Quantum anharmonic oscillator

[hilbert2]

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.

 

[DrClaude]

I don't think there are any simple creation and annihilation operators for an anharmonic oscillator. See

L. Friedlander, Creation and annihilation operators for anharmonic oscillators, Proc. Amer. Math. Soc. 113, 425 (1991)

and

http://www.physics.buffalo.edu/gonsalves/aqm/lectures/07/lec-07.pdf

 

[hilbert2]

Thanks for the links.

 

[stevendaryl]

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 $J$ as follows:
$J = \frac{1}{2 \pi}\int \sqrt{2m(E - V(x))} dx$

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

$\omega = \dfrac{\partial E}{\partial J}$

Finally, you create a new angle variable $\theta$ to be just

$\theta = \omega t$

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

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

$J = n \hbar$

Since $E$ can be computed from $J$, this gives you an indirect quantization of $E$.

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

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

$a^\dagger = \sqrt{J} e^{i \theta}$
$a = e^{-i \theta} \sqrt{J}$

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

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