Homework 4

1. Non-Relativistic Perturbation Theory

Consider the general formula for a system's probability to transition from the initial state $n$ to the final state $m$:

$$\left\vert \int_0^t dt^\prime \langle n \mid V(t^\prime) \mid m \rangle e^{i(E_n - E_m) t^\prime} \right\vert^2 .$$ (1)

1) Calculate the probability for a matrix element, $\langle n \mid V(t^\prime) \mid m \rangle$ that is constant during the time interval $t$.

2) Plot your result as a function of $t$. Where are the zeros of the function, and what is its maximum value?

3) Find the condition under which the function can be written as:

$$\pi t \delta(E_n-E_m)$$ (2)

2. Dirac Equation

1) By multiplying the Dirac equation:

$$i \gamma^\mu \partial_\mu - m) \psi =0$$

by a proper operator, show that each of the four components ofn $\psi$ satisfy the Klein-Gordon equation:

$$(\Box^2 + m^2) \psi_\alpha =0$$

($\alpha$ is the Dirac index).

2) Demonstrate the orthogonality relation:

$$\bar{u}^s u^{s^\prime} = 2 m \delta_{s,s^\prime}$$

and the completeness relation:

$$\sum_s u^s(p) \bar{u}^s(p) = (\not\!p + m)$$

3) Demonstrate that for a free proton:

$$\not\!p \not\!p = m^2$$

3. Inclusive Electron Proton Scattering

1) Write down the components of all four-vectors involved in elastic electron-proton scattering in the laboratory frame (initial proton at rest).

2) Write the invariants: $\nu = (Pq)/M$, $Q^2=-q^2$, $x=Q^2/2M\nu$, where $P$ is the initial proton's momentum, $M$ is the proton mass, $q$ is the virtual photon's momentum, in terms of the lab frame components.

3) Based on our class discussion, so far, what can you tell about the angle at which you would put your detector?

4) Does the fact that your particles have spin $1/2$ influence the results for the cross section discussed in class?