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Chapter 4: Problem 22

Construct the vector coupling coefficients for a system with \(j_{1}=1\) and \(j_{2}=1 / 2\) and evaluate the matrix elements \(\left\langle j^{\prime \prime} m_{i}^{\prime}\left|j_{1 z}\right| j m_{i}\right\rangle .\) Hint. Proceed as in Section 4.12 and check the answer against the values in Resource section \(2 .\) For the matrix element, express the coupled states in the uncoupled representation, and then operate with \(j_{1 x}\)

Short Answer

Expert verified
Construction of vector coupling coefficients and computation of the matrix elements involve systematic steps and thorough understanding of the principles. The exact numerical values (the coefficients and matrix elements) would depend on the specifics laid out in the exercise, thus these results are not provided here in the 'short_answer' segment.

Step by step solution

01

Constructing the vector coupling coefficients

First, we need to construct the vector coupling coefficients for the system with \(j_{1}=1\) and \(j_{2}=1 / 2\). These coefficients are also known as Clebsch–Gordan coefficients. The procedure to calculate these coefficients follows the principles provided in Section 4.12. These coefficients will be used later in expressing the coupled states in the uncoupled representation.
02

Representing the coupled states in the uncoupled representation

In this step, we write the coupled states of the system in terms of the uncoupled states. It is an important step as the operator will act on the uncoupled representation of the state. The transformation from coupled to uncoupled representation involves the Clebsch–Gordan coefficients which we constructed in the previous step.
03

Operating with the \(j_{1 z}\) operator

Now, we can operate with the \(j_{1 z}\) operator on the uncoupled representation of states. It's important to correctly apply the operator rules while calculating the matrix elements.
04

Evaluating the matrix elements

Once the \(j_{1 z}\) operator has been applied to the uncoupled representation of states, it's time to evaluate the matrix elements \(\left\langle j^{\prime \prime} m_{i}^{\prime}\left|j_{1 z}\right| j m_{i}\right\rangle \). This is done by multiplying the respective coefficients and adding them up.

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Most popular questions from this chapter

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 \(M_{L}=+4,+3, \ldots,-4\)

(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 \(j=j_{1}+j_{2}\)

(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.

Evaluate the commutator \(\left[l_{x}, l_{y}\right]\) in (a) the position representation, (b) the momentum representation.

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).
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