Elmag decays of polarized neutrons

[afi]

Hi,
I'm not sure what to do in this problem.

The interaction between magnetic moment of the neutron and the photon is described by the interaction Hamiltonian $$H=-\bold{ m B}$$. Where m is the magnetic moment, and B is the magnetic field corresponding to the presence of a photon.
Find the total photon emission rate for the transition from |^> (up-state) to |,> (down-state) using Fermis Golden Rule for electromagnetic processes by including a summation of all possible directions of the emitted photon as well as the corresponding polarization directions of the emitted photon.

If the Hamiltonian is given by $$H=-\frac{e}{m}\bold{ A p}$$ where A is the vector potential and p the impulse and m the mass, then I know what to do, but the magnetic moment $$m= g(mu) S \frac{2 \pi}{h}$$, where S is the spin operator, confuses me.

 

[AndyW]

Hello,

Thank you for your question. The interaction between the magnetic moment of a neutron and a photon is a fundamental process in quantum mechanics. In order to calculate the total photon emission rate for the transition from the up-state to the down-state, we can use Fermi's Golden Rule for electromagnetic processes.

First, we need to understand the Hamiltonian given in the problem. The Hamiltonian is a mathematical representation of the energy of a system. In this case, it describes the interaction between the magnetic moment of the neutron and the photon. The Hamiltonian is given by H=-\bold{mB}, where m is the magnetic moment and B is the magnetic field corresponding to the presence of a photon.

To calculate the total photon emission rate, we need to take into account all possible directions of the emitted photon as well as the corresponding polarization directions. This can be done by using the matrix element, which is a measure of the probability of a transition between two quantum states. The matrix element for this process can be written as <\downarrow|H|\uparrow>, where |\uparrow> and |\downarrow> represent the up-state and down-state, respectively.

Next, we need to calculate the transition probability using Fermi's Golden Rule. This rule states that the transition probability is proportional to the square of the matrix element and the density of final states. In this case, the density of final states is given by the number of possible directions and polarizations of the emitted photon.

Now, let's take a closer look at the magnetic moment of the neutron. As you mentioned, the magnetic moment is given by m=g(mu)S\frac{2\pi}{h}, where g is the Landé g-factor, mu is the nuclear magneton, S is the spin operator, and h is Planck's constant. This expression can be simplified to m=gS\frac{2\pi}{h}, since mu is a constant.

Finally, we can combine all of these elements to calculate the total photon emission rate for the transition from the up-state to the down-state. The final expression is given by:

R = \frac{2\pi}{h}g^2|\langle\downarrow|S|\uparrow\rangle|^2 N_{ph},

where N_{ph} is the number of possible directions and polarizations of the emitted photon.

I hope this helps to clarify the problem for you. If you have any further questions, please