# A fermion oscillator interacted with a boson oscillator!

#### dhg19861106

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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#### tom.stoer

Sorry, I do not see how a fermion oscillator should emerge from the boson oscillator. Can you please be a bit more specific?

#### dhg19861106

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.

#### samalkhaiat

You should know that the configuration space of the fermion oscillator is $R^{2}_{a}$. That is, you need two real anticommuting dynamical variables to describe the fermion oscillator. The action for such oscillator is given by
$$S = (i/2)\int dt (X_{1}\dot{X}_{1} + X_{2}\dot{X}_{2} + X_{1}X_{2}-X_{2}X_{1})$$
Form this, you find the Hamiltonian;
$$H = -iX_{1}X_{2}$$

#### Cthugha

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.

#### dhg19861106

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?

#### dhg19861106

I know that the Hamiltonin have the form =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.

#### samalkhaiat

Couple them by potential;
$$H=(1/2)\left(\dot{q}^{2}+[V'(q)]^{2}\right) -iV''(q)X_{1}X_{2}$$

#### dhg19861106

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?

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