/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 42 A particle is in the ground stat... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 35: Problem 42

A particle is in the ground state of an infinite square well. What's the probability of finding the particle in the left-hand third of the well?

Short Answer

Expert verified
So, the probability of finding the particle in the left-hand third of the well is \(1/2 - \sqrt{3}/4\) or approximately 0.134.

Step by step solution

01

Identifying the Wave Function

The wave function for the ground state of a particle in an infinite square well of width \(L\) is given by \(\psi(x) = \sqrt{2/L} \sin(\pi x/L)\). Here, \(L\) is the width of the well, and \(x\) is the position of the particle within the well.
02

Setup the Probability Integral

The probability of finding the particle in a particular range within the well is given by the integral of the absolute square of the wave function over that range. So to find the probability of finding the particle in the left-hand third of the well, we integrate from \(x = 0\) to \(x = L/3\). Hence, the probability \(P\) is given by \(P = \int_{0}^{L/3} |\psi(x)|^2 dx\).
03

Evaluate the Integral

Now substitute the wave function into the integral and evaluate it. This leaves us with the integral \(P = \int_{0}^{L/3} (2/L) \sin^2(\pi x/L) dx = 2/L \int_{0}^{L/3} \sin^2(\pi x/L) dx\). By substitution \(u = \pi x/L\), we get \(P = 2/\pi \int_{0}^{\pi/3} \sin^2(u) du\). Since \(\sin^2(u) = 1/2 - 1/2 \cos(2u)\), the integral becomes \(P = 1/\pi [\frac{1}{2}u - \frac{1}{4}\sin(2u)]|_{0}^{\pi/3}.\)
04

Calculate the Probability

Substitute upper and lower limits into the above equation, which leaves us with \(P = 1/\pi [(1/2) \cdot (\pi/3) - (1/4)\sin(2\pi/3)] - 0 = 1/2 - \sqrt{3}/4\).

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

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

Your roommate is taking Newtonian physics, while you've moved on to quantum mechanics. He claims that QM can't be right, because he didn't see any evidence of quantized energy levels in a mass-spring harmonic oscillator experiment. You reply by calculating the spacing between energy levels of this system, which consists of a \(1-\mathrm{g}\) mass on a spring with \(k=80 \mathrm{N} / \mathrm{m}\) What is that spacing, and how does this help your argument?

The table below lists the wavelengths emitted as electrons in identical square-well potentials drop from various states \(n\) to the ground state. Determine a quantity that, when you plot \(\lambda\) against it, should yield a straight line. Make your plot, establish a best-fit line, and use your line to determine the width of the square well. $$\begin{array}{|l|c|c|c|c|c|} \hline \text { Initial state, } n & 4 & 5 & 7 & 8 & 10 \\ \hline \text { Wavelength, } \lambda(\mathrm{nm}) & 1110 & 674 & 354 & 281 & 169 \\ \hline \end{array}$$

Find the probability that a particle in an infinite square well is located in the central one-fourth of the well for the quantum states \(n=(a) 1,(b) 2,(c) 5,\) and \((d) 20 .\) (e) What's the classical probability in this situation?

Does quantum tunneling violate energy conservation? Explain.

A particle is confined to a two-dimensional box whose sides are in the ratio \(1: 2 .\) Are any of its energy levels degenerate? If so, give an example. If not, why not?
See all solutions

Recommended explanations on Physics Textbooks

Astrophysics

Read Explanation

Electricity and Magnetism

Read Explanation

Fundamentals of Physics

Read Explanation

Force

Read Explanation

Translational Dynamics

Read Explanation

Particle Model of Matter

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.