# Quantum anharmonic oscillator

• hilbert2

#### hilbert2

Gold Member
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.

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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 $J$ 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.