# Creation and annihilation operators

1. Mar 27, 2013

### LagrangeEuler

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?

2. Mar 27, 2013

### LayMuon

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.

3. Mar 27, 2013

### Bill_K

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.

4. Mar 27, 2013

### LagrangeEuler

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?

5. Mar 27, 2013

### Bill_K

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.

6. Mar 28, 2013

### Bill_K

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

The book itself is available online here.

Last edited: Mar 28, 2013