Login Registrieren

Warning: foreach() argument must be of type array|object, bool given in /var/www/html/web/app/themes/studypress-core-theme/template-parts/header/mobile-offcanvas.php on line 20

Chapter 8: Q3CQ (page 338)

Summarize the connection between angular momentum quantization and the stem-Gerlach experiment.

Short Answer

Expert verified

The quantized magnetic moment of the electron manifests as a splitting of the atomic beam.

Step by step solution

01

Summarize the experiment

The Stern-Gerlach experiment was to test the Bohr-Sommerfeld hypothesis that the direction of the angular momentum of the silver atom is quantized.

Electrons rotating in the hydrogen atom possess quantized angular momentum, which gives quantized magnetic moment. In the Stem-Gerlach experiment, silver atoms are sent through a non-uniform magnetic field.

02

Explanation

This impacts a force proportional to the magnetic moment. Here, a quantized magnetic moment of the electron manifests as a splitting of the atomic beam.

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 lithium atom has three electrons. These occupy individual particle states corresponding to the sets of four quantum numbers given by .

$(n, l, m_{l}, m_{j}) = (1, 0, 0, + \frac{1}{2}), (1, 0, 0, - \frac{1}{2}) and (2, 0, 0, + \frac{1}{2})$

Using $ψ_{1, 0, 0} (r_{j}) ↑, ψ_{1, 0, 0} (r_{j}) ↓, and ψ_{2, 0, 0} (r_{j}) ↑$ to represent the individual-particle states when occupied by particle $j$ . Apply the Slater determinant discussed in Exercise 42 to find an expression for an antisymmetric multiparticle state. Your answer should be sums of terms like .

$ψ_{1, 0, 0} (r_{1}) ↑, ψ_{1, 0, 0} (r_{2}) ↓, and ψ_{2, 0, 0} (r_{3}) ↑$

Question: In nature, lithium exists in two isotopes: lithium-6, with three neutrons in its nucleus, and lithium-7, with four as individual atoms, would these behave as Bosons or as Fermions? Might a gas of either behave as a gas of Bosons? Explain.

What is the minimum possible energy for five (non-interacting) spin $- \frac{1}{2}$ particles of massmin a one dimensional box of length L ? What if the particles were spin-1? What if the particles were spin $- \frac{3}{2}$ ?

Slater Determinant: A convenient and compact way of expressing multi-particle states of anti-symmetric character for many fermions is the Slater determinant:

$| \begin{matrix} ψ_{n_{1}} (x_{1}) m_{31} & ψ_{n_{2}} (x_{1}) m_{32} & ψ_{n_{3}} (x_{1}) m_{33} & \cdot \cdot \cdot & ψ_{n_{N}} (x_{1}) m_{s N} \\ ψ_{n_{1}} (x_{2}) m_{11} & ψ_{n_{2}} (x_{2}) m_{32} & ψ_{n_{3}} (x_{2}) m_{33} & \cdot \cdot \cdot & ψ ψ_{n_{1}} (x_{2}) m_{s N} \\ ψ_{n_{3}} (x_{3}) m_{31} & ψ_{n_{2}} (x_{3}) m_{12} & ψ_{n_{3}} (x_{3}) m_{33} & ψ_{n_{N}} (x_{3}) m_{s N} \\ \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot & \cdot \cdot \cdot \\ ψ_{n_{1}} (x_{N}) m_{11} & ψ_{n_{2}} (x_{N}) m_{32} & ψ_{n_{3}} (x_{N}) m_{33} & \cdot \cdot \cdot & ψ_{n_{N}} (x_{N}) m_{s N} \end{matrix} |$

It is based on the fact that for N fermions there must be Ndifferent individual-particle states, or sets of quantum numbers. The ith state has spatial quantum numbers (which might be $n_{i}, ℓ_{i}$ , and $m_{f i}$ ) represented simply by $n_{i}$ and spin quantum number $m_{s i}$ . Were it occupied by the ith particle, the slate would be $ψ_{n i} (x_{j}) m_{s i}$ a column corresponds to a given state and a row to a given particle. For instance, the first column corresponds to individual particle state $ψ_{n} (x_{j}) m_{s 1}$ . Where jprogresses (through the rows) from particle 1 to particle N. The first row corresponds to particle I. which successively occupies all individual-particle states (progressing through the columns). (a) What property of determinants ensures that the multiparticle state is 0 if any two individual particle states are identical? (b) What property of determinants ensures that switching the labels on any two particles switches the sign of the multiparticle state?

The Zeeman effect occurs in sodium just as in hydrogen-sodium's lone $3 s$ valence electron behaves much as hydrogen's 1.5. Suppose sodium atoms are immersed in a $0.1 T$ magnetic field.

(a) Into how many levels is the $3 P_{1 / 2}$ level split?

(b) Determine the energy spacing between these states.

(c) Into how many lines is the $3 P_{1 / 2}$ to $3 s_{1 / 2}$ spectral line split by the field?

(d) Describe quantitatively the spacing of these lines.

(e) The sodium doublet $(589.0 n m a n d 589.6 n m)$ is two spectral lines. $3 P_{3 / 2} \to 3 s_{1 / 2}$ and $3 P_{1 / 2} \to 3 s_{1 / 2}$ . which are split according to the two differentpossible spin-orbit energies in the 3Pstate (see Exercise 60). Determine the splitting of the sodium doublet (the energy difference between the two photons). How does it compare with the line splitting of part (d), and why?

See all solutions

Recommended explanations on Physics Textbooks

Torque and Rotational Motion

Read Explanation

Scientific Method Physics

Read Explanation

Astrophysics

Read Explanation

Waves Physics

Read Explanation

Measurements

Read Explanation

Fundamentals of Physics

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