Suppose you have an electron in the infinite square well

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Discussion Overview

The discussion revolves around the behavior of an electron in an infinite square well, particularly focusing on the nature of its wave function and whether it can be considered an eigenstate of the Hamiltonian. Participants explore the implications of initial conditions, external potentials, and the assumptions made in practical calculations involving electrons.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions whether the wave function of an electron in an infinite square well is necessarily an eigenstate of the Hamiltonian, given that it could be a linear combination of eigenstates due to external influences.
  • Another participant counters that practical calculations do not always assume electrons are in eigenstates, citing recent Nobel prize-winning work as an example.
  • A further contribution emphasizes that Hamiltonian eigenstates lack physical time dependence, suggesting that if systems were always in such states, no dynamics would occur.
  • One participant agrees that in the context of oscillations in solids, it is common to assume electrons are in eigenstates of the Hamiltonian as an equilibrium assumption, but raises a question about the behavior of an electron when an external potential is suddenly turned off.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding eigenstates in practical calculations, indicating that multiple competing perspectives remain unresolved.

Contextual Notes

The discussion highlights the dependence on initial conditions and external potentials, as well as the implications of assuming eigenstates in various contexts, without resolving the underlying assumptions or mathematical details.

aaaa202
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Suppose you have an electron in the infinite square well. The system is completely isolated from the rest of the world and has been its entire lifetime.
Do we then know that the wave function describing the electron is an eigenstate of the Hamiltonian? The question arose because I was given a problem where the intial wave function was a linear combination of eigenstates, which I guess is perfectly possible. But then I thought: Did God create this state or was it created because we had some potential from outside which put it there and was then turned off.
It seems that for all practical calculations you always assume that the electrons are described by their corresponding eigenstates (free electron gas etc. etc.)
 
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aaaa202 said:
It seems that for all practical calculations you always assume that the electrons are described by their corresponding eigenstates (free electron gas etc. etc.)
Certainly not. The new Nobel prize in chemistry is a good counterexample for relevant work which is not based on this assumption.
 
aaaa202 said:
Suppose you have an electron in the infinite square well. The system is completely isolated from the rest of the world and has been its entire lifetime.
Do we then know that the wave function describing the electron is an eigenstate of the Hamiltonian? The question arose because I was given a problem where the intial wave function was a linear combination of eigenstates, which I guess is perfectly possible. But then I thought: Did God create this state or was it created because we had some potential from outside which put it there and was then turned off.
It seems that for all practical calculations you always assume that the electrons are described by their corresponding eigenstates (free electron gas etc. etc.)

Hamiltonian eigenstates have no physical time dependence whatsoever (there's a time dependent phase, but cancels out of all physical calculations). If everything were always in such an eigenstate, nothing would ever happen. So it's a good thing that's not the case!
 
But do you agree with me that for oscillations in solids, free electrons in solid you generally take the approach that the electrons are in eigenstates of the Hamiltonian for a start - it's sort of an equilibrium assumption.
Actually: Suppose an electron is in an eigenstate of some weird potential and you suddenly turn the potential off. Will the electron state then seek towards an eigenstate of the free electrons.
 

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