April 7, 1998

PHY 442 - Problem Set # 16

Consider the Dirac equation for the free particle:

$$i \hbar \frac{\partial \Psi}{\partial t} = \left( {\bf{\alpha \cdot p}}c + \beta mc^2 \right) \Psi .$$

Use the results on page 92 of Sakurai's text to find four normalized plane wave eigenstates of this equation. Use these eigenstates to determine the matrix elements of the following operators. When appropriate, diagonalize the matrices to find eigenstates of the operators.

1.

${\bf{\alpha}}$.

2.

$\beta$.

3.

$\Sigma_z$.

4.

${\bf{\Sigma \cdot p}}$.