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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.

##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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