Symmetric Potential: Reasons for Eigenstate Solutions

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SUMMARY

The discussion centers on the relationship between eigenstate solutions of the Schrödinger Equation (SE) and symmetric potentials. It is established that the Hamiltonian and the space reflection operator commute, which leads to the conclusion that they share common eigenstates that are either symmetric or antisymmetric. However, the conversation raises a critical question regarding constant potentials, which, despite being symmetric, yield plane wave solutions that do not conform to this symmetry. This highlights the distinction between bound and unbound states in quantum mechanics.

PREREQUISITES
  • Understanding of the Schrödinger Equation (SE)
  • Familiarity with quantum mechanics concepts such as eigenstates and Hamiltonians
  • Knowledge of symmetry operations in quantum systems
  • Basic principles of potential energy in quantum mechanics
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  • Explore the implications of the commutation of operators in quantum mechanics
  • Study the characteristics of bound versus unbound states in quantum systems
  • Investigate the role of constant potentials in quantum mechanics
  • Learn about the mathematical formulation of plane wave solutions in quantum mechanics
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Students and professionals in quantum mechanics, physicists studying wave functions, and anyone interested in the properties of eigenstates in symmetric potentials.

jostpuur
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I never learned this in the lectures (maybe I was sleeping), but now I think I finally realized what is the reason that eigenstate solutions of SE with a symmetric potential are either symmetric or antisymmetric. Is the argument this:

"The Hamiltonian and the space reflection operator commute, therefore they have common eigenstates" ?

If it is this, can somebody explain me why does a constant potential (which is also symmetric) have plane wave solutions, that are not symmetric or antisymmetric.
 
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Any quantum particle in a constant potential is in an un-bound state, i.e. a plain plane wave.
 

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