Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 7

(a) Confirm that the Pauli matrices \\[ \sigma_{x}=\left(\begin{array}{ll} 0 & 1 \\ 1 & 0 \end{array}\right) \quad \sigma_{y}=\left(\begin{array}{rr} 0 & -\mathrm{i} \\ \mathrm{i} & 0 \end{array}\right) \quad \boldsymbol{\sigma}_{z}=\left(\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right) \\] satisfy the angular momentum commutation relations when we write \(s_{q}=\frac{1}{2} \hbar \sigma_{q},\) and hence provide a matrix representation of angular momentum. (b) Why does the representation correspond to \(s=1 / 2\) ? Hint. For the second part, form the matrix representing \(s^{2}\) and establish its eigenvalues.

Short Answer

Expert verified
The Pauli matrices satisfy the angular momentum commutation relations when \(s_{q}=\frac{1}{2} \hbar \sigma_{q}\) is used and hence provide a matrix representation of angular momentum which corresponds to \(s=1 / 2\) due to the eigenvalues of the matrix representation of \(s^{2}\).

Step by step solution

01

Analyzing the commutation relations

The commutation relation for the angular momentum operators is given by \[\[s_{i}, s_{j}\] = \mathrm{i} \hbar \epsilon_{ijk} s_{k}\] where \(\[[s_{i}, s_{j}]\] = s_{i} s_{j} - s_{j} s_{i}\) is the commutator of the operators, \(\epsilon_{ijk}\) is the Levi-Civita symbol, and \(s_{k}\) is the angular momentum operator. Substituting \(s_{q}=\frac{1}{2} \hbar \sigma_{q}\) into the commutation relation, it can be confirmed that the equation is satisfied only when the commutation relations for the Pauli matrices, which are \(\[[\sigma_{i}, \sigma_{j}]\] = 2 \mathrm{i} \epsilon_{ijk} \sigma_{k}\) are used.
02

Matrix representations

The right side of the angular momentum commutation relation can be transformed using the relation for the Pauli matrices. The equation becomes \[\[s_{i}, s_{j}\] = \frac{\hbar^2}{4} \[[\sigma_{i}, \sigma_{j}]\] = \frac{\hbar^2}{4} * 2i * \epsilon_{ijk} \sigma_{k} = \frac{\hbar^2}{2} * i * \epsilon_{ijk} \sigma_{k}\] Which means that this forms a matrix representation of angular momentum.
03

Verifying the representation

For the second part, form the matrix representing \(s^{2} = s_{x}^{2} + s_{y}^{2} + s_{z}^{2}\). Substituting the terms for \(s_{i}\), we have that \(s^{2} = (\frac{\hbar}{2} \sigma_{x})^2 + (\frac{\hbar}{2} \sigma_{y})^2 + (\frac{\hbar}{2} \sigma_{z})^2 \). Note that the square of each Pauli matrix is the identity matrix, so the right side of this equation simplifies to \(\frac{3}{4} \hbar^2\). The eigenvalues for the operator \(s^{2}\) must be of the form \(s(s+1)\hbar^2\). We can solve the equation \(s(s+1)\hbar^2 = \frac{3}{4} \hbar^2\) to find that the value of \(s\) that satisfies this is \(s = \frac{1}{2}\), therefore the representation corresponds to \(s=1/2\).
04

Summing up

We have shown that we can describe the angular momentum operators through the Pauli matrices and the representation corresponds to \(s=1/2\). The Pauli matrices thus provide a correct matrix representation for the angular momentum in quantum mechanics.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Suppose that in place of the actual angular momentum commutation rules, the operators obeyed \(\left[l_{x}, l_{y}\right]=-\mathrm{i} \hbar l_{z^{*}}\) What would be the roles of \(l_{2}=l_{x} \pm l_{y} ?\)

(a) Demonstrate that if \(\left[j_{1 q}, j_{2 q^{\prime}}\right]=0\) for all \(q, q^{\prime},\) then (b) Go on to show that if \(j_{1} \times j_{1}=i \hbar j_{1}\) and \(j_{1} \times j_{2}=-j_{2} \times j_{1}\) \(j_{2} \times j_{2}=i \hbar j_{2},\) then \(j \times j=\) ihj where \(j=j_{1}+j_{2}\)

What is the expectation value of the \(z\) -component of orbital angular momentum of electron 1 in the \(\left|G, M_{L}\right\rangle\) state of the configuration \(\mathrm{d}^{2}\) ? Hint. Express the coupled state in terms of the uncoupled states, find \(\left\langle\mathrm{G}, M_{L}\left|l_{1 z}\right| \mathrm{G}, M_{L}\right\rangle\) in terms of the vector coupling coefficients, and evaluate it for \(M_{L}=+4,+3, \ldots,-4\)

Determine what total angular momenta may arise in the following composite systems: (a) \(j_{1}=3, j_{2}=4 ;\) (b) the orbital momenta of two electrons (i) both in p-orbitals, (ii) both in d-orbitals, (iii) the configuration \(\mathrm{p}^{1} \mathrm{d}^{1} ;\) (c) the spin angular momenta of four electrons. Hint. Use the ClebschGordan series, eqn \(4.42 ;\) apply it successively in (c).

Calculate the matrix elements (a) \(\left\langle 0,0\left|l_{z}\right| 0,0\right\rangle\) (b) \(\left\langle 2,1\left|l_{+}\right| 2,0\right\rangle\) (c) \(\left\langle 2,2\left|l_{+}^{2}\right| 2,0\right\rangle,\) (d) \(\left\langle 2,0\left|l_{+} l_{-}\right| 2,0\right\rangle\) (e) \(\left\langle 2,0\left|l \downarrow_{+}\right| 2,0\right\rangle,\) and (f) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)
See all solutions

Recommended explanations on Chemistry Textbooks

Ionic and Molecular Compounds

Read Explanation

Chemistry Branches

Read Explanation

Physical Chemistry

Read Explanation

Nuclear Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

The Earths Atmosphere

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free