Physics for Scientist and Engineers: A Strategic Approach

Randall Knight

Chapter 40

Wave Functions and Uncertainty - all with Video Answers

Educators

Chapter Questions

Problem 1

An experiment has four possible outcomes, labeled A to D. The probability of $A$ is $P_{A}=40 \%$ and of $B$ is $P_{B}=30 \% .$ Outcome C is twice as probable as outcome D. What are the probabilities $P_{\mathrm{C}}$ and $P_{\mathrm{D}} ?$

João Bravo

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Problem 2

Suppose you toss three coins into the air and let them fall on the floor. Each coin shows either a head or a tail.
a. Make a table in which you list all the possible outcomes of this experiment. Call the coins $\mathrm{A}, \mathrm{B},$ and $\mathrm{C} .$
b. What is the probability of getting two heads and one tail? Explain.
c. What is the probability of getting at least two heads?

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Problem 3

Suppose you draw a card from a regular deck of 52 cards.
a. What is the probability that you draw an ace?
b. What is the probability that you draw a spade?

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Problem 4

You are dealt 1 card each from 1000 decks of cards. What is the expected number of picture cards (jacks, queens, and kings)?

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Problem 5

Make a table in which you list all possible outcomes of rolling two dice. Call the dice A and B. What is the probability of rolling (a) a $7,$ (b) any double, and (c) a 6 or an 87 You can give the probabilities as fractions, such as $3 / 36.$

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Problem 6

In one experiment, 2000 photons are detected in a $0.10-\mathrm{mm}-$ wide strip where the amplitude of the electromagnetic wave is 10 V/m. How many photons are detected in a nearby 0.10-mmwide strip where the amplitude is $30 \mathrm{V} / \mathrm{m} ?$

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Problem 7

In one experiment, 6000 photons are detected in a $0.10-\mathrm{mm}-$ wide strip where the amplitude of the electromagnetic wave is 200 V/m. What is the wave amplitude at a nearby 0.20-mm-wide strip where 3000 photons are detected?

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Problem 8

$1.0 \times 10^{10}$ photons pass through an experimental apparatus. How many of them land in a 0.10 -mm-wide strip where the probability density is $20 \mathrm{m}^{-1} ?$

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Problem 9

When $5 \times 10^{12}$ photons pass through an experimental apparatus, $2.0 \times 10^{9}$ land in a 0.10 -mm-wide strip. What is the probability density at this point?

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Problem 10

What are the units of $\psi$ ? Explain.

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Problem 11

shows the probability density for an electron that has passed through an experimental apparatus. If $1.0 \times 10^{6}$ electrons are used, what is the expected number that will land in a 0.010 -mmwide strip at (a) $x=0.000 \mathrm{mm}$ and (b) $2.000 \mathrm{mm} ?$ (FIGURE CANNOT COPY)

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Problem 12

In an interference experiment with electrons, you find the most intense fringe is at $x=7.0 \mathrm{cm} .$ There are slightly weaker fringes at $x=6.0$ and $8.0 \mathrm{cm},$ still weaker fringes at $x=4.0$ and $10.0 \mathrm{cm},$ and two very weak fringes at $x=1.0$ and $13.0 \mathrm{cm} .$ No electrons are detected at $x<0 \mathrm{cm}$ or $x>14 \mathrm{cm}$
a. Sketch a graph of $|\psi(x)|^{2}$ for these electrons.
b. Sketch a possible graph of $\psi(x)$
c. Are there other possible graphs for $\psi(x) ?$ If so, draw one.

João Bravo

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Problem 13

shows the probability density for an electron that has passed through an experimental apparatus. What is the probability that the electron will land in a 0.010 -mm-wide strip
at (a) $x=0.000 \mathrm{mm},$ (b) $x=0.500 \mathrm{mm},$ (c) $x=1.000 \mathrm{mm},$ and
(d) $x=2.000 \mathrm{mm} ?$(FIGURE CANNOT COPY)

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Problem 14

Is a graph of $|\psi(x)|^{2}$ for an electron.
a. What is the value of $a ?$
b. Draw a graph of the wave function $\psi(x)$. (There is more than one acceptable answer.)
c. What is the probability that the electron is located between $x=1.0 \mathrm{nm}$ and $x=2.0 \mathrm{nm} ?$

