| A 1.0-mm-diameter sphere bounces back and forth between two walls at $x=0$ mm and $x=100\text{mm}$. The collisions are perfectly elastic, and the sphere repeats this motion over and over with no loss of speed. At a random instant of time, what is the probability that the center of the sphere is

1. At exactly $x=50.0\text{mm}?$

2. Between $x=49.0\text{mm}$ and $x=51.0\text{mm}?$

3. At $x\ge 75\text{mm}?$

|| INT Ultrasound pulses with a frequency of 1.000 MHz are transmitted into water, where the speed of sound is $1500\text{m}\text{/}\text{s}$. The spatial length of each pulse is 12 mm.

1. How many complete cycles are contained in one pulse?

2. What range of frequencies must be superimposed to create each pulse?

|| FIGURE P39.28 shows a pulse train. The period of the pulse train is $T=2\text{Δ}t$, where $\text{Δ}t$ is the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an electronics system with a 200 kHz bandwidth? (This is the bandwidth allotted to each FM radio station.)

FIGURE P39.28

|| INT Consider a single-slit diffraction experiment using electrons. (Single-slit diffraction was described in Section 33.4.) Using Figure 39.5 as a model, draw

1. A dot picture showing the arrival positions of the first 40 or 50 electrons.

2. A graph of $\left|\psi \left(x\right)\right|{}^2$ for the electrons on the detection screen.

3. A graph of $\psi \left(x\right)$ for the electrons. Keep in mind that $\psi$, as a wave-like function, oscillates between positive and negative.

|| An experiment finds electrons to be uniformly distributed over the interval $0\text{cm}\le x\le 2\text{cm,}$ with no electrons falling outside this interval.

1. Draw a graph of $\left|\psi \left(x\right)\right|{}^2$ for these electrons.

2. What is the probability that an electron will land within the interval 0.79 to 0.81 cm?

3. If ${10}^6$ electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm?

4. What is the probability density at $x=0.80\text{cm}?$

|| FIGURE P39.31 shows the wave function of a particle confined between $x=0\text{nm}$ and $x=1.0\text{nm}$. The wave function is zero outside this region.

1. Determine the value of the constant $c$, as defined in the figure.

2. Draw a graph of the probability density $P\left(x\right)=\left|\psi \left(x\right)\right|{}^2$.

3. Draw a dot picture showing where the first 40 or 50 particles might be found.

4. Calculate the probability of finding the particle in the interval $0\text{nm}\le x\le 0.25\text{nm}$.

FIGURE P39.31

|| FIGURE P39.32 shows $\left|\psi \left(x\right)\right|{}^2$ for the electrons in an experiment.

1. Is the electron wave function normalized? Explain.

2. Draw a graph of $\psi \left(x\right)$ over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)

3. What is the probability that an electron will be detected in a 0.0010-cm-wide region at $x=0.00\text{cm}?$ At $x=0.50\text{cm}?$ At $x=0.999\text{cm}?$

4. If ${10}^4$ electrons are detected, how many are expected to land in the interval $-0.30\text{cm}\le x\le 0.30\text{cm}?$

FIGURE P39.32

|| FIGURE P39.33 shows the probability density for finding a particle at position $x$.

1. Determine the value of the constant $a$, as defined in the figure.

2. At what value of $x$ are you most likely to find the particle? Explain.

3. Within what range of positions centered on your answer to part b are you 75% certain of finding the particle?

4. Interpret your answer to part c by drawing the probability density graph and shading the appropriate region.

FIGURE P39.33

||| FIGURE P39.34 shows the wave function of a particle confined between $x=-4.0\text{mm}$ and $x=4.0\text{mm}$. The wave function is zero outside this region.

1. Determine the value of the constant $c$, as defined in the figure.

2. Draw a graph of the probability density $P\left(x\right)=\left|\psi \left(x\right)\right|{}^2$.

3. Draw a dot picture showing where the first 40 or 50 particles might be found.

4. Calculate the probability of finding the particle in the interval $-2.0\text{mm}\le x\le 2.0\text{mm}$.

FIGURE P39.34

||| CALC An electron that is confined to $x\ge 0\text{nm}$ has the normalized wave function

where $x$ is in nm.

