91Ó°ÊÓ

FIGURE Q$40.1$shows the de Broglie waves of three equal-mass particles. Which one is moving most slowly? Explain.

The electrons in a rigid box emit photons of wavelength$1484nm$during the$3s2$ transition.

a. What kind of photons are they—infrared, visible, or ultraviolet?

b. How long is the box in which the electrons are confined.

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

The graph in FIGURE EX40.16 shows the potential-energy function U(x of a particle. Solution of the Schrödinger equation finds that the n=3 level has $E_3=0.5eV$and that the n=6 level has $E_6=2.0eV$.

a. Redraw this figure and add to it the energy lines for the n=3 and n=6 states.

b. Sketch the n=3 and n=6 wave functions. Show them as oscillating about the appropriate energy line.

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

An electron in a rigid box absorbs light. The longest wavelength in the absorption spectrum is$600nm$. How long is the box?

What is the probability that an electron will tunnel through a $0.45\mathrm{nm}$gap from a metal to a STM probe if the work function is $4.0\mathrm{eV}$?

A particle confined in a rigid one-dimensional box of length $10\mathrm{fm}$has an energy level ${\mathrm{E}}_{\mathrm{n}}=32.9\mathrm{MeV}$and an adjacent energy level ${\mathrm{E}}_{\mathrm{n}+1}=51.4\mathrm{MeV}$.

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 $\mathrm{n}+1→\mathrm{n}$transition? Compare this to a typical visible-light wavelength.

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

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

a. Sketch graphs of ${\left|\mathrm{ψ}\left(\mathrm{x}\right)\right|}^2$. Label the points $\mathrm{x}=0$and $\mathrm{x}=\mathrm{L}$.

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 ${\left|\mathrm{ψ}\left(\mathrm{x}\right)\right|}^2$graphs, if the probability of finding the particle in the left one-third of the box is less than, equal to, or greater than $\frac{1}{3}$. Explain your reasoning.

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

A neutron is confined in a $10\mathrm{fm}$-diameter nucleus. If the nucleus is modeled as a one-dimensional rigid box, what is the probability that a neutron in the ground state is less than $2.0\mathrm{fm}$ from the edge of the nucleus?