Momentum to Energy Representation in 3D Box

[mkrems]

Consider a 3D particle in a square box. One can represent a complete set of quantum states by indexing them with their momentum component quantum numbers. So a state would be |p_x p_y p_z>. If one goes to the energy basis, two momentum states (say |1 0 0> and |0 1 0>) will correspond to the same energy. Does this mean the energy eigenstates are not a complete set due to energy degeneracies? Is there a loss if information here?

 

[reilly]

First, Fouriers theorem says you've got a complete set. There are many situations in which a system has energy degeneracy -- like a gas with identical molecules, or a free fermion particle for which the energy is independent of spin.

Regards,
Reilly Atkinson

 

[denian]

Yes, the energy eigenstates in this representation are not a complete set due to energy degeneracies. This means that there are multiple momentum states that correspond to the same energy, and therefore, the energy eigenstates cannot uniquely describe the system. This does result in a loss of information, as the momentum states contain additional information about the system that is not captured by the energy eigenstates alone. However, this representation can still be useful in certain cases, such as when studying systems with symmetries that lead to degeneracies in the energy spectrum.