It is well-known that the angular momentum operator Lz is the generator of rotations. However, while the operator exp(-i Omega t Lz / hbar) performs a rotation of vectors R (position) and P (momentum), it doesn't transform the kinetic term to the rotating frame (because it commutes with the kinetic energy P^2/2m).
I need to find a unitary operator that transforms P^2/2m to P^2/2m-Omega.L
The operator you wrote, A=exp(-i Omega t Lz / hbar), is what you need.
Directly applying i\hbar \partial/(\partial t) to A\psi for \psi being the hamiltonian eigenstate, you'll see that its (\psi state) energy transforms correctly (=to the correct energy value in the rotating frame).
The paradox (as it appears in your question) is due to the way you wrote your kinetic energy (tacitly assuming non-rotating state) and the need to notice that P^2 from the point of view of the original frame is not the same as the P^2 in the rotating frame.
Just take a look at Landau-Lifshitz vol. 1 (Mechanics), chapter on non-inertial ref. frames. Machinery is the same here as in classical mechanics.
- Badges
- Science topic
- Similar topics
- Physical Sciences
- Atomic, Molecular and Optical Physics