A proton confined in a one-dimensional box emits a 2.0 MeVgamma-ray photon in a quantum

The fi rst-order x-ray diffraction of monochromatic x rays from a crystal occurs at angle 01 The crystal is then compressed, causing a slight reduction in its volume. Does 01 increase, decrease, or stay the same? Explain.

Explain the reasoning by which we claim that the stopping potential V.iop measures the maximum kinetic energy of the electrons in a photoelectric-effect experiment.

How does Einstein's explanation account for each of these characteristics of the photoelectric e ffect? a. The photoelectric cutTent is zero for frequenc ies below some threshold. b. The photoelectric current increases with increasing light intensity. c. The photoelectric current is independent of D.V for D.V ~ IV. d. The photoelectric current decreases slowly as D.V becomes more negative. e. The stopping potential is independent of the light intensity. Which of these canno1 be expla ined by classical physics? Explain.

A current is detected in a photoelectric-effect experiment when the cathode is illuminated with green light. Will a current necessarily be detected if the cathode is illuminated with blue light? With red light?

Figure Q28.5 shows the typical photoelectric behavior of a metal as the a node-cathode potential di fference D.V is varied. a. Why do the c urves become horizontal for D.V ~ IV? Shouldn't the current increase as the potential difference increases? Explain. b. Why doesn't the current immediate ly drop to zero for D.V < 0 V? Shouldn' t D.V < 0 V prevent the electrons from reaching the anode? Explain. c. The current is zero for D.V < - 2.0 V. Where do the electrons go? Are no electrons emitted if D.V < - 2.0 V? Or if they are, why is there no current? Explain.

In the photoelectric effect expe1iment, as illustrated by Figure Q28.6, a current is measured while light is shining on the cathode. But this does not appear to be a complete circuit, so how can there be a current? Explain.

Metal surfaces on spacecraft in bright sunlight develop a net electric charge. Do they develop a negative or a positive charge? Explain.

Metal I has a larger work function than metal 2. Both are illuminated with the same short-wavelength ultraviolet light. Do electrons from metal I have a higher speed, a lower speed, or the same speed as electrons from metal 2? Explain.

A gold cathode is illuminated with light of wavelength 250 nm. It is found that the cutTent is zero when D.V = 1.0 V. Would the current change if a. The light intensity is doubled? b. The anode-cathode potential difference is increased to D.V = 5.5 V?

Three laser beams have wavelengths A 1 = 400 nm, A2 = 600 nm, and A3 = 800 nm. The power of each laser beam is I W. a. Rank in order, from largest to smallest, the photon energies E1, E2, and 3 ir these three laser beams. Explain. b. Rank in order, from largest to smallest, the number of photons per secood N1, .N2 , and N3 delivered by the three laser beams. Explain.

When we say that a photon is a "quantum of light," what does that mean? What is quantized?

A solar cell is illuminated with 2.0 W ollight. Ts the maximum possible current greater with 2.0 W of red light or 2.0 W of blue light?

An investigator is measuring the current in a photoelectJic effect experiment. The cathode is illuminated by light of a single wavelength. What happens to the current if the wavelength of the light is reduced by a factor of two whi le keeping the intensity constant?

An investigator is measuring the current in a photoelectJic effect experiment. The cathode is illuminated by light of a single wavelength. What happens to the current if the wavelength of the light is reduced by a factor of two whi le keeping the intensity constant?

An electron and a proton are accelerated from rest through potential di fferences of the same magnitude. Afterward, which particle has the larger de Broglie wavelength? Explain.

A neutron is shot straight up with an initial speed of I 00 m/s. As it rises, does its de Broglie wavelength increase, decrease, or not change? Explain.

Electrons are accelerated from rest through an 8000 V potential difference. By what factor would their de Broglie wavelength increase if they were instead accelerated through a 2000 V potential?

Can an electron with a de Broglie wavelength of 2 ,m pass through a slit that is I ,m wide? Explain.

a. For the allowed energies of a particle in a box to be large, should the box be very big or very small? Explain. b. Which is likely to have larger values for the allowed energies: an atom in a molecule, an electron in an atom, or a proton in a nucleus? Explain.

Figure Q28.20 shows the standing de Broglie wave of a particle in a box. a. What is the quantum number? b. Can you determine from this picture whether the "classical" pait icle is moving to the right or to the left? If so, which is it? If not, why not?

A particle j n a box of length La has E 1 = 2 e V. The same particle in a box of length Lb has 2 = 50 eV. What is the ratio L ,!Lb?

Imagine that the horizontal box of Figure 28. 18 is instead oriented vertically. Also imagine t~e box to be on a neutron star where the gravitational field is so strong that the particle in the box slows significantly, nearly stopping, before it hits the top of the box. Make a qualitative sketch of the n = 3 de Broglie standing wave of a particle in this box. Hint: The nodes are not uni formly spaced.

Figure Q28.23 shows a standi ng de Broglie wave. a. Does this standing wave represent a particle that travels back and forth between the boundaries with a constant speed or a changing speed? Explain. b. If the speed is changing, at which end is the partic le moving faster and at which end is it moving slower?

The molecules in the rods and cones in the eye are tuned to absorb photons of particular energies. The retinal molecule, like many molecules, is a long chain. Electrons can freely move along one stretch of the chain but are reflected at the ends, thus behaving like a particle in a one-dimensional box. The absorption of a photon lifts an electron from the ground state into the first excited state. Do the molecules in a red cone (which are tuned to absorb red light) or the molecules in a blue cone (tuned to absorb blue light) have a longer "box"?

