Existance of ladder operators for a system

[espen180]

I have only heard about the use of ladder operators in connection with the harmonic oscillator and spin states. However, I would expect them to be useful in other systems as well.

For example, can we find ladder operators for the discrete states of the hydrogen atom, or any other system with discrete energy levels? Are there established conditions that guarantee their existence?

Thanks for any help.

 

[Simon Bridge]

You also see them as creation and annihilation operators in quantum optics.
The ladder operator makes sense where there is fixed step-size between states. You can try for a counter example by trying to derive a ladder operator for non-harmonic potentials.

 

[Bill_K]

They always exist. For any Hamiltonian with eigenstates |n> you can always define an operator, a*|n> = |n+1>. But the question is, can a* be written in a simple form.

Generally, a simple form will exist for problems with symmetry, for then the operators will yield an algebraic solution. The hydrogen atom is one example of this, see here.