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Problem 15

is a graph of $|\psi(x)|^{2}$ for a neutron.
a. What is the value of $a ?$
b. Draw a graph of the wave function $\psi(x)$. (There is more than one acceptable answer.)
c. What is the probability that the neutron is located at a position with $|x| \geq 2 \mathrm{fm} ?$

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Numerade Educator

Problem 16

Shows the wave function of an electron.
a. What is the value of $c ?$
b. Draw a graph of $|\psi(x)|^{2}$
c. What is the probability that the electron is located between $x=-1.0 \mathrm{nm}$ and $x=1.0 \mathrm{nm} ?$
(FIGURE CANNOT COPY)

João Bravo

Numerade Educator

Problem 17

Shows the wave function of a neutron.
a. What is the value of $c ?$
b. Draw a graph of $|\psi(x)|^{2}$
c. What is the probability that the neutron is located between $x=-1.0 \mathrm{mm}$ and $x=1.0 \mathrm{mm} ?$

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Problem 18

What minimum bandwidth is needed to transmit a pulse that consists of 100 cycles of a 1.00 MHz oscillation?

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Problem 19

A radio-frequency amplifier is designed to amplify signals in the frequency range $80 \mathrm{MHz}$ to $120 \mathrm{MHz}$. What is the shortestduration radio-frequency pulse that can be amplified without distortion?

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Problem 20

Sound waves of $498 \mathrm{Hz}$ and $502 \mathrm{Hz}$ are superimposed at a temperature where the speed of sound in air is $340 \mathrm{m} / \mathrm{s}$. What is the length $\Delta x$ of one wave packet?

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Problem 21

A 1.5 - $\mu$ m-wavelength laser pulse is transmitted through a 2.0-GHz-bandwidth optical fiber. How many oscillations are in the shortest-duration laser pulse that can travel through the fiber?

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Problem 22

What is the position uncertainty, in $\mathrm{nm}$, of an electron whose velocity is known to be between $3.48 \times 10^{5} \mathrm{m} / \mathrm{s}$ and $3.58 \times$ $10^{5} \mathrm{m} / \mathrm{s} ?$

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Problem 23

Andrea, whose mass is $50 \mathrm{kg}$, thinks she's sitting at rest in her 5.0 -m-long dorm room as she does her physics homework. Can Andrea be sure she's at rest? If not, within what range is her velocity likely to be?

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Problem 24

A thin solid barrier in the $x y$ -plane has a $10-\mu$ m-diameter circular hole. An electron traveling in the z-direction with $v_{x}=0 \mathrm{m} / \mathrm{s}$ passes through the hole. Afterward, is $v_{x}$ still zero? If not, within what range is $v_{x}$ likely to be?

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Problem 25

A proton is confined within an atomic nucleus of diameter $4.0 \mathrm{fm} .$ Use a one-dimensional model to estimate the smallest range of speeds you might find for a proton in the nucleus.

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Problem 26

A 1.0 -mm-diameter sphere bounces back and forth between two walls at $x=0 \mathrm{mm}$ and $x=100 \mathrm{mm} .$ 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
a. At exactly $x=50.0 \mathrm{mm} ?$
b. Between $x=49.0 \mathrm{mm}$ and $x=51.0 \mathrm{mm} ?$
c. At $x \geq 75$ mm?

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Problem 27

A radar antenna broadcasts clectromagnetic waves with a period of 0.100 ns. What range of frequencies would need to be superimposed to create a 1.0 -ns-long radar pulse?

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Problem 28

Ultrasound pulses of with a frequency of $1.000 \mathrm{MHz}$ are transmitted into water, where the speed of sound is $1500 \mathrm{m} / \mathrm{s}$ The spatial length of each pulse is $12 \mathrm{mm}$.
a. How many complete cycles are contained in one pulse?
b. What range of frequencies must be superimposed to create each pulse?

