A fermion oscillator interacted with a boson oscillator

In summary, the Jaynes-Cummings model is a simple model for the interaction of a single bosonic photon mode with a single fermionic two-level system.
  • #1
dhg19861106
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It is known to all that the Hamiltonin H=p^2/m+x^2 can describe the boson and fermion particle, but how can embody the fermion properties when a fermion oscillator interacted with a boson oscillator? what is their interaction form?
 
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  • #2
Sorry, I do not see how a fermion oscillator should emerge from the boson oscillator. Can you please be a bit more specific?
 
  • #3
Thank you for your reply, I just want to build a theoretical model for a fermion interacted with a boson in dissipative system. But I don't know how to build it.
 
  • #4
You should know that the configuration space of the fermion oscillator is [itex]R^{2}_{a}[/itex]. That is, you need two real anticommuting dynamical variables to describe the fermion oscillator. The action for such oscillator is given by
[tex]S = (i/2)\int dt (X_{1}\dot{X}_{1} + X_{2}\dot{X}_{2} + X_{1}X_{2}-X_{2}X_{1})[/tex]
Form this, you find the Hamiltonian;
[tex]H = -iX_{1}X_{2}[/tex]
 
  • #5
That is all a bit vague and allows for a lot of approaches.

One of the simplest models is the Jaynes-Cummings model which is used for the interaction of a single bosonic photon mode (which is of course dissipative e.g. in the presence of a cavity) with a single fermionic two-level system (atom, quantum dot or whatever).

I do not know whether this is what you are looking for, but it might provide a starting point to find similar models.
 
  • #6
Thank you for the reply. It's a good information and advice for me. I will take care of it. I still want to know what the exact form about the interaction of boson and fermion or how to build it?
 
  • #7
I know that the Hamiltonin have the form:H=1/2(p^2+q^2)-i*x(1)*x(2) when taken into accout the spin of fermion. But I didn't have the ability to build the interaction team. Welcome to give me a advice. x(i) is the odd Grassmann number.
 
  • #8
Couple them by potential;
[tex]H=(1/2)\left(\dot{q}^{2}+[V'(q)]^{2}\right) -iV''(q)X_{1}X_{2}[/tex]
 
  • #9
It's a valueable information. I appreciate it. Let me consider more complicated model, just like a boson interacted with its respective boson bath and a fermion interacted with fermion bath. Could I build the coupled team with potential?
 

1. What is a fermion oscillator and a boson oscillator?

A fermion oscillator is a type of quantum mechanical system that follows the rules of fermionic statistics, meaning that it obeys the Pauli exclusion principle and can only be occupied by one fermion at a time. A boson oscillator, on the other hand, follows the rules of bosonic statistics and can have multiple bosons occupying the same state simultaneously.

2. How do fermion and boson oscillators interact?

Fermion and boson oscillators can interact through the exchange of virtual particles. This interaction is known as the Fermi-Bose interaction and is responsible for a wide range of phenomena in quantum mechanics, such as superconductivity and superfluidity.

3. What are some real-world applications of fermion and boson oscillators?

Fermion and boson oscillators have many applications in modern technology, including in the development of quantum computers, lasers, and transistors. They also play a crucial role in understanding the behavior of matter at the atomic and subatomic level.

4. How are fermion and boson oscillators affected by temperature?

The behavior of fermion and boson oscillators is highly dependent on temperature. At very low temperatures, fermion oscillators can exhibit properties such as superconductivity, while boson oscillators can form a Bose-Einstein condensate. At higher temperatures, both types of oscillators can exhibit thermal fluctuations.

5. What is the significance of studying the interaction between fermion and boson oscillators?

The interaction between fermion and boson oscillators is crucial for understanding the fundamental laws of quantum mechanics and the behavior of matter at a microscopic level. It also has important implications for the development of new technologies and our understanding of the universe as a whole.

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