Homework on Review of Special Relativity and Dirac Equation

Level 1

1) The virtual photon exchanged in the $ep$ scattering diagram discussed in class is space-like. Explain what this means and give an example of a similar process involving a time-like exchange.

2) Show that for an electron at Jlab energies ($E_e= 6 GeV$), its mass can be disregarded in writing the energy-momentum relation. What percentage error would one have if muons ($\mu^\pm$) were considered? How long does it take an electron to travel the length of one linac track $l = 7/8$ mile? How long would a muon take?

(the values of the particles' masses in multiple of the $eV$ units, can be found on the PDG data group website).

3) Consider the nuclear reaction:

$$D + D \rightarrow ^3He + n$$

where $D$ represents the deuteron, and $n$ the neutron. If the energy $E=3.3$ MeV is released, what is the difference between the combined initial masses, and the combined masses of $^3He$ and the neutron?

4) Two events happen in a particle accelerator at a distance of $0.45$ m, and separated in time by $6.8 \times 10^{-9}$ s. What are the distance and time between the events as seen in the frame of a proton moving at $0.92c$ along a line between the two.

5) Making use of the anticommutation relation of the gamma matrices show that: $Tr(\gamma^\mu\gamma^\nu) = 4 g^{\mu\nu}$.

Level 2

1) By using Dirac's Equation and its conjugate, demonstrate that the covariant flux density is: $j^\mu = \bar{\psi}\gamma^\mu \psi$.

2) Demonstrate that: (a) the angular momentum, ${\bf L} = {\bf r} \times {\bf p}$,

and (b) the total angular momentum, ${\bf J} = {\bf L} + {\bf S}$ with:

$${\bf S} = (1/2) \left(\begin{array}{cc} \sigma & 0 \\ 0 & \sigma \end{array} \right)$$

are conserved.

3) Show that:

$$\gamma^\mu q_\mu \gamma^\nu p_\nu = q_\mu p^\mu - i \sigma^{\mu \nu}q_\mu p_\nu$$

with $\sigma^{\mu \nu} = i/2 [\gamma^\mu,\gamma^\nu ]$.

4) By showing that $(\sigma \cdot \hat{{\bf p}})^2 = 1$, demonstrate that the matrix:

$$\Sigma \cdot \hat{{\bf p}} = \left(\begin{array}{cc} \sigma \... ...}} & 0 \\ 0 & \sigma \cdot \hat{{\bf p}} \end{array} \right)$$

where $\hat{{\bf p}} = {\bf p}/\mid {\bf p} \mid$, commutes with the free particle Hamiltonian, $H$ ( $\sigma \cdot \hat{{\bf p}}$ is the particle's helicity).