1. What is the probability of finding the electron in a 0.010-nm-wide region at $x=1.0\text{nm}?$

2. What is the probability of finding the electron in the interval $0.50\text{nm}\le x\le 1.50\text{nm}?$

|| CALC Consider the electron wave function

where $x$ is in cm.

1. Determine the normalization constant $c$.

2. Draw a graph of $\psi \left(x\right)$ over the interval $-2\text{cm}\le x\le$ 2 cm. Provide numerical scales on both axes.

3. Draw a graph of $\left|\psi \left(x\right)\right|{}^2$ over the interval $-2\text{ cm}\le x\le 2\text{ cm}$. Provide numerical scales.

4. If ${10}^4$ electrons are detected, how many will be in the interval $0.00\text{cm}\le x\le 0.50\text{cm}?$

|| CALC Consider the electron wave function

1. Determine the normalization constant $c$. Your answer will be in terms of $L$.

2. Draw a graph of $\psi \left(x\right)$ over the interval $-L\le x\le 2L$.

3. Draw a graph of $\left|\psi \left(x\right)\right|{}^2$ over the interval $-L\le x\le 2L$.

4. What is the probability that an electron is in the interval $0\le x\le L\text{/}3?$

||| CALC A particle is described by the wave function

where $L=2.0\text{mm}$.

1. Sketch graphs of both the wave function and the probability density as functions of $x$.

2. Determine the normalization constant $c$.

3. Calculate the probability of finding the particle within 1.0 mm of the origin.

4. Interpret your answer to part b by shading the region representing this probability on the appropriate graph in part a.

|| CALC The probability density for finding a particle at position $x$ is

and zero elsewhere.

1. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about the relationship between $a$ and $b$?

2. Determine values for $a$ and b.

3. Draw a graph of the probability density over the interval $-2\text{mm}\le x\le 2\text{mm}$.

4. What is the probability that the particle will be found to the left of the origin?

|| A pulse of light is created by the superposition of many waves that span the frequency range $f_0-\frac{1}{2}\text{ Δ}f\le f\le f_0+\frac{1}{2}\text{ Δ}f$, where $f_0=c\text{/}\text{λ}$ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of 600 nm and lasts a mere 6.0 fs $(1\text{ fs}=1\text{ femtosecond }={10}^{-15}\text{ s})$.

1. What is the center frequency of this pulse of light?

2. How many cycles, or oscillations, of the light wave are completed during the 6.0 fs pulse?

3. What range of frequencies must be superimposed to create this pulse?

4. What is the spatial length of the laser pulse as it travels through space?

5. Draw a snapshot graph of this wave packet.

||| What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electron’s speed is no more than $10\text{m}\text{/}\text{s}?$ Give your answer to one significant figure, in $\mu \text{m}$.

|| INT You learned in Chapter 37 that, except for hydrogen, the mass of a nucleus with atomic number $Z$ is larger than the mass of the $Z$ protons. The additional mass was ultimately discovered to be due to neutrons, but prior to the discovery of the neutron it was suggested that a nucleus with mass number $A$ might contain $A$ protons and $(A-Z)$ electrons. Such a nucleus would have the mass of $A$ protons, but its net charge would be only Ze.

1. We know that the diameter of a nucleus is approximately 10 fm. Model the nucleus as a one-dimensional box and find the minimum range of speeds that an electron would have in such a box. Give your answer to one significant figure. Use a classical analysis, even though a relativistic analysis would be more appropriate.

2. What does your answer imply about the possibility that the nucleus contains electrons? Explain.

||| Heavy nuclei often undergo alpha decay in which they emit an alpha particle (i.e., a helium nucleus). Alpha particles are so tightly bound together that it’s reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted.

1. A ${}^{238}\text{U}$ nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a ${}^{238}\text{U}$ nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have?

2. The probability that a nucleus will undergo alpha decay is proportional to the frequency with which the alpha particle reflects from the walls of the nucleus. What is that frequency (reflections/s) for a maximum-speed alpha particle within a ${}^{238}\text{U}$ nucleus?

|| INT Physicists use laser beams to create an atom trap in which atoms are confined within a spherical region of space with a diameter of about 1 mm. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately 1 nK, which is extremely close to absolute zero, but it would be interesting to know if this temperature is close to any limit set by quantum physics. We can explore this issue with a one-dimensional model of a sodium atoms in a 1-mm-long box.