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Problem 29

shows a pulse train. The period of the pulse train is $T=2 \Delta t,$ where $\Delta t$ is the duration of each pulse. What is the maximum pulse-transmission rate (pulses per second) through an electronics system with a $200 \mathrm{kHz}$ bandwidth? (This is the bandwidth allotted to each FM radio station.) (FIGURE CANNOT COPY)

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Problem 30

Consider a single-slit diffraction experiment using electrons. (single-slit diffraction was described in Section $22.4 .)$ Using Figure 40.5 as a model, draw
a. A dot picture showing the arrival positions of the first 40 or 50 electrons.
b. A graph of $|\psi(x)|^{2}$ for the electrons on the detection screen.
c. A graph of $\psi(x)$ for the electrons. Keep in mind that $\psi,$ as a wave-like function, oscillates between positive and negative.

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Problem 31

An experiment finds electrons to be uniformly distributed over the interval $0 \mathrm{cm} \leq x \leq 2 \mathrm{cm},$ with no electrons falling outside this interval.
a. Draw a graph of $|\psi(x)|^{2}$ for these electrons.
b. What is the probability that an electron will land within the interval 0.79 to $0.81 \mathrm{cm} ?$
c. If $10^{6}$ electrons are detected, how many will be detected in the interval 0.79 to $0.81 \mathrm{cm} ?$
d. What is the probability density at $x=0.80 \mathrm{cm} ?$

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Problem 32

In an experiment with 10,000 electrons, which land symmetrically on both sides of $x=0,5000$ are detected in the range $-1.0 \mathrm{cm} \leq x \leq+1.0 \mathrm{cm}, 7500$ are detected in the range $-2.0 \mathrm{cm} \leq x \leq+2.0 \mathrm{cm},$ and all 10,000 are detected in the range $-3.0 \mathrm{cm} \leq x \leq+3.0 \mathrm{cm} .$ Draw a graph of a probability density that is consistent with these data. (There may be more than one acceptable answer.)

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Problem 33

shows $|\psi(x)|^{2}$ for the electrons in an experiment. a. Is the electron wave function normalized? Explain.
b. Draw a graph of $\psi(x)$ over this same interval. Provide a numerical scale on both axes. (There may be more than one acceptable answer.)
c. What is the probability that an electron will be detected in a $0.0010-\mathrm{cm}-$ wide region at $x=0.00 \mathrm{cm} ? \mathrm{At} x=0.50 \mathrm{cm} ? \mathrm{At}$
$x=0.999 \mathrm{cm} ?$
d. If $10^{4}$ electrons are detected, how many are expected to land in the interval $-0.30 \mathrm{cm} \leq x \leq 0.30 \mathrm{cm} ?$
(FIGURE CANNOT COPY)

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Numerade Educator

Problem 34

Shows the wave function of a particle confined between $x=0$ nm and $x=1.0$ nm. The wave function is zero outside this region.
a. Determine the value of the constant $c,$ as defined in the figure.
b. Draw a graph of the probability density $P(x)=|\psi(x)|^{2}$
c. Draw a dot picture showing where the first 40 or 50 particles might be found.
d. Calculate the probability of finding the particle in the interval $0.0 \mathrm{nm} \leq x \leq 0.3 \mathrm{nm}$

João Bravo

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Problem 35

shows the wave function of a particle confined between $x=-4.0 \mathrm{mm}$ and $x=4.0 \mathrm{mm} .$ The wave function is zero outside this region.
a. Determine the value of the constant $c,$ as defined in the figure.
b. Draw a graph of the probability density $P(x)=|\psi(x)|^{2}.$
c. Draw a dot picture showing where the first 40 or 50 particles might be found.
d. Calculate the probability of finding the particle in the interval $-2.0 \mathrm{mm} \leq x \leq 2.0 \mathrm{mm}.$
(FIGURE CANNOT COPY)

João Bravo

Numerade Educator

Problem 36

Shows the probability density for finding a particle at position $x$
a. Determine the value of the constant $a$, as defined in the figure.
b. At what value of $x$ are you most likely to find the particle? Explain.
c. Within what range of positions centered on your answer to part b are you $75 \%$ certain of finding the particle?
d. Interpret your answer to part $c$ by drawing the probability density graph and shading the appropriate region.

