Chapter 24, Relativistic Electron Theory Video Solutions, Quantum mechanics

Problem 1

If $\mathbf{A}$ and $\mathbf{B}$ are proportional to the unit $4 \times 4$ matrix, derive expansion formulas for the matrix products $(\boldsymbol{\alpha} \cdot \mathbf{A})(\boldsymbol{\alpha} \cdot \mathbf{B})$ and $(\boldsymbol{\alpha} \cdot \mathbf{A})(\boldsymbol{\Sigma} \cdot \mathbf{B})$ in terms of $\boldsymbol{\alpha}$ and $\boldsymbol{\Sigma}$ matrices in analogy with formula (16.59).

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03:02

Problem 2

. If a field theory of massless spin one-half particles (neutrinos) is developed, so that the $\beta$ matrix is absent, show that the conditions (24.30) and (24.31) are solved by $2 \times 2$ Pauli matrices, $\boldsymbol{\alpha}= \pm \boldsymbol{\sigma}$. Work out the details of the resulting two-component theory with particular attention to the helicity properties. Is this theory invariant under spatial reflection?

Katie Mcalpine

Numerade Educator

01:30

Problem 3

Develop the outlines of relativistic quantum field theory for neutral spinless bosons with mass. What modifications are indicated when the particles are charged?

Zachary Warner

Numerade Educator

Problem 4

Show that the vector operator
$$
\mathbf{Q}=\beta \boldsymbol{\Sigma}+(1-\boldsymbol{\beta}) \mathbf{\Sigma} \cdot \hat{\mathbf{p}} \hat{\mathbf{p}}
$$
satisfies the same commutation relations as $\Sigma$ and that it commutes with the free Dirac particle Hamiltonian. Show that the eigenvalues of any component of $\mathbf{Q}$ are $\pm 1$.
Apply the unitary transformation
$$
\exp \left[i(\theta / 2)\left(-p_y Q_x+p_x Q_y\right) / \sqrt{p_x^2+p_y^2}\right]
$$
to the spinors (24.92) and (24.93), and prove that the resulting spinors are eigenstates of $H$ with sharp momentum and definite value of $Q_z$. Show that these states are the relativistic analogues of the nonrelativistic momentum eigenstates with "spin up" and "spin down."

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10:26

Problem 5

Assume that the potential energy $-\mathbf{e} \phi(\mathbf{r})$ in the Dirac Hamiltonian (24.175) is a square well of depth $V_0$ and radius $a$. Determine the continuity condition for the Dirac wave function $\psi$ at $r=a$, and derive a transcendental equation for the minimum value of $V_0$ which just binds a particle of mass $m$ for a given value of $a$.

Guilherme Barros

Numerade Educator

Problem 6

Solve the relativistic Schrödinger equation for a spinless particle of mass $m$ and charge $-e$ in the presence of the Coulomb field of a point nucleus with charge Ze. Compare the fine structure of the energy levels with the corresponding results for the Dirac electron.

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Problem 7

Consider a neutral spin one-half Dirac particle with mass and with an intrinsic magnetic moment, and assume the Hamiltonian
$$
H=c \boldsymbol{\alpha} \cdot \frac{\hbar}{i} \nabla+\beta m c^2+\lambda B \beta \Sigma_z
$$
in the presence of a uniform constant magnetic field along the $z$ axis. Determine the important constants of the motion, and derive the energy eigenvalues. Show that orbital and spin motions are coupled in the relativistic theory but decoupled in a nonrelativistic limit. The coefficient $\lambda$ is a constant, proportional to the gyromagnetic ratio.

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Problem 8

If a Dirac electron is moving in a uniform constant magnetic field pointing along the $z$ axis, determine the energy eigenvalues and eigenspinors.

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