Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 4: Problem 29

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

Short Answer

Expert verified
The z-component of the electron's spin angular momentum can be in either the spin-up state \(\alpha\) with a probability of \(25\%\) or the spin-down state \(\beta\) with a probability of \(75\%\).

Step by step solution

01

Identify the Coefficients

The spin state of the electron is given by \( (1 / 4)^{1 / 2} \alpha-(3 / 4)^{1 / 2} \beta \). The coefficients are the square root of \( 1/4 \) and the square root of \( 3/4 \). So, the coefficient for spin-up state \(\alpha\) is \(1/2\) and the coefficient for spin-down state \(\beta\) is \(\sqrt{3}/2\).
02

Calculate the Probabilities of Each Outcome

The probability for each outcome is the square of its respective coefficient. Therefore, the probability of measuring the electron in spin-up state \(\alpha\) is \( (1/2)^2 = 1/4 \) or \(25\% \). The probability of measuring the electron in spin-down state \(\beta\) is \( (\sqrt{3}/2)^2 = 3/4 \) or \(75\% \).
03

Identify the Possible Outcomes

The two possible outcomes of measuring the z-component of the spin are \(\alpha\) (spin-up) with a probability of \(25\%\) and \(\beta\) (spin-down) with a probability of \(75\% \).

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

Using the Pauli matrix representation, reduce each of the operators (a) \(s_{x} s_{y}\) (b) \(s_{x} s_{y}^{2} s_{z}^{2},\) and (c) \(s_{x}^{2} s_{y}^{2} s_{z}^{2},\) to a single spin operator.

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) \(\left\langle 2,0\left|l^{2} l_{z} l^{2}+\right| 2,0\right\rangle\)

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).

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 \(j=j_{1}+j_{2}\)
See all solutions

Recommended explanations on Chemistry Textbooks

Physical Chemistry

Read Explanation

Chemical Reactions

Read Explanation

Kinetics

Read Explanation

Organic Chemistry

Read Explanation

Nuclear 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