Suppose you have an electron in the infinite square well

In summary, the conversation discusses the concept of Hamiltonian eigenstates and their relevance in practical calculations involving electrons. The question arises whether the initial wave function of an electron in an isolated system is an eigenstate of the Hamiltonian, and if not, what factors influence the creation of this state. It is mentioned that for practical calculations, it is often assumed that electrons are described by their corresponding eigenstates. However, it is also noted that this assumption is not always true, as evidenced by recent work in chemistry. The conversation also touches on the idea of time dependence in Hamiltonian eigenstates and how they relate to oscillations in solids.
  • #1
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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  • #2
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.
 
  • #3
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!
 
  • #4
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.
 

FAQ: Suppose you have an electron in the infinite square well

1. What is an infinite square well and how does it relate to electrons?

An infinite square well is a theoretical model used in quantum mechanics to represent a confined space where a particle, such as an electron, can move freely within. It is often used to demonstrate the behavior of particles in a potential well and their quantized energy levels.

2. How does the position of the electron change in an infinite square well?

In an infinite square well, the position of the electron is restricted to a specific region within the well. As the electron moves, it can only occupy discrete energy levels, meaning it can only exist at certain points within the well. This is in contrast to classical mechanics, where a particle can exist at any point within a given space.

3. What is the significance of the infinite square well in understanding quantum mechanics?

The infinite square well is a key concept in quantum mechanics as it helps to illustrate the principles of quantization and the discrete nature of energy levels. It also serves as a simple yet powerful model to explain the behavior of particles in confined spaces, which is essential in many areas of physics and technology.

4. How does the energy of the electron change in an infinite square well?

In an infinite square well, the energy of the electron is quantized, meaning it can only have certain discrete values. As the electron moves, it can only transition between these energy levels, and cannot have any energy values in between. This is an important aspect of quantum mechanics as it differs from classical mechanics, where energy can exist at any value.

5. Can the infinite square well model be applied to other particles besides electrons?

Yes, the infinite square well model can be applied to other particles, such as protons or neutrons, as long as they are confined to a potential well. However, the specific energy levels and behavior may vary depending on the particle's mass and the dimensions of the well.

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