High energy electron in very deep potential well

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Discussion Overview

The discussion revolves around the behavior of a high energy electron in a deep potential well, focusing on the conditions under which the electron remains bound and the implications of its energy relative to the well's boundaries. The scope includes theoretical considerations of quantum mechanics and the nature of eigenstates in relation to potential wells.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that as long as the electron's energy is lower than the well, it remains bound, even if it is considered "high energy" in the context of an infinite potential well.
  • There is a question regarding the behavior of an electron with energy above the highest bound-state energy but below the top of the well, with some participants asserting that such an electron cannot exist in an eigenstate of the Hamiltonian.
  • One participant proposes that an electron in this energy range might exist in a superposition of bound and scattering states, leading to an expectation value for energy that does not correspond to an eigenstate.
  • Another participant seeks clarification on why an electron with energy above the bound state lacks an eigenstate and questions how it can possess an energy expectation value if no such eigenstate exists.
  • It is noted that to have a definite energy, a state must be an eigenstate of the Hamiltonian, but superpositions of different eigenstates can yield an expectation value that is not an eigenstate energy.

Areas of Agreement / Disagreement

Participants express differing views on the existence of eigenstates for electrons with energies above the highest bound state but below the well's top. The discussion remains unresolved regarding the implications of these energy states and the nature of superpositions.

Contextual Notes

The discussion highlights limitations in understanding the relationship between energy eigenstates and expectation values, particularly in the context of potential wells and superposition states. Assumptions regarding the definitions of bound and scattering states are also present but not fully explored.

blackwater
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Why should a high energy electron have to remain in a deep potential well?
 
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As long as the electron energy is lower than the well, the electron will remain bound. If you are thinking of the infinite potential well, "high energy" is still less than infinity.
 
Orodruin said:
As long as the electron energy is lower than the well, the electron will remain bound.

What if the electron has an energy above the highest bound-state energy but below the top of the well?
 
king vitamin said:
What if the electron has an energy above the highest bound-state energy but below the top of the well?
It cannot. There is no such eigenstate of the Hamiltonian. It might have such an expectation value for the energy. Then it needs to be in a superposition of bound and scattering states.
 
Why an electron having energy above bound state but below top of the well will have no eigenstate of the Hamiltonian? If it has no such energy eigenstate how can it have energy expectation value?

Please Explain.
 
In order to have a definite energy, a state needs to be an eigenstate of the Hamiltonian. There are states which are superpositions of different eigenstates. These states have some probabilities of being in the different eigenstates, meaning that the expectation of the energy does not have to be an energy of an eigenstate.
 

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