João Bravo

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Problem 37

An electron that is confined to $x \geq 0$ nm has the normalized wave function $$\psi(x)=\left\{\begin{array}{ll}0 & x<0 \mathrm{nm} \\\left(1.414 \mathrm{nm}^{-1 / 2}\right) e^{-x/(1.0 \mathrm{mm})} & x \geq 0 \mathrm{nm}\end{array}\right.$$
where $x$ is in $\mathrm{nm}$.
a. What is the probability of finding the electron in a $0.010-\mathrm{nm}-$ wide region at $x=1.0 \mathrm{nm} ?$
b. What is the probability of finding the electron in the interval $0.50 \mathrm{nm} \leq x \leq 1.50 \mathrm{nm} ?$

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Problem 38

A particle is described by the wave function $$\psi(x)=\left\{\begin{array}{ll}
c e^{x / L} & x \leq 0 \mathrm{mm} \\c e^{-x / L} & x \geq 0 \mathrm{mm}\end{array}\right.$$
where $L=2.0 \mathrm{mm}$
a. Sketch graphs of both the wave function and the probability density as functions of $x$
b. Determine the normalization constant $c$
c. Calculate the probability of finding the particle within
$1.0 \mathrm{mm}$ of the origin.
d. Interpret your answer to part $b$ by shading the region representing this probability on the appropriate graph in part a.

João Bravo

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Problem 39

Consider the electron wave function $$\psi(x)=\left\{\begin{array}{ll}c \sqrt{1-x^{2}} & |x| \leq 1 \mathrm{cm} \\0 & |x| \geq 1 \mathrm{cm}\end{array}\right.$$
where $x$ is in $\mathrm{cm}$.
a. Determine the normalization constant $c$
b. Draw a graph of $\psi(x)$ over the interval $-2 \mathrm{cm} \leq x \leq 2 \mathrm{cm} .$ Provide numerical scales on both axes.
c. Draw a graph of $|\psi(x)|^{2}$ over the interval $-2 \mathrm{cm} \leq x \leq$
2 cm. Provide numerical scales.
d. If $10^{4}$ electrons are detected, how many will be in the interval
$0.00 \mathrm{cm} \leq x \leq 0.50 \mathrm{cm} ?$

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Problem 40

Consider the electron wave function $$\psi(x)=\left\{\begin{array}{ll} c \sin \left(\frac{2 \pi x}{L}\right) & 0 \leq x \leq L \\0 & x<0 \text { or } x>L\end{array}\right.$$
a. Determine the normalization constant $c .$ Your answer will be in terms of $L$
b. Draw a graph of $\psi(x)$ over the interval $-L \leq x \leq 2 L$
c. Draw a graph of $|\psi(x)|^{2}$ over the interval $-L \leq x \leq 2 L$
d. What is the probability that an electron is in the interval $0 \leq x \leq L / 3 ?$

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Problem 41

The probability density for finding a particle at position $x$ is $$P(x)=\left\{\begin{array}{lr}
\frac{a}{(1-x)} & -1 \mathrm{mm} \leq x<0 \mathrm{mm} \\b(1-x) & 0 \mathrm{mm} \leq x \leq 1 \mathrm{mm}\end{array}\right.$$
and zero elsewhere.
a. You will learn in Chapter 41 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 $b ?$
b. Draw a graph of the probability density over the interval $$-2 \mathrm{mm} \leq x \leq 2 \mathrm{mm}$$
c. Determine values for $a$ and $b$.
d. What is the probability that the particle will be found to the left of the origin?

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Problem 42

A pulse of light is created by the superposition of many waves that span the frequency range $f_{0}-\frac{1}{2} \Delta f \leq f \leq f_{0}+\frac{1}{2} \Delta f,$ where $f_{0}=c / \lambda$ is called the center frequency of the pulse. Laser technology can generate a pulse of light that has a wavelength of $600 \mathrm{nm}$ and lasts a mere $6.0 \mathrm{fs}\left(1 \mathrm{fs}=1 \text { femtosecond }=10^{-15} \mathrm{s}\right)$
a. What is the center frequency of this pulse of light?
b. How many cycles, or oscillations, of the light wave are completed during the $6.0 \mathrm{fs}$ pulse?
c. What range of frequencies must be superimposed to create this pulse?
d. What is the spatial length of the laser pulse as it travels through space?
e. Draw a snapshot graph of this wave packet.

