Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 27

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

Short Answer

Expert verified
Expectation value for an electron's z-component of orbital angular momentum in the \( \left|G, M_{L}\right\rangle \) state is given by \( \left\langle G M_{L} \right| l_{1z} \left| G M_{L}\rangle \). The specific value will be determined by evaluating this expression for different \( M_{L} \) values ranging from +4 to -4.

Step by step solution

01

Express in terms of uncoupled states

The \(|G, M_{L}\rangle\) state of the configuration \(\mathrm{d}^{2}\) can be expressed in terms of the uncoupled states as: \( |G, M_{L}\rangle = \sum_{m_1 , m_2 } \langle l_1 m_1, l_2 m_2 | G M_{L}\rangle |l_1 m_1\rangle |l_2 m_2\rangle \), where \(l_1\) and \(l_2\) are the orbital angular momenta of each electron, and \(m_1\) and \(m_2\) are their respective z-components.
02

Find expectation value

The expectation value of the z-component of the angular momentum for the first electron can be found using the formula for expectation values in quantum mechanics: \( \left\langle G M_{L} \right| l_{1z} \left| G M_{L}\rangle = \sum_{m_1, m_2} | \langle l_1 m_1, l_2 m_2 | G M_{L}\rangle |^2 m_1 \). It's crucial to remember that the z-component of the angular momentum for one of the electrons, say the first electron, is equal to \( m_1 \), the z-component quantum number.
03

Evaluate for M_{L} Values

Finally, plug in the values of \( M_{L} =+4,+3,...,-4 \) into the expectation value formula and evaluate.

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

(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.

What are the possible outcomes of a single measurement of the \(z\) -component of spin angular momentum of an electron in the spin state \((1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta ?\)

Consider a system of two electrons that can have either paired or unpaired spins (e.g. a biradical). The energy of the system depends on the relative orientation of their spins. Show that the operator \(\left(h J / \hbar^{2}\right) s_{1} \cdot s_{2}\) distinguishes between singlet and triplet states. The system is now exposed to a magnetic field in the \(z\) -direction. Because the two electrons are in different environments, they experience different local fields and their interaction energy can be written \(\left(\mu_{\mathrm{B}} / \hbar\right) \times\) \(B\left(g_{1} s_{1 z}+g_{2} s_{2 z}\right)\) with \(g_{1} \neq g_{2} ; \mu_{5}\) is the Bohr magneton and \(g\) is the electron \(g\) -value, quantities discussed in Chapter 13 Establish the matrix of the total hamiltonian, and demonstrate that when \(h J \gg \mu_{\mathrm{B}} \mathscr{B},\) the coupled representation is "bctter', but that when \(\mu_{B} D\) : wh \(J\), the uncoupled representation is 'better', Find the eigenvalues of the system in each case. Hint. Use the vector coupling coefficients in Resource section 2 to determine hamiltonian matrix elements.
See all solutions

Recommended explanations on Chemistry Textbooks

Nuclear Chemistry

Read Explanation

Chemical Analysis

Read Explanation

Chemistry Branches

Read Explanation

Kinetics

Read Explanation

Organic Chemistry

Read Explanation

Inorganic Chemistry

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