The wave function of a particle is c1x2 = c b 11 + x2 2 -1

What is quantum mechanics?

What is a wave function?

What role does probability play?

How are waves and particles reconciled?

What is Heisenbergs uncertainty principle?

Suppose you roll a die 30 times. What is the expected number of 1s and 6s?

In an experiment, 6000 out of 600,000 photons are detected in a 1.0-mm-wide strip located at position x = 50 cm. What is the probability density at x = 50 cm?

The figure shows the detection of photons in an optical experiment. Rank in order, from largest to smallest, the square of the amplitude function of the electromagnetic wave at positions A, B, C, and D.

FIGURE 39.9 shows the wave function of a particle confined within the region between x = 0 nm and x = L = 1.0 nm. The wave function is zero outside this region. a. Determine the value of the constant c that makes this a normalized wave function. b. Draw a graph of the probability density P1x2. 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 a region of width dx = 0.01 nm at positions x1 = 0.05 nm, x2 = 0.50 nm, and x3 = 0.95 nm.

This is the wave function of a neutron. At what value of x is the neutron most likely to be found?

A short-wave radio station broadcasts at a frequency of 10.000 MHz. What is the range of frequencies of the waves that must be superimposed to broadcast a radio-wave pulse lasting 0.800 ms?

The value of the constant a is a. a = 2.0 mm-1 b. a = 1.0 mm-1 c. a = 0.5 mm-1 d. a = 2.0 mm-1/2 e. a = 1.0 mm-1/2 f. a = 0.5 mm-1/2

A 1.0@mm@diameter dust particle 1m 10-15 kg2 is confined within a 10@mm@long box. Can we know with certainty if the particle is at rest? If not, within what range is its velocity likely to be found?

What minimum bandwidth must a medium have to transmit a 100-ns-long pulse? a. 1 MHz b. 10 MHz c. 100 MHz d. 1000 MHz

What range of velocities might an electron have if confined to a 0.1-nm-wide region, about the size of an atom?

Which of these particles, A or B, can you locate more precisely? 0 x c(x) A 0 x c(x) B

A particle is described by the wave function c1x2 = b 0 x 6 0 ce-x/L x 0 where L = 1 nm. a. Determine the value of the constant c. b. Draw graphs of c1x2 and the probability density P1x2. c. If 106 particles are detected, how many are expected to be found in the region x 1 nm?

FIGURE Q39.1 shows the probability density for photons to be detected on the x-axis. a. Is a photon more likely to be detected at x = 0 m or at x = 1 m? Explain. b. One million photons are detected. What is the expected number of photons in a 1-mm-wide interval at x = 0.50 m?

What is the difference between the probability and the probability density?

For the electron wave function shown in FIGURE Q39.3, at what position or positions is the electron most likely to be found? Least likely to be found? Explain.

FIGURE Q39.4 shows the dot pattern of electrons landing on a screen. a. At what value or values of x is the electron probability density at maximum? Explain. b. Can you tell at what value or values of x the electron wave function c1x2 is most positive? If so, where? If not, why not?

What is the value of the constant a in FIGURE Q39.5?

FIGURE Q39.6 shows wave packets for particles 1, 2, and 3. Which particle can have its velocity known most precisely? Explain.

An experiment has four possible outcomes, labeled A to D. The probability of A is PA = 40, and of B is PB = 30,. Outcome C is twice as probable as outcome D. What are the probabilities PC and PD?

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 A, B, and C. b. What is the probability of getting two heads and one tail? c. What is the probability of getting at least two heads?

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

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?

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 8? You can give the probabilities as fractions, such as 3/36

In one experiment, 6000 photons are detected in a 0.10-mmwide 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?

In one experiment, 2000 photons are detected in a 0.10-mmwide 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 V/m?

1.0 * 1010 photons pass through an experimental apparatus. How many of them land in a 0.10-mm-wide strip where the probability density is 20 m-1 ?

When 5 * 1012 photons pass through an experimental apparatus, 2.0 * 109 land in a 0.10-mm-wide strip. What is the probability density at this point?

