2D hydrogen atom
The Hamiltonian for a non-relativistic electron (electric charge ) in a hydrogen atom is:
where is the canonical momentum to the radial distance , and is the canonical momentum of the azimuthal angle .
Take the action-angle coordinates:
For the radial coordinate :
where we are integrating between the two classical turning points ()
Using EBK quantization :
- ;\quad m=0,1,2,\cdots }
- ;\quad n_{r}=0,1,2,\cdots }
and by making the spectrum of the 2D hydrogen atom [7] is recovered :
- ;\quad n=1,2,3,\cdots }
Note that for this case almost coincides with the usual quantization of the angular momentum operator on the plane . For the 3D case, the EBK method for the total angular momentum is equivalent to the Langer correction.