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Problem 43

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 $10 \mathrm{m} / \mathrm{s} ?$

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Problem 44

A small speck of dust with mass $1.0 \times 10^{-13} \mathrm{g}$ has fallen into the hole shown in FIGURE 40.44, 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 decpest hole of this width from which it would have a good chance to escape? (FIGURE CANNOT COPY)

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Problem 45

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 $1 \mathrm{mm}$. The scientists have been able to cool the atoms in an atom trap to a temperature of approximately
$1 \mathrm{nK},$ 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 atom in a 1.0 -mm-long box.
a. Estimate the smallest range of speeds you might find for a sodium atom in this box.
b. 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 $v_{\max }$ of the atoms in the trap is half the value you found in part a. Use this $v_{\operatorname{mat}}$ to estimate the temperature of the atoms when they've been cooled to the limit set by the uncertainty principle.

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Problem 46

You learned in Chapter 38 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.
a. 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.
b. What does your answer imply about the possibility that the nucleus contains electrons? Explain.

João Bravo

Numerade Educator

Problem 47

a. Starting with the expression $\Delta f \Delta t \approx 1$ for a wave packet, find an expression for the product $\Delta E \Delta t$ for a photon.
b. Interpret your expression. What does it tell you?
c. 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 42 that the time any particular atom spends in the excited state before emitting a photon is unpredictable, but the average lifetime \Deltat of many atoms can be determined. You can think of $\Delta t$ as being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is $\Delta t \approx 10$ 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.
d. What is the fractional uncertainty $\Delta E / E$ in the photon's energy?

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Problem 48

shows $1.0-\mu \mathrm{m}$ diameter dust particles $(m=$ $\left.1.0 \times 10^{-15} \mathrm{kg}\right)$ in a vacuum chamber. The dust particles are released from rest above
a $1.0-\mu$ m-diameter hole, fall through the hole (there's just barely room for the particles to go through), and land on a detector at distance $d$ below.
a. If the particles were purely classical, they would all land in the same $1.0-\mu \mathrm{m}$ -diameter circle. But quantum effects don't allow this. If $d=1.0 \mathrm{m},$ by how much does the diameter of the circle in which most dust particles land exceed $1.0 \mu \mathrm{m} ?$ Is this increase in diameter likely to be detectable?
b. Quantum effects would be noticeable if the detection-circle diameter increased by $10 \%$ to $1.1 \mu \mathrm{m}$. At what distance $d$ would the detector need to be placed to observe this increase in the diameter?

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Problem 49

The wave function of a particle is $$\psi(x)=\sqrt{\frac{b}{\pi\left(x^{2}+b^{2}\right)}}$$ where $b$ is a positive constant. Find the probability that the particle is located in the interval $-b \leq x \leq b.$

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Problem 50

The wave function of a particle is $$\psi(x)=\left\{\begin{array}{lr} \frac{b}{\left(1+x^{2}\right)} & -1 \mathrm{mm} \leq x<0 \mathrm{mm} \\ c(1+x)^{2} & 0 \mathrm{mm} \leq x \leq 1 \mathrm{mm} \end{array}\right.$$
and zero elsewhere.
a. You will learn in Chapter 41 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 $c ?$
b. Draw graphs of the wave function and the probability density over the interval $-2 \mathrm{mm} \leq x \leq 2 \mathrm{mm}$
c. What is the probability that the particle will be found to the right of the origin?

João Bravo

Numerade Educator

Problem 51

Consider the electron wave function $$\psi(x)=\left\{\begin{array}{ll}
c x & |x| \leq 1 \mathrm{nm} \\\frac{c}{x} & |x| \geq 1 \mathrm{nm}\end{array}\right.$$
where $x$ is in $\mathrm{nm}$.
a. Determine the normalization constant $c$
b. Draw a graph of $\psi(x)$ over the interval -5 nm $\leq x \leq 5$ nm. Provide numerical scales on both axes.
c. Draw a graph of $|\psi(x)|^{2}$ over the interval $-5 \mathrm{nm} \leq x \leq$
5 nm. Provide numerical scales.
d. If $10^{6}$ electrons are detected, how many will be in the interval $-1.0 \mathrm{nm} \leq x \leq 1.0 \mathrm{nm} ?$

João Bravo

Numerade Educator