# Fermi's Golden Rule

Fermi's Golden rule is used to calculate the probability (per unit time) of a quantum mechanical transition between two particles ( a and b) in an initial quantum state to two particles ( c and d) in a final state .

a + b c + d

where a is the incoming particle and b is the target particle.

Although Fermi first coined the term "Golden Rule", Dirac developed most of the machinery.

Let represent the Flux of particle per unit time through a unit area normal to the beam. Then

where

= density of particles in the incident beam
= velocity of a relative to b.

The probability the incident particle will hit a target particle is given by the cross-section times the number of target particles per unit area .

The number of interactions therefor will be given by

Number of Interactions per unit area per unit second=

The transition rate per target particle is

Solving the above for the Cross section \sigma we have

A calculation of the transition rate W si equivalent to calculating the cross section of the scattering process.

Fermi's Golden rule says that the transition rate is given by a transition matrix element (or "Amplitude") weighted by the phase space and Plank's constant such that

(Phase Space) = \frac{0.693}{t_{1/2}}
transition half life.

## Transition Amplitude

where

= initial quantum state of the system which is an eigenstate of the time independent ("steady state") Hamiltonian ()
= final quantum state of system after a transition
= the part of the total Hamiltonian () which describes the interaction responsible for the transition.
= Unperturbed ("steady state") Hamiltonian
= total Hamiltonian describing the quantum mechanical system
integration over all space

The off diagonal elements of the matrix tell you the transition probablility.

### Stationary State

The stationary state system is given by the solutions of the schrodinger equation for H_0

where

= energy eigen values

### Interaction Hamiltonian

= Interaction Hamiltonian = perturbation to which causes a transition/interaction. The time dependent schrodinger equation for this perturbation is:

One can write the solution in terms of a linear combination of the basis functions from the Stationary State solution.

You jsut need to find the components of the basis eigenfunctions (eigenvectors) using the Schrodinger equation

Because

The second term on the left hand side is canceled with the term on the right hand side

multiple both sides by and integrate over the whole volume

where

Integrate the above equation

If we make the following assumptions

1.) The quantum state is just before the particle interacts via

2.) acts over a very small time ( strong force range is )

3.) is so weak that the initial quantum state is not substantially altered by the interaction. Or other states are very weakly involved ( prefers one state over all the others)

= 1

Then

= amplitude for making a transition from to

### Transition Probability

The probability of making a transition from t state to the state is given by the magnitude of the amplitude

and

What does this probability function look like?

Notice
The probability of a transition gets smaller if the energy difference between the state gets bigger.
It illustrates the uncertainty principle . is the Half Width of the peak. In the above picture .
Another indication of the uncertainty principle is oberved if you let then above function goes to a delta function
For Most of the above function lies in the interval

The probability of transition (The total transition probability to all states) is given by adding up all the probabilities to individual states.

If we assume the states are all clustered together as in a continuum then

constant

This assumption isn't so bad when you realize that since the above function lies over a finite energy interval and that that finite energy interval contains similar states.

Since we assume a continuum the summation becomes an integral such that

Where the Energy is an explicit function of state number .

Using the chain rule we can recast the integral

let

Then

and

=:
Density of states

## Single Particle decay

Consider the case when a single particle decays into multiple fragments (several other particles)

where

= probability per second that the particle will decay
= a symmetry factor of for every group of identical particles in the final state
= 4-momentum of the particle. ;
= conservation of 4-momentum
Note
= invariant under Lorentz transformations

### Example: Pi-zero () decay

we are interested in calculating

Consider the decay of a neutral pion () into two photons ().

The two gammas are identical particles so

Since the pion is initially at rest (or we can go to its rest frame and then Lorentz boost back to the lab frame)

Because photons have no mass,  :

Integrating over :

If ( the transition does not depend on the momentum vector directions)

then

with the additional conditions that and which must be applied when evaluating
Units check
= energy eigenvalues squared
transition probability per unit time
Caveat
Sometimes will depend on the momentum vector directions in which case the integral must be done after evaluating the matrix element amplitude. An example of this is when the transition is spin dependent as in the hyperfine interaction or polarization based transitions.

### Example: Two -Body decay (fission fragments)

Now consider a more general case in which the decay daughters have mass:

We don't know if the two daughter particles are identical so leave as a funtion.

We can still do the calculation in the mother particles rest frame.

Now that the daughters have mass we need:

Recast the delta function:

Upon integrating over

After integrating over the delta function gives you

Once again , if the transition amplitude does not depend on the vector directions of and then you can integrate over angles and get a .

Let

Then

substituting

where

and

= momentum when

or

## 2 Body scattering in CM frame

Lets consider the case when two particles ( and ) collide and are transformed into two separate particle ( and ). An interaction happens during the collision which changes the two particles into two other particles.

Calculate the differential cross section, in the center of momentum frame, assuming that is the amplitude for this collision.

The general expression for the cross section via Fermi's golden rule is similar to the above pion decay example if you consider m_1 and m_2 forming an intermediate state and is given as

In the CM frame

or

substituting the above and

:

into the cross ection equation we have

integrating over we have

Unfortunately depends on scattering angles. For some interactions it even depends on angles (5th structure function, single spin asymmetries).

If you write the cross section in terms of a differential cross section in solid angle then only the momentum part of the integral remains.

This integral looks just like the 2-Body decay problem if you let

Note: In Center of Momentum frame and

so

where and

Units
Notice that the number of integrals is equal to the number of particles in the interaction. The units of change with the number of integrals such that