Momentum to Energy Representation in 3D Box

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SUMMARY

The discussion centers on the representation of quantum states in a 3D box, specifically addressing the relationship between momentum states and energy eigenstates. Reilly Atkinson highlights that two momentum states, |1 0 0> and |0 1 0>, correspond to the same energy level, raising questions about the completeness of energy eigenstates due to energy degeneracies. The conversation references Fourier's theorem, affirming the existence of a complete set of states despite potential information loss in energy representation. The implications of energy degeneracy in systems with identical particles or free fermions are also noted.

PREREQUISITES
  • Quantum mechanics fundamentals
  • Understanding of momentum and energy eigenstates
  • Fourier's theorem in quantum systems
  • Concept of energy degeneracy in quantum systems
NEXT STEPS
  • Study the implications of energy degeneracy in quantum mechanics
  • Explore Fourier's theorem applications in quantum state representation
  • Investigate the behavior of identical particles in quantum systems
  • Learn about the role of spin in energy levels of fermions
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Students and professionals in quantum mechanics, physicists exploring particle behavior in confined systems, and researchers studying energy states and degeneracies in quantum systems.

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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?
 
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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
 

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