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Quantum anharmonic oscillator

  1. Jun 10, 2014 #1


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    Let's say I have an anharmonic 1D oscillator that has the hamiltonian

    ##H=\frac{p^2}{2m}+\frac{1}{2}kx^2+\lambda x^4##

    or some other hamiltonian with higher than second-order terms in the potential energy. Is it possible, in general, to find raising and lowering operators for such a system? I mean operators that commute with the hamiltonian in such a way that they transform an eigenstate of ##H## into another eigenstate that has a higher or lower eigenvalue (of course the eigenvalues will not be evenly spaced in the anharmonic situation). I remember reading an article where someone solved the Morse oscillator system with some kind of generalized raising and lowering operators, but for some reason I can't access the full text anymore.

    How would I go about constructing an example of an oscillator hamiltonian that is anharmonic and for which the creation and annihilation operators can be found?

    EDIT: Yes, of course such operators exist for any quantum system, but usually they are not a simple function of the x and p operators.
    Last edited: Jun 10, 2014
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  3. Jun 10, 2014 #2


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  4. Jun 10, 2014 #3


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    Thanks for the links.
  5. Jun 13, 2014 #4


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    Action-Angle Variables

    There is an approximate way to get raising and lowering operators for an arbitrary potential that uses the heuristic of the Bohr-Sommerfeld quantization rule. Unfortunately, it's only an approximation which is good in the semiclassical limit. I don't know how to systematically get better and better approximations.

    The Bohr-Sommerfeld quantization rule is this:

    First, using classical dynamics, you define the action variable [itex]J[/itex] as follows:
    [itex]J = \frac{1}{2 \pi}\int \sqrt{2m(E - V(x))} dx[/itex]

    This gives [itex]J[/itex] as a function of [itex]E[/itex]. If you invert this, to get [itex]E[/itex] as a fuction of [itex]J[/itex], then you can define an angular frequency [itex]\omega[/itex] via:

    [itex]\omega = \dfrac{\partial E}{\partial J}[/itex]

    Finally, you create a new angle variable [itex]\theta[/itex] to be just

    [itex]\theta = \omega t[/itex]

    The pair [itex]\theta, J[/itex] act as a generalized coordinate and corresponding momentum. In principle, you can figure out how to compute [itex]\theta, J[/itex] from the usual coordinates [itex]x, p[/itex]. The advantage to the [itex]\theta, J[/itex] description is that the dynamics are particularly simple: the energy only depends on [itex]J[/itex], and so (by the hamilton equations of motion), the time dependence of [itex]\theta[/itex] is trivial.

    To relate this to quantum mechanics, Bohr and Sommerfeld proposed the quantization rule:

    [itex]J = n \hbar[/itex]

    Since [itex]E[/itex] can be computed from [itex]J[/itex], this gives you an indirect quantization of [itex]E[/itex].

    The problem with this approach is that it's only an approximation. The rule [itex]J = n \hbar[/itex] is only good in the limit where [itex]n \gg 1[/itex]. For the harmonic oscillator, it gives [itex]E = n \hbar \omega[/itex] rather than [itex]E = (n + 1/2) \hbar \omega[/itex]

    Anyway, action-angle variables gives a heuristic starting point for the raising and lowering operators [itex]a[/itex] and [itex]a^\dagger[/itex]:

    [itex]a^\dagger = \sqrt{J} e^{i \theta}[/itex]
    [itex]a = e^{-i \theta} \sqrt{J}[/itex]

    Since [itex]J[/itex] is the momentum canonical to [itex]\theta[/itex], we have the quantization rule [itex][J, \theta] = -i \hbar[/itex]. If [itex]|n\rangle[/itex] is an eigenstate of [itex]J[/itex] with eigenvalue [itex]n \hbar[/itex], then [itex]a^\dagger |n\rangle[/itex] is an eigenstate with eigenvalue [itex](n+1) \hbar[/itex].

    So [itex]a^\dagger = \sqrt{J} e^{i \theta}[/itex] and [itex]a = e^{-i \theta} \sqrt{J}[/itex] [itex]a = e^{-i \theta} \sqrt{J}[/itex] give starting points for raising and lowering operators for an arbitrary potential. However, knowing the classical forms of [itex]J[/itex] and [itex]\theta[/itex] as functions of [itex]p[/itex] and [itex]x[/itex] doesn't uniquely determine the quantum forms, because of operator ordering ambiguity (which can be thought of as the source of the [itex]1/2 \hbar \omega[/itex] error in the harmonic oscillator case).

    Action-angle variables is a very rich subject in classical mechanics, although not a lot has been done with it in quantum mechanics, except in the old quantum theory.
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