Proving Fermion Occupation with Commutation Relations

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In summary, using the commutation relations for bosons and fermions, it can be proven that the number operator yields the values 0 and 1 for fermions, and any non-negative values for bosons. This is because, in the case of fermions, applying the number operator on a state with n fermions leads to an absurd eigenvalue of (-n), showing that the maximum number of fermions in a single state is 1. However, for bosons, there is no restriction on the number of particles in a single state, leading to any non-negative eigenvalues.
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Homework Statement

prove , using appropriate commutation relations that the number operator yields the values 0 and 1 for fermions , and any non - negative values for bosons.



Homework Equations

the commutation relations for bosons and fermions.



The Attempt at a Solution

the boson case is solved.
the fermions-i can understand this thing intuitively ,because on applying the number operator on a state containing n fermions in a single state , and then using commutation relation i am getting (-n) as eigenvalue which is absurd. but i do not know how to give a more compact proof.
 
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The problem is not precisely stated. The number operator for fermions can as well have eigenvalue 100 - can't you have 100 of electrons? Of course you can. Probably what you mean is the number of fermions in a given state. In that case you indeed use anticommutation relations, but you end up with the eigenstate being equal to its minus, not with the eigenvalue equal to its minus!
 
  • #3
yes. i meant single state.
i have understood my mistake. thanks.
 

FAQ: Proving Fermion Occupation with Commutation Relations

1. What is "Proving Fermion Occupation with Commutation Relations"?

"Proving Fermion Occupation with Commutation Relations" is a mathematical method used to determine the occupation of fermions, which are particles with half-integer spin, in a given system. This method involves using commutation relations, which describe how operators representing physical quantities interact with each other.

2. Why is proving fermion occupation important?

Understanding the occupation of fermions in a system is crucial for understanding the behavior of many physical systems, such as atoms, molecules, and solid materials. It allows scientists to make predictions and develop theories about the properties and behavior of these systems.

3. How does proving fermion occupation with commutation relations work?

This method involves using the commutation relations of fermion creation and annihilation operators to determine the occupation number of each energy level in a system. By solving a system of equations, the occupation of each energy level can be determined.

4. What are the challenges of using this method?

One of the main challenges of using this method is the complexity of the equations involved. It requires a strong understanding of quantum mechanics and mathematical skills to be able to solve the equations and interpret the results accurately.

5. Are there any real-world applications of proving fermion occupation?

Yes, there are many real-world applications of this method, particularly in the fields of condensed matter physics and quantum chemistry. It can be used to study the properties of materials and molecules, as well as to investigate the behavior of systems at the atomic and subatomic level.

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