find:

    [tex]<i|a|j>[/tex]

    This is in the n basis. This is a matrix element. It gives a particluar element in a matrix. As you can see i,j can go from negnative infinity to infinity, and they label your rows and columns of your matrix. But elements like [tex]<n|n+j>[/tex] ( for any j nonzero) are orthogonal so they go to zero. So you only expect states off by one to be connected. Your matrices for the a and a dagger will have one off diagonal. These are the states where the operator connects the next higher or lower state.

    the other matrix elements proceed in a similar manner but with a different basis. Try it and let me know if you have any questions.
     
  7. Nov 15, 2004 #6
    So would the one for a look like this?

    0 1 1 1 ....1
    1 0 1 1 ....1
    .
    .
    .
    1 1 1 1 ....0

    I am still confused. Could you write down one explicitly so I can get a mental image as to what I am looking for?

    Thanks,
    James
     
  8. Nov 15, 2004 #7
    I really need this urgently....sorry to be pushy
     
  9. Nov 15, 2004 #8

    ZapperZ

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    Staff Emeritus
    Science Advisor
    Education Advisor

    There appears to be a more fundamental problem here beyond just this specific problem.

    If you are given an operator A, and a set of basis functions |i>, do you know how to find the matrix A^? (A with a hat, since I'm too lazy to do LaTex formatting.)

    Zz.
     
  10. Nov 15, 2004 #9
    How can an operator be equal to a state??

    it's not a=k[n> or whatever. It's a[n>=k[n-1>, where k is the constant you wrote. Operators are not states. I think what you are trying to ask is how do we obtain algebraicaly the energy eigenkets and eigenvalues for a particle in a harmonic potencial. What you do is the following:

    You defined the N operator as N=a'a, where the useing the notation d' is the adjoint of d. Then you observe that H=N + 1/2 therefore diagonalizing N is the same as diagonalizing H i.e., solving the problem. From this, clearly the states [n> are energy eigenkets with eigenvalues n +1/2.

    Hope that helped!
     
  11. Nov 15, 2004 #10

    dextercioby

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    Homework Helper

    Good point.Probably a lack of notation,or,worse,a confusion mixed with ignorance.

    I think the original question was pretty clear.There's no need to elaborate and mislead people.He was basically interested of solving the QM simple 1D harmonic oscillator in the original matrix version of QM (Born,Heisenberg,Jordan-1925),knowing it in the more abstract and elegant version of Dirac-Von Neumann (1926-1935).

    Hints for solving the problem (original one,not one invented):
    1.Write equations correctly.E.g.
    [tex]\hat{a}|n>= \sqrt{n}|n-1>[/tex]
    [tex]\hat{a}^{\dagger}|n>= \sqrt{n+1}|n+1>[/tex]
    2.Use the fact that the standard basis is orthormal:
    [tex] <i|j> =\delta_{ij} [/tex]
    3.Apply all bra's corresponding to all kets in the standard basis on the 2 relations written above and generate operators' matrices.
    4.Use the matrices for the creation+annihilation operators deduced at 3. together with the expressions for the Hamilton and number operators to find the latters' matrices.

    Good luck!!
     
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