1. Estimate the smallest range of speeds you might find for a sodium atom in this box. The mass of a sodium atom is 23 u.

2. Even if we do our best to bring a group of sodium atoms to rest, individual atoms will have speeds within the range you found in part a. Because there’s a distribution of speeds, suppose we estimate that the root-mean-square speed $v_{\text{rms}}$ of the atoms in the trap is half the value you found in part a. Use this $v_{\text{rms}}$ to estimate the temperature of the atoms when they’ve been cooled to the limit set by the uncertainty principle.

|| A small speck of dust with mass $1\times {10}^{-13}\text{g}$ has fallen into the hole shown in FIGURE P39.45 and appears to be at rest. According to the uncertainty principle, could this particle have enough energy to get out of the hole? If not, what is the deepest hole of this width from which it would have a good chance to escape?

FIGURE P39.45

||

1. INT Starting with the expression $\text{Δ}f\text{Δ}t\approx 1$ for a wave packet, find an expression for the product $\text{Δ}E\text{Δ}t$ for a photon.

2. Interpret your expression. What does it tell you?

3. The Bohr model of atomic quantization says that an atom in an excited state can jump to a lower-energy state by emitting a photon. The Bohr model says nothing about how long this process takes. You’ll learn in Chapter 41 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime $\text{Δ}t$ of many atoms can be determined. You can think of $\text{Δ}t$ as being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is $\text{Δ}t\approx 10\text{ns}$. Consider an atom that emits a photon with a 500 nm wavelength as it jumps down from an excited state. What is the uncertainty in the energy of the photon? Give your answer in eV.

4. What is the fractional uncertainty $\text{Δ}E\text{/}E$ in the photon’s energy?

||| CALC The probability density of finding a particle somewhere along the $x$-axis is 0 for $x<1\text{mm}$. At $x=1\text{mm,}$ the probability density is $c$. For $x\ge 1\text{mm,}$ the probability density decreases by a factor of 8 each time the distance from the origin is doubled. What is the probability that the particle will be found in the interval $2\text{mm}\le x\le 4\text{mm}?$

||| CALC The wave function of a particle is

where $b$ is a positive constant. Find the probability that the particle is located in the interval $-b\le x\le b$.

||| CALC The wave function of a particle is

and zero elsewhere.

1. You will learn in Chapter 40 that the wave function must be a continuous function. Assuming that to be the case, what can you conclude about the relationship between $b$ and $c$?

2. Draw graphs of the wave function and the probability density over the interval $-2\text{mm}\le x\le 2\text{mm}$.

3. What is the probability that the particle will be found to the right of the origin?

||| CALC Consider the electron wave function

where $x$ is in nm.

1. Determine the normalization constant $c$.

2. Draw a graph of $\psi \left(x\right)$ over the interval $-5\text{nm}\le x\le 5\text{nm}$. Provide numerical scales on both axes.

3. Draw a graph of $\left|\psi \left(x\right)\right|{}^2$ over the interval $-5\text{ nm}\le x\le 5\text{ nm}$. Provide numerical scales.

4. If ${10}^6$ electrons are detected, how many will be in the interval $-1.0\text{nm}\le x\le 1.0\text{nm}?$

||| Soot particles, from incomplete combustion in diesel engines, are typically 15 nm in diameter and have a density of $1200{\text{kg/m}}^3$. FIGURE CP39.51 shows soot particles released from rest, in vacuum, just above a thin plate with a $0.5\text{-}\mu \text{m-diameter}$ hole—roughly the wavelength of visible light. After passing through the hole, the particles fall distance d and land on a detector. If soot particles were purely classical, they would fall straight down and, ideally, all land in a $0.5\text{-}\mu \text{m-diameter}$ circle. Allowing for some experimental imperfections, any quantum effects would be noticeable if the circle diameter were $2\mu m$. How far would the particles have to fall to fill a circle of this diameter?

FIGURE CP39.51