Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 40: Q. 6 (page 1174)

Consider a quantum harmonic oscillator.

a. What happens to the spacing between the nodes of the wave function as |x| increases? Why?

b. What happens to the heights of the antinodes of the wave function as |x| increases? Why?

c. Sketch a reasonably accurate graph of the n=8 wave function of a quantum harmonic oscillator.

Short Answer

Expert verified

Electron's quantum state after partition isn=6

Step by step solution

01

Step 1. Given information

Energy of particle confined in a box of length L=

En=n2h28mL2

Here,

En= energy of nthstate,

h= plank's constant,

m=mass of particle,

L=length of box and

n denotes the state of particle (harmonics) (n=1,2,3 ...)

02

Step 2. Finding the electron's quantum state

From the law of conservation of energy,

the energy in 4nmsection should be equal to the energy in 12nmsection.

For 4nm

E2=(2)2h28m(4nm)2

For 12nm

En=n2h28m(12nm)2

(2)2h28m(4nm)2=n2h28m(12nm)2

(2)2(4nm)2=n2(12nm)2

n2=(2)2(12nm)2(4nm)2

n=6

Thus, the electron's quantum state after partition =6

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

Suppose that ψ1(x)and ψ2(x)are both solutions to the Schrödinger equation for the same potential energy U(x). Prove that the superposition ψ(x)=Aψ1(x)+Bψ2(x)is also a solution to the Schrödinger equation.

A proton’s energy is 1.0MeV below the top of a 10fm-wide energy barrier. What is the probability that the proton will tunnel through the barrier?

An electron has a 0.0100 probability (a 1.00% chance) of tunneling through a potential barrier. If the width of the barrier is doubled, will the tunneling probability be0.0050,0.0025or0.0001? Explain.

Figure 40.27a modeled a hydrogen atom as a finite potential well with rectangular edges. A more realistic model of a hydrogen atom, although still a one-dimensional model, would be the electron + proton electrostatic potential energy in one dimension:

U(x)=-e24πε0x

a. Draw a graph of U(x) versus x. Center your graph at x=0.

b. Despite the divergence at x=0, the Schrödinger equation can be solved to find energy levels and wave functions for the electron in this potential. Draw a horizontal line across your graph of part a about one-third of the way from the bottom to the top. Label this line E2, then, on this line, sketch a plausible graph of the n=2wave function.

c. Redraw your graph of part a and add a horizontal line about two-thirds of the way from the bottom to the top. Label this line E3, then, on this line, sketch a plausible graph of the n=3 wave function.

| FIGURE EX40.4 shows the wave function of an electron in a rigid box. The electron energy islocalid="1650137157775" 12.0eV. What is the energy, in localid="1650137162096" eV, of the next higher state?
See all solutions

Recommended explanations on Physics Textbooks

Nuclear Physics

Read Explanation

Fundamentals of Physics

Read Explanation

Radiation

Read Explanation

Oscillations

Read Explanation

Translational Dynamics

Read Explanation

Fields in 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