What are the units of c? Explain

In an interference experiment with electrons, you find the most intense fringe is at x = 7.0 cm. There are slightly weaker fringes at x = 6.0 and 8.0 cm, still weaker fringes at x = 4.0 and 10.0 cm, and two very weak fringes at x = 1.0 and 13.0 cm. No electrons are detected at x 6 0 cm or x 7 14 cm. a. Sketch a graph of 0 c1x2 0 2 for these electrons. b. Sketch a possible graph of c1x2. c. Are there other possible graphs for c1x2? If so, draw one

FIGURE EX39.12 shows the probability density for an electron that has passed through an experimental apparatus. If 1.0 * 106 electrons are used, what is the expected number that will land in a 0.010-mm-wide strip at (a) x = 0.000 mm and (b) 2.000 mm?

FIGURE EX39.13 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 mm, (b) x = 0.500 mm, (c) x = 1.000 mm, and (d) x = 2.000 mm?

FIGURE EX39.14 is a graph of 0 c1x2 0 2 for an electron. a. What is the value of a? b. Draw a graph of the wave function c1x2. (There is more than one acceptable answer.) c. What is the probability that the electron is located between x = 1.0 nm and x = 2.0 nm?

FIGURE EX39.15 is a graph of 0 c1x2 0 2 for a neutron. a. What is the value of a? b. Draw a graph of the wave function c1x2. (There is more than one acceptable answer.) c. What is the probability that the neutron is located at a position with 0 x 0 2 fm?

FIGURE EX39.17 shows the wave function of an electron. a. What is the value of c? b. Draw a graph of 0 c1x2 0 2 . c. What is the probability that the electron is located between x = -1.0 nm and x = 1.0 nm?

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

A 1.5@mm@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?

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

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

A thin solid barrier in the xy-plane has a 10@mm@diameter circular hole. An electron traveling in the z-direction with vx = 0 m/s passes through the hole. Afterward, is it certain that vx is still zero? If not, within what range is vx likely to be?

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

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

What is the minimum uncertainty in position, in nm, of an electron whose velocity is known to be between 3.48 * 105 m/s and 3.58 *105 m/s?

A 1.0-mm-diameter sphere bounces back and forth between two walls at x = 0 mm and x = 100 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 mm? b. Between x = 49.0 mm and x = 51.0 mm? c. At x 75 mm?

Ultrasound pulses with a frequency of 1.000 MHz are transmitted into water, where the speed of sound is 1500 m/s. The spatial length of each pulse is 12 mm. a. How many complete cycles are contained in one pulse? b. 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 t, where 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.)

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 a. A dot picture showing the arrival positions of the first 40 or 50 electrons. b. A graph of 0 c1x2 0 2 for the electrons on the detection screen. c. A graph of c1x2 for the electrons. Keep in mind that c, as a wave-like function, oscillates between positive and negative.

An experiment finds electrons to be uniformly distributed over the interval 0 cm x 2 cm, with no electrons falling outside this interval. a. Draw a graph of 0 c1x2 0 2 for these electrons. b. What is the probability that an electron will land within the interval 0.79 to 0.81 cm? c. If 106 electrons are detected, how many will be detected in the interval 0.79 to 0.81 cm? d. What is the probability density at x = 0.80 cm?

FIGURE P39.31 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 P1x2 = 0 c1x2 0 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 nm x 0.25 nm.

FIGURE P39.32 shows 0 c1x2 0 2 for the electrons in an experiment. a. Is the electron wave function normalized? Explain. b. Draw a graph of c1x2 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-cm-wide region at x = 0.00 cm? At x = 0.50 cm? At x = 0.999 cm? d. If 104 electrons are detected, how many are expected to land in the interval -0.30 cm x 0.30 cm?

FIGURE P39.33 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.

FIGURE P39.34 shows the wave function of a particle confined between x = -4.0 mm and x = 4.0 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 P1x2 = 0 c1x2 0 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 mm x 2.0 mm

An electron that is confined to x 0 nm has the normalized wave function c1x2 = b 0 x 6 0 nm 11.414 nm-1/2 2e-x/11.0 nm2 x 0 nm where x is in nm. a. What is the probability of finding the electron in a 0.010-nmwide region at x = 1.0 nm? b. What is the probability of finding the electron in the interval 0.50 nm x 1.50 nm?

