High energy electron in very deep potential well

In summary: Therefore, an electron with an energy above the highest bound-state energy but below the top of the well cannot have a definite energy state and will be in a superposition of bound and scattering states. In summary, an electron with high energy can only remain in a deep potential well if its energy is lower than the well. If its energy is above the highest bound-state energy but below the top of the well, it cannot have a definite energy state and will be in a superposition of bound and scattering states. This is because in order for a state to have a definite energy, it needs to be an eigenstate of the Hamiltonian, and in this case, there is no such eigenstate.
  • #1
blackwater
3
0
Why should a high energy electron have to remain in a deep potential well?
 
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  • #2
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.
 
  • #3
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?
 
  • #4
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.
 
  • #5
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.
 
  • #6
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.
 

Related to High energy electron in very deep potential well

1. What is a "high energy electron"?

A high energy electron is an electron that possesses a large amount of energy, typically measured in electron volts (eV). This energy can be imparted to the electron through various means such as acceleration or absorption of photons.

2. What is a "very deep potential well"?

A very deep potential well is a region in space where the potential energy is significantly lower than the surrounding areas. In the context of an electron, this refers to a region where the electron experiences a strong attractive force towards the center, resulting in a deep potential well.

3. How does a high energy electron behave in a very deep potential well?

In a very deep potential well, a high energy electron will experience a strong attractive force towards the center, causing it to become trapped in the well. This results in the electron's energy levels becoming quantized and its wave function becoming localized within the well.

4. What are the applications of studying high energy electrons in very deep potential wells?

Studying high energy electrons in very deep potential wells can provide valuable insights into the behavior of particles in extreme conditions, such as those found in the early universe or in certain types of solid-state devices. This research can also have practical applications in fields such as quantum computing and energy production.

5. How does the depth of the potential well affect the behavior of high energy electrons?

The depth of the potential well has a direct impact on the behavior of high energy electrons. A deeper potential well will result in a stronger attractive force, causing the electron to become more tightly bound and its energy levels to become more closely spaced. This can also affect the probability of the electron tunneling out of the well, which is an important phenomenon in quantum mechanics.

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