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Chapter 40: Q. 38 (page 1176)

Show that the constant b used in the quantum-harmonic-oscillator wave functions (a) has units of length and (b) is the classical turning point of an oscillator in the n=1ground state.

Short Answer

Expert verified

All the statements are proved.

Step by step solution

01

Part (a) Step 1: Given information 

We have given,

System of quantum harmonic oscillator.

We have to find the unit of constant b.

02

Simplify

Since we know that the value of b is given by,

b=h2πmω

Where

h has unit J.s and m has unit of kg and w is in per second then,

bunit=ML2T-1MT-112bunit=L

Hence proved.

03

Part (b) Step 1: Given information

We have to show that b is classical turning point for ground state.

04

Simplify

To show that the classical turning point is b is ,

E=12kx2=12mω2A2A=2Emω2A=2mω2(n+1)hω2πA=(2n+1)h2πmωfor,n=0A=h2πmω=bb=x

Hence proved.

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Most popular questions from this chapter

A particle of mass m has the wave functionψx=Axexp-x2a2 when it is in an allowed energy level with E=0.

a. Draw a graph of ψxversusx.

b. At what value or values of xis the particle most likely to be found?

c. Find and graph the potential-energy function Ux.

A particle confined in a rigid one-dimensional box of length 10fmhas an energy level En=32.9MeVand an adjacent energy level En+1=51.4MeV.

a. Determine the values of n and n + 1.

b. Draw an energy-level diagram showing all energy levels from 1 through n + 1. Label each level and write the energy beside it.

c. Sketch the n + 1 wave function on the n + 1 energy level.

d. What is the wavelength of a photon emitted in the n+1→ntransition? Compare this to a typical visible-light wavelength.

e. What is the mass of the particle? Can you identify it?

Sketch the n=4 wave function for the potential energy shown in FIGURE EX40.13.

a. Determine the normalization constant A1for the n=1ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of b.

b. Write an expression for the probability that a quantum harmonic oscillator in its n=1ground state will be found in the classically forbidden region.

c. (Optional) Use a numerical integration program to evaluate your probability expression of part b.

Consider a particle in a rigid box of length L. For each of the states n=1,n=2,and n=3:

a. Sketch graphs of ψ(x)2. Label the points x=0and x=L.

b. Where, in terms of L, are the positions at which the particle is most likely to be found?

c. Where, in terms of L, are the positions at which the particle is least likely to be found?

d. Determine, by examining your ψ(x)2graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than 13. Explain your reasoning.

e. Calculate the probability that the particle will be found in the left one-third of the box
See all solutions

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