Creation and annihilation operators

In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
 
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In one dimensional problems in QM only in case of the potential ##V(x)=\frac{m\omega^2x^2}{2}## creation and annihilation operator is defined. Why? Why we couldn't define same similar operators in cases of other potentials?
I guess mathematics wouldn't work and you would not be able to reduce those operators into entities with commutation operators leading to simple algebraic relations. Plus this harmonic potential corresponds to free fields, and this "freeness" i guess is the physical reason why things are simple and tractable and meaningful.
 

Bill_K

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Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.
 
Stepping operators can also be used to solve the Coulomb problem. Also the n-dimensional harmonic oscillator. In principle such operators always exist, but only in a few problems are they simple enough to write down in useful form.

I suppose that. That they always could be written. But why we that speak about phonons only in that problem. Why we don't give a name of excited states in some other problem.

Stepping operators can also be used to solve the Coulomb problem.
Do you have reference for this?
 

Bill_K

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The stepping operators for the H atom form the generators of an SO(4) symmetry group, and are also related to its separability in parabolic coordinates. Here's a paper that talks about it.
 

Bill_K

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Also see the reference given in this earlier thread, post #2, courtesy of dextercioby.

The book itself is available online here.
 
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