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More ladder operators

  1. Mar 20, 2008 #1
    Is there a simple expression for the ladder operators, in terms of x and [itex]-i\hbar\partial_x[/itex], for the infinite potential well? After some attempts, I couldn't figure out any nice operators that would map functions like this

    [tex]
    \sin\frac{\pi n x}{L} \mapsto \sin\frac{\pi(n\pm 1)x}{L}.
    [/tex]
     
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  3. Mar 20, 2008 #2

    Hurkyl

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    What's wrong with just using this to define a pair of operators? Such sinusoids form a basis for the state space, don't they? Is expressing it in terms of those operators a necessity?


    If you really need to write things in terms of those operators, it seems straightforward by using the angle addition formulas for trigonometric functions. You'll probably have to use operators like [itex]\sin X[/itex] and [itex]\cos X[/itex], though.
     
    Last edited: Mar 20, 2008
  4. Mar 20, 2008 #3
    I'm not sure.

    Yes.

    I'm not sure.

    Better to know, than to not know! If some nice form for the operators exists, I have no intention to ignore it.

    It didn't seem straightforward when I tried it.

    [tex]
    \exp\frac{i\pi n x}{L}\; -\; \exp\frac{-i\pi nx}{L} \quad\mapsto\quad
    \exp\frac{i\pi n x}{L}\exp\frac{i\pi x}{L} \;-\; \exp\frac{-i\pi nx}{L}\exp\frac{-i\pi x}{L}
    [/tex]

    Multiplying by some function doesn't do this. There is always some problems with cross terms. Derivatives seem problematic, because they bring [itex]\propto n[/itex] factors down from the exponents.
     
  5. Mar 20, 2008 #4

    Hurkyl

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    Incidentally, does [itex]\partial_x[/itex] even count as an operator here? When applied to a state, it usually produces something that isn't a state!


    Well, by far the nicest form of these operators is given by the expression you wrote! Let [itex]\psi_n[/itex] denote those wavefunctions, and write everything in that basis. Then, the raising operator is simply given by

    [tex]A \psi_n := \psi_{n+1}[/tex]
     
  6. Mar 20, 2008 #5
    It seems natural to work in the space spanned by

    [tex]
    \exp\frac{i\pi kx}{L},\quad k\in\mathbb{Z}
    [/tex]

    and consider the physical state space as a subspace. It is allowable to go outside temporarily.

    I didn't take a closer look at the infinite matrices yet. It could be I'm returning to this later or soon.

    hmhmh.... or is it allowable? :biggrin: I'm not sure really... The insistence on writing everything in terms of x and p is getting strange of course...
     
    Last edited: Mar 20, 2008
  7. Mar 20, 2008 #6

    Hurkyl

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    I want to say that you can write the "Number" operator as

    [tex]N = \sqrt{ \partial_x^2 }[/tex]

    where the square root one is the one induced by the real number square root. (And this would solve your problems with n appearing) Unfortunately, the reasonability of this expression is on the boundary between my knowledge and my optimism. :frown:
     
  8. Mar 20, 2008 #7

    reilly

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    The sines are complete, in configuration space. Further they are eigenstates of the free Hamiltonian. When in doubt, brute force is sometime a good way to go. Lets call one of these eigenstates | S,n> so that <x|S, n> =sin (n pi x/L). Your ladder operator for going up then becomes

    Sum over n{|S,n+1><S,n|} -- i've neglected constants

    A similar expression holds for the n-> n-1 operator, which is adjoint to the above operator..

    You also might be able to do this by converting to an oscillator basis; there's a standard generating function for Hermite polynomials that might do the trick.

    Regards,
    Reilly Atkinson
     
  9. Mar 20, 2008 #8

    reilly

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    Hurykyl's Number Operator

    Your operator shows up, for example, in classical E&M. The standard approach is to go to a Fourier representation, and end up with deleta functions and principle parts. Discussed in Mandel and Wolf, Optical Coherence ...., page 223.
    Regards,
    Reilly Atkinson
     
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