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Quarks And Leptons. An Introductory Course In Modern Particle Physics

Francis Halzen, Alan D. Martin

Chapter 5

The Dirac Equation - all with Video Answers

Educators

Chapter Questions

01:44

Problem 1

Prove that the $\alpha_{i}$ and $\beta$ are hermitian, traceless matrices of even dimensionality, with eigenvalues $\pm 1$.

Nick Johnson
Nick Johnson
Numerade Educator
01:32

Problem 2

Operate on (5.7) with $\gamma^{\nu} \partial_{\nu}$ and show that each of the four components $\psi_{i}$ satisfies the Klein-Gordon equation
$$
\left(\square^{2}+m^{2}\right) \psi_{i}=0 .
$$

Chai Santi
Chai Santi
Numerade Educator
02:29

Problem 3

Calculate the $\lambda=+\frac{1}{2}$ helicity eigenspinor of an electron of momentum $\mathbf{p}^{\prime}=(p \sin \theta, 0, p \cos \theta)$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:06

Problem 4

Confirm the desired result that the Dirac equation describes " "intrinsic" angular momentum (\equiv spin)- $\frac{1}{2}$ particles.

Narayan Hari
Narayan Hari
Numerade Educator
02:25

Problem 5

For a nonrelativistic electron of velocity $v$, use ( $5.24$ ) to show that $u_{A}$ is larger than $u_{B}$ by a factor of the order $v / c$. In nonrelativistic problems, $\psi_{A}$ and $\psi_{B}$ are referred to as the "large" and "small" components of the electron wavefunction $\psi$.

Prabhakar Kumar
Prabhakar Kumar
Numerade Educator
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Problem 6

In representation (5.4) and (5.8) of the $\gamma$-matrices, show that a possible choice of $C$ is
$$
C \gamma^{0}=i \gamma^{2}=\left(\begin{array}{llll}
& & -1 & 1 \\
& -1 & & \\
1 & & &
\end{array}\right)
$$

Victor Salazar
Victor Salazar
Numerade Educator
01:11

Problem 7

Use (5.41) to show that
$$
\bar{u}^{(s)} u^{(s)}=2 m, \quad \bar{v}^{(s)} v^{(s)}=-2 m
$$

Raj Bala
Raj Bala
Numerade Educator
01:02

Problem 8

Show that }(\boldsymbol{\sigma} \cdot \mathbf{p})^{2}=|\mathbf{p}|^{2}

Raj Bala
Raj Bala
Numerade Educator
01:07

Problem 9

Derive the completeness relations }

Suzanne W.
Suzanne W.
Numerade Educator
01:33

Problem 10

Show that } p p=p^{2} .

Adriano Chikande
Adriano Chikande
Numerade Educator
02:27

Problem 11

Show that
$$
\Lambda_{+}=\frac{\not+m}{2 m}, \quad \Lambda_{-}=\frac{-p+m}{2 m}
$$
project over positive and negative energy states, respectively. Recall that the projection operators must satisfy
$$
\Lambda_{\pm}^{2}=\Lambda_{\pm} \quad \text { and } \quad \Lambda_{+}+\Lambda_{-}=1 .
$$

Bobby Barnes
Bobby Barnes
University of North Texas
04:23

Problem 12

For a proper infinitesimal Lorentz transformation show that an } S \text { which satisfies (5.57)

Robert Zaballa
Robert Zaballa
Numerade Educator
07:38

Problem 13

Show that the operators
$$
P_{R} \equiv \frac{1}{2}\left(1+\gamma^{5}\right), \quad P_{L} \equiv \frac{1}{2}\left(1-\gamma^{5}\right)
$$
have the appropriate properties to be (right- and left-hand) projection operators, that is,
$$
P_{i}^{2}=P_{i}, \quad P_{L}+P_{R}=1, \quad P_{R} P_{L}=0 .
$$
Here, $\gamma^{5}$ is called the chirality operator.
For a massive fermion, we define the projections $\frac{1}{2}\left(1 \pm \gamma^{5}\right) u$ to be right- and left-handed components of $u$.

Tamara Worner
Tamara Worner
Numerade Educator
04:20

Problem 14

For a massive fermion, show that handedness is not a good quantum number. That is, show that $\gamma^{5}$ does not commute with the Hamiltonian. However, verify that helicity is conserved but is frame dependent. In particular, show that the helicity is reversed by "overtaking" the particle concerned.

Hafiz Shahzaib
Hafiz Shahzaib
Numerade Educator
01:07

Problem 15

Working in the Dirac-Pauli representation of $\gamma$-matrices, (5.51), show that at high energies

Raj Bala
Raj Bala
Numerade Educator