Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 8

Compare orbital angular momentum with spin angular momentum of an electron in the hydrogen atom.

Short Answer

Expert verified
Orbital angular momentum and spin angular momentum, both quantized, arise from different aspects of an electron's behavior. The former is due to the electron’s motion around the nucleus, and its direction is undetermined while the magnitude depends on the azimuthal quantum number. The latter, in contrast, is an intrinsic property of the electron, always with a magnitude of \( \sqrt{3}/2 \hbar \), and can only be up or down in orientation.

Step by step solution

01

Understanding Orbital Angular Momentum

Orbital angular momentum emerges from the motion of the electron around the nucleus. Its magnitude is quantized and given by \( \sqrt{l(l+1) \hbar } \), where l is the azimuthal quantum number and \(\hbar\) is the reduced Planck’s constant. The direction, however, is not completely determined - it's known only that the angular momentum vector points somewhere on a cone.
02

Understanding Spin Angular Momentum

Spin angular momentum, on the other hand, is an intrinsic property of electrons. It's similar to the angular momentum of a spinning particle, although it can't be explained via classical mechanics. The magnitude of spin angular momentum is quantized and given by \( \sqrt{s(s+1) \hbar} \), with s always equalling to 1/2 for electrons (spin quantum number), making its magnitude always equal to \( \sqrt{3}/2 \hbar \). The direction of the spin can be either up or down with respect to an external magnetic field.
03

Comparison

So in comparison, both orbital and spin angular momenta are quantized but the quantization comes from different sources. For orbital, it comes from the restricted motion of the electron (defined by quantized energy levels), while for spin, it's an inherent quantum property of the electron. Moreover, the direction of orbital angular momentum is undetermined while the direction of spin can only be up or down, showing a difference in terms of their orientation properties.

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

The yellow light from a sodium-vapor street lamp is produced by a transition of sodium atoms from a \(3 p\) state to a \(3 s\) state. If the difference in energies of those two states is \(2.10 \mathrm{eV},\) what is the wavelength of the yellow light?

What is spin-orbit coupling?

What is the minimum frequency of a photon required to ionize: (a) a \(\mathrm{He}^{+}\) ion in its ground state? (b) \(\mathrm{A} \mathrm{Li}^{2+}\) ion in its first excited state?

An X-ray tube accelerates an electron with an applied voltage of \(50 \mathrm{kV}\) toward a metal target. (a) What is the shortest-wavelength X-ray radiation generated at the target? (b) Calculate the photon energy in eV. (c) Explain the relationship of the photon energy to the applied voltage.

NGC1763 is an emission nebula in the Large Magellanic Cloud just outside our Milky Way Galaxy. Ultraviolet light from hot stars ionize the hydrogen atoms in the nebula. As protons and electrons recombine, light in the visible range is emitted. Compare the energies of the photons involved in these two transitions.
See all solutions

Recommended explanations on Physics Textbooks

Wave Optics

Read Explanation

Dynamics

Read Explanation

Magnetism

Read Explanation

Fundamentals of Physics

Read Explanation

Physics of Motion

Read Explanation

Classical Mechanics

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