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?
 
149
0
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

Science Advisor
Insights Author
4,155
194
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

Science Advisor
Insights Author
4,155
194
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

Science Advisor
Insights Author
4,155
194
Also see the reference given in this earlier thread, post #2, courtesy of dextercioby.

The book itself is available online here.
 
Last edited:

Related Threads for: Creation and annihilation operators

  • Last Post
Replies
1
Views
662
  • Last Post
Replies
2
Views
1K
Replies
2
Views
5K
Replies
2
Views
7K
  • Last Post
Replies
16
Views
8K
Replies
2
Views
502
Replies
3
Views
209
Replies
2
Views
1K

Hot Threads

Top