|| CALC Suppose that ${\psi }_1\left(x\right)$ and ${\psi }_2\left(x\right)$ are both solutions to the Schrödinger equation for the same potential energy $U\left(x\right)$. Prove that the superposition $\psi \left(x\right)=A{\psi }_1\left(x\right)+B{\psi }_2\left(x\right)$ is also a solution to the Schrödinger equation.

|| A $2.0\text{-}\mu \text{m-diameter}$ water droplet is moving with a speed of $1.0\mu \text{m}\text{/}\text{s}$ in a $20\text{-}\mu \text{m-long}$ box.

1. Estimate the particle’s quantum number.

2. Use the correspondence principle to determine whether quantum mechanics is needed to understand the particle’s motion or if it is “safe” to use classical physics.

|| 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:

1. Draw a graph of $U\left(x\right)$ versus $x$. Center your graph at $x=0$.

2. 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 $E_2$, then, on this line, sketch a plausible graph of the $n=2$ wave function.

3. 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 $E_3$, then, on this line, sketch a plausible graph of the $n=3$ wave function.

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

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

2. 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 $\lambda {}_{n\to m}$ to indicate the transition.

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

4. 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.

5. 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?

||

1. Derive an expression for $\lambda {}_{2\to 1}$, the wavelength of light emitted by a particle in a rigid box during a quantum jump from $n=2$ to $n=1$.

2. In what length rigid box will an electron undergoing a $2\to 1$ transition emit light with a wavelength of 694 nm? This is the wavelength of a ruby laser.

|| CALC Show that the normalization constant $A_n$ for the wave functions of a particle in a rigid box has the value given in Equation 40.26.

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

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

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

3. Sketch the $n+1$ wave function on the $n+1$ energy level.

4. What is the wavelength of a photon emitted in the $n+1\to n$ transition? Compare this to a typical visible-light wavelength.

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

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

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

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

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

4. Determine, by examining your $\left|\psi \left(x\right)\right|{}^2$ 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.

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

|| In most metals, the atomic ions form a regular arrangement called a crystal lattice. The conduction electrons in the sea of electrons move through this lattice. FIGURE P40.34 is a one-dimensional model of a crystal lattice. The ions have mass $m$, charge $e$, and an equilibrium separation $b$.

1. Suppose the middle charge is displaced a very small distance $(x\ll b)$ from its equilibrium position while the outer charges remain fixed. Show that the net electric force on the middle charge is given approximately by

   $$F=-{\displaystyle \frac{e^2}{b^3\pi {\epsilon }_0}}x$$

   In other words, the charge experiences a linear restoring force.

2. Suppose this crystal consists of aluminum ions with an equilibrium spacing of 0.30 nm. What are the energies of the four lowest vibrational states of these ions?

3. What wavelength photons are emitted during quantum jumps between adjacent energy levels? Is this wavelength in the infrared, visible, or ultraviolet portion of the spectrum?

FIGURE P40.34

|| For the quantum-well laser of Figure 40.16, estimate the probability that an electron will be found within one of the GaAlAs layers rather than in the GaAs layer. Explain your reasoning.

||| For a particle in a finite potential well of width $L$ and depth $U_0$, what is the ratio of the probability $\text{Prob(in}\delta x\text{at }x=L+\eta )$ to the probability $\text{Prob(in}\delta x\text{at }x=L)?$

|| A typical electron in a piece of metallic sodium has energy $-E_0$ compared to a free electron, where $E_0$ is the 2.36 eV work function of sodium.

1. At what distance beyond the surface of the metal is the electron’s probability density 10% of its value at the surface?

2. How does this distance compare to the size of an atom?

|| INT 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 $n=1$ ground state.

||

1. CALC Determine the normalization constant $A_1$ for the $n=1$ ground-state wave function of the quantum harmonic oscillator. Your answer will be in terms of $b$.

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

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

Hint: It helps to simplify the integral by making a change of variables to $u=x\text{/}b$.

||

1. INT Derive an expression for the classical probability density $P_{\text{class}}\left(x\right)$ for a simple harmonic oscillator with amplitude $A$.

2. Graph your expression between $x=-A$ and $x=+A$.

3. Interpret your graph. Why is it shaped as it is?

||

1. Derive an expression for the classical probability density $P_{\text{class}}\left(y\right)$ for a ball that bounces between the ground and height $h$. The collisions with the ground are perfectly elastic.

2. Graph your expression between $y=0$ and $y=h$.

3. Interpret your graph. Why is it shaped as it is?

|| CALC A particle of mass $m$ has the wave function

when it is in an allowed energy level with $E=0$.

1. Draw a graph of $\psi \left(x\right)$ versus $x$.

2. At what value or values of $x$ is the particle most likely to be found?

3. Find and graph the potential-energy function $U\left(x\right)$.

|| Figure 40.17 showed that a typical nuclear radius is 4.0 fm. As you’ll learn in Chapter 42, a typical energy of a neutron bound inside the nuclear potential well is $E_n=-20\text{MeV}$. To find out how “fuzzy” the edge of the nucleus is, what is the neutron’s penetration distance into the classically forbidden region as a fraction of the nuclear radius?

|| Even the smoothest mirror finishes are “rough” when viewed at a scale of 100 nm. When two very smooth metals are placed in contact with each other, the actual distance between the surfaces varies from 0 nm at a few points of real contact to $\approx 100\text{nm}$. The average distance between the surfaces is $\approx 50\text{nm}$. The work function of aluminum is 4.3 eV. What is the probability that an electron will tunnel between two pieces of aluminum that are 50 nm apart? Give your answer as a power of 10 rather than a power of $e$.

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

||| In a nuclear physics experiment, a proton is fired toward a $Z=13$ nucleus with the diameter and neutron energy levels shown in Figure 40.17. The nucleus, which was initially in its ground state, subsequently emits a gamma ray with wavelength $1.73\times {10}^{-4}\text{nm}$. What was the minimum initial speed of the proton?

Hint: Don’t neglect the proton-nucleus collision.

||| CALC A neutron is confined in a 10-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 fm from the edge of the nucleus?

|||

1. What is the probability that an electron will tunnel through a 0.50 nm air gap from a metal to a STM probe if the work function is 4.0 eV?

2. The probe passes over an atom that is 0.050 nm “tall.” By what factor does the tunneling current increase?

3. If a 10% current change is reliably detectable, what is the smallest height change the STM can detect?

||| Tennis balls traveling faster than 100 mph routinely bounce off tennis rackets. At some sufficiently high speed, however, the ball will break through the strings and keep going. The racket is a potential-energy barrier whose height is the energy of the slowest string-breaking ball. Suppose that a 100 g tennis ball traveling at 200 mph is just sufficient to break the 2.0-mm-thick strings. Estimate the probability that a 120 mph ball will tunnel through the strings without breaking them. Give your answer as a power of 10 rather than a power of $e$.