Consider the electron wave function c1x2 = b c 21 - x2 0 x 0 1 cm 0 0 x 0 1 cm where x is in cm. a. Determine the normalization constant c. b. Draw a graph of c1x2 over the interval -2 cm x 2 cm. Provide numerical scales on both axes. c. Draw a graph of 0 c1x2 0 2 over the interval -2 cm x 2 cm. Provide numerical scales. d. If 104 electrons are detected, how many will be in the interval 0.00 cm x 0.50 cm?

Consider the electron wave function c1x2 = c c sin1 2px L 2 0 x L 0 x 6 0 or x 7 L a. Determine the normalization constant c. Your answer will be in terms of L. b. Draw a graph of c1x2 over the interval -L x 2L. c. Draw a graph of 0 c1x2 0 2 over the interval -L x 2L. d. What is the probability that an electron is in the interval 0 x L/3?

A particle is described by the wave function c1x2 = b cex/L x 0 mm ce-x/L x 0 mm where L = 2.0 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 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

The probability density for finding a particle at position x is P1x2 = a 11 - x2 -1 mm x 6 0 mm b11 - x2 0 mm x 1 mm and zero elsewhere. a. 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? b. Determine values for a and b. c. Draw a graph of the probability density over the interval -2 mm x 2 mm. d. 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 f0 - 1 2 f f f0 + 1 2 f, where f0 = c/l 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 11 fs = 1 femtosecond =10-15 s2. 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 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.

What is the smallest one-dimensional box in which you can confine an electron if you want to know for certain that the electrons speed is no more than 10 m/s?

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 1A - Z2 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.

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 its reasonable to think of an alpha particle as a single unit within the nucleus from which it is emitted. a. A 238 U nucleus, which decays by alpha emission, is 15 fm in diameter. Model an alpha particle within a 238 U nucleus as being in a one-dimensional box. What is the maximum speed an alpha particle is likely to have? b. 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 U nucleus?

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 onedimensional 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 theres a distribution of speeds, suppose we estimate that the root-mean-square speed vrms of the atoms in the trap is half the value you found in part a. Use this vrms to estimate the temperature of the atoms when theyve been cooled to the limit set by the uncertainty principle

Soot particles, from incomplete combustion in diesel engines, are typically 15 nm in diameter and have a density of 1200 kg/m3 . FIGURE P39.45 shows soot particles released from rest, in vacuum, just above a thin plate with a 0.50@mm@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 0.50@mm@diameter circle. Allowing for some experimental imperfections, any quantum effects would be noticeable if the circle diameter were 2000 nm. How far would the particles have to fall to fill a circle of this diameter?

A small speck of dust with mass 1.0 * 10-13 g has fallen into the hole shown in FIGURE P39.46 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?

a. Starting with the expression f t 1 for a wave packet, find an expression for the product E 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. Youll 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 t of many atoms can be determined. You can think of t as being the uncertainty in your knowledge of how long the atom spends in the excited state. A typical value is t 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 E/E in the photons energy?

The probability density of finding a particle somewhere along the x-axis is 0 for x 6 1 mm. At x = 1 mm, the probability density is c. For x 1 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 mm x 4 mm?

The wave function of a particle is c1x2 = B b p1x2 + b2 2 where b is a positive constant. Find the probability that the particle is located in the interval -b x b.

The wave function of a particle is c1x2 = c b 11 + x2 2 -1 mm x 6 0 mm c11 + x22 0 mm x 1 mm and zero elsewhere. a. 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? b. Draw graphs of the wave function and the probability density over the interval -2 mm x 2 mm. c. What is the probability that the particle will be found to the right of the origin?

Consider the electron wave function c1x2 = c cx 0 x 0 1 nm c x 0 x 0 1 nm where x is in nm. a. Determine the normalization constant c. b. Draw a graph of c1x2 over the interval -5 nm x 5 nm. Provide numerical scales on both axes. c. Draw a graph of 0 c1x2 0 2 over the interval -5 nm x 5 nm. Provide numerical scales. d. If 106 electrons are detected, how many will be in the interval -1.0 nm x 1.0 nm?