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Chapter 40: Q. 32 (page 1176)

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?

Short Answer

Expert verified

here,n=4and the energy diagram is given below.

Wavelength is not in visible range and the particle is seem to be proton.

Step by step solution

01

Part (a) step 1: Given information

We have given,

En=32.9MeVEn+1=51.4MeV

we have to find the value of n and n+1.

02

Simplify

Since the energy is one dimensional potential box is given by,

En=h2n28mL2En+1=h2(n+1)28mL2

Divided the both equation

En+1En=(n+1)2n251.432.9=(n+1)2n21.25n=n+10.25n=1n=4n+1=5

03

Part (b) step 1: Given information

We have to draw the energy level diagram.

04

Simplify

05

Part (c) Step 1: Given information

We have to sketch the wave functions.

06

Simplify

07

Part (d) Step 1: Given information

We have to find the wavelength of the wave.

08

Simplify

We know that we can write,

λn→m=hcEn-Emλ5→4=1240eV.nm(51.4-32.9)×106eVλ=67×10-15m

Its is not a visible wavelength.

09

Part (e) Step 1: Given information

We have to find the mass of the particle.

10

Simplify

Since,

E4=32.9MeVE4=16×(6.62×10-34J.s)22m×10-30mm≈10-27kg

This particle can be proton.

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Most popular questions from this chapter

Model an atom as an electron in a rigid box of length 0.100nm, roughly twice the Bohr radius.

a. What are the four lowest energy levels of the electron?

b. Calculate all the wavelengths that would be seen in the emission spectrum of this atom due to quantum jumps between these four energy levels. Give each wavelength a label λn→mto indicate the transition.

c. Are these wavelengths in the infrared, visible, or ultraviolet portion of the spectrum?

d. The stationary states of the Bohr hydrogen atom have negative energies. The stationary states of this model of the atom have positive energies. Is this a physically significant difference? Explain.

e. Compare this model of an atom to the Bohr hydrogen atom. In what ways are the two models similar? Other than the signs of the energy levels, in what ways are they different?

Verify that the n=1 wave function ψ1(x) of the quantum harmonic oscillator really is a solution of the Schrödinger equation. That is, show that the right and left sides of the Schrödinger equation are equal if you use the ψ1(x) wave function.

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.

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.

An electron confined in a harmonic potential well emits a 1200nm photon as it undergoes a 3→2quantum jump. What is the spring constant of the potential well?
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