Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 1: Problem 4

Show that if a component of angular momentum of the electron of a hydrogen atom is known to be \(2 \hbar\) within 5 per cent error, its angular orbital position in the plane perpendicular to that component cannot be specified at all.

Short Answer

Expert verified
The large uncertainty \( \Delta \phi \geq 5 \) indicates that the angular orbital position cannot be specified.

Step by step solution

01

Understanding the given condition

The problem states that the component of angular momentum of an electron in a hydrogen atom is known to be \(L_z = 2 \hbar\) with a 5% error. This error implies uncertainty \(\Delta L_z\). We need to calculate this uncertainty first.
02

Calculate the uncertainty in angular momentum

The uncertainty in the angular momentum component \(L_z\) is given as 5% of \(2 \hbar\). Calculate \Delta L_{\text{z}}\ as follows:\( \Delta L_z = 0.05 \times (2 \hbar) = 0.1 \hbar \)
03

Applying the Heisenberg Uncertainty Principle

The Heisenberg Uncertainty Principle in terms of angular momentum and angular position \(\phi\) states that:\( \Delta L_z \Delta \phi \geq \frac{\hbar}{2} \)Given \(\Delta L_z = 0.1 \hbar \), we need to substitute this into the equation.
04

Solve for the uncertainty in the angular position

Using the uncertainty principle equation, substitute the values:\( 0.1 \hbar \Delta \phi \geq \frac{\hbar}{2} \)Dividing both sides by 0.1 \hbar:\( \Delta \phi \geq \frac{1}{2 \times 0.1} = 5 \)
05

Interpretation of the result

The result \( \Delta \phi \geq 5 \) suggests a very large uncertainty in the angular position \(\phi\), implying that the electron's angular position in the plane perpendicular to the angular momentum component cannot be determined. Essentially, the position is entirely uncertain.

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Momentum
Quantum mechanics is a branch of physics that deals with the behavior of particles on very small scales, such as atoms and subatomic particles. It fundamentally differs from classical mechanics as it incorporates the principles of uncertainty and wave-particle duality In quantum mechanics, particles like electrons have properties of both particles and waves, and their behavior is described by wave functions. These functions provide probabilities of finding a particle in a particular state or location.

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 perfectly elastic ping-pong ball is dropped in vacuum from a height equal to ten times its radius onto a perfectly elastic fixed sphere of the same radius. \(\quad\) Neglecting effects due to earth motion, estimate the largest number of bounces against the fixed sphere that the ball can be expected to make under optimum conditions of release.

A beam of monoenergetic electrons is used to excite a particular level of an atom in a Franck-Hertz experiment. If this level is of short duration, owing to radiation back to the ground state, show that the inelastically scattered electrons that have lost energy to produce the excited level will not all be expected to have the same final energy. If the excited level lasts about \(10^{-10}\) second, what is the order of magnitude of the electron energy spread, measured in electron-volts?
See all solutions

Recommended explanations on Physics Textbooks

Medical Physics

Read Explanation

Electromagnetism

Read Explanation

Modern Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Circular Motion and Gravitation

Read Explanation

Rotational Dynamics

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