Science fi ction movies often use devices that transport people and objects rapidly from one position to another. To "beam" people in this fashion means taking them apart atom by atom, carefully measuring each position, and then sending the atoms in a beam to the desired fin al location where they reassemble. How do the principles of quantum mechanics pose problems for this futuristic means of transportation?

A light sensor is based on a photodiode that requires a minimum photon energy of 1.7 eV to c reate mobile electrons. What is the longest wavelength of electromagnetic radiation that the sensor can detect? A. 500 nm B. 730 nm C. 1200 nm D. 2000 nm

In a photoelectric effect experiment, the frequency of the light is increased while the intensity is held constant. As a result, A. There are more electrons. B. The electrons are faster. C. Both A and B. D. Neither A nor B.

In a photoelectric effect experiment, the intensity of the light is increased while the frequency, which is above the threshold frequency, is held constant. As a result, A. There are more electrons. B. The electrons are faster. C. Both A and B. D. Nei ther A nor B.

In the photoelectric e ffect, electrons are never emitted from a metal ifthe frequency of the incoming light is be low a certain threshold value. This is because A. Photons of lower-frequency light don' t have enough energy to eject an electron. B. The electric fi eld of low-frequency light does not vibrate the electrons rapidly enough to eject them. C. The number of photons in low-frequency light is too small to eject electrons. D. Low-frequency light does not penetrate far enough into the metal to eject electrons.

Visible light has a wavelength of about 500 nm. A typical radio wave has a wavelength of about 1.0 m. How many photons of the radio wave are needed to equal the energy of one photon of visible light? A. 2,000 c. 200,000 B. 20,000 D. 2,000,000

Two radio stations have the same power output from their antennas. One broadcasts AM at a frequency of 1000 kHz and one broadcasts FM at a frequency of JOO MHz. Which statement is true? A. The FM station emits more photons per second. B. The AM station emits more photons per second. C. The two stations emit the same number of photons per second.

Light consisting of 2.7 eV photons is incident on a piece of potassium, which has a work fun ction of2.3 eV. What is the maximum ki netic energy of the ejected electrons? A. 2.3 eV B. 2.7 eV C. 5.0 eV D. 0.4 eV

How many photons does a (A = 633 nm) emit in I second? A. 1.2 X 1019 c. 8.0 x 1016 D. 0.4 eV 5.0 mW helium-neon laser B. 4.0 X !018 D. 1.6 X 1016

You shoot a beam of e lectrons through a double slit to make an interference pattern. After noting the properties of the pattern, you then double the speed of the e lectrons. What effect would this have? A. The fringes would get closer together. B. The fri nges would get farther apart. C. The positions of the fringes would not change.

Photon Pin Figure Q28.35 moves an electron from energy level n = I to energy level n = 3. The electron jumps down ton = 2, emitting photon Q, and then jumps down ton = l, emitting photon R. The spacing between energy levels is drawn to scale. What is the correct relationship among the wavelengths of the photons? A. Ar < Ao < AR C. Ao < Ap < AR B. AR < Ap < Ao D. AR < Ao < Ap

What is the length of a box in which the minimum energy of an electron is 1.5 x 10- 1 8 J?

What is the length of a one-dimensional box in which an electron in then= I state has the same energy as a photon with a wavelength of 600 nm?

An electron confined in a one-dimensional box is observed, al different times, to have energies of 12 eV, 27 eV, and 48 eV. What is the length of the box?

The nucleus of a typical atom is 5.0 fm ( I fm = I0- 15 m) in diameter. A very simple model of the nucleus is a onedimensional box in which protons are confined. Estimate the energy of a proton in the nucleus by finding the first three allowed energies of a proton in a 5.0-fm -long box.

The allowed energies of a quantum system are 1.0 e V, 2.0 e V, 4.0 eV, and 7.0 eV. What wavelengths appear in the system's emission spectrum?

Figure P28.4 I is an energy-level diagram for a quantum system. What wavelengths appear in the system's emission spectrum?

The allowed energies of a quantum system are 0.0 eV, 4.0 eV, and 6.0eV. a. Draw the system's energy-level diagram. Label each level with the e nergy and the quantum nui:nber. b. What wavelengths appear in the system's emission spectrum?

A quantum system has three energy levels, so three wavelengths appear in its emission spectrum. The sh01test observed wavelength is 248 nm; light with a 4 14 nm wavelength is also observed. What is the third wavelength?

The allowed energies of a quantum system are 0.0 e V, 1.5 e V, 3.0 eV, and 6.0 eV. How many different wavelengths appear in the emission spectrum?

The speed of an electron is known to be between 3.0 X I 06 m/s and 3.2 x I06 m/s. Estimate the smallest possible uncertainty in its position.

What is the smallest box in which you can confine an e lectron if you want to know for certain that the electron's speed is no more than IO mis?

A spherical virus has a diameter of 50 nm. It is contained inside a long, nan-ow cell of length I x I 0-4 111. What uncertainty INT does this imply for the veloc ity of the virus along the length of the cell? Assume the virus has a density equal to that of water.

A thin solid barrier in the xy-plane has a I 0-,m-diameter ci rcular hole. An electron traveling in the z-direction with Vx = 0 mis passes through the hole. Afterward, is vx still zero? If not, within what range is vx likely to be?

proton is confined within an atomic nucleus of diameter 4 fm ( I fm = 10- 15 m). Estimate the smallest range of speeds you might find for a proton in the nucleus.

X rays with a wavelength of0.0700 nm diffract from a crystal. Two adjacent angles of x-ray diffraction are 45.6 and 2 1.0. What is the distance in nm between the atomic planes responsible for the diffraction?

Potassium and gold cathodes are used in a photoelectriceffect experiment. For each cathode, find: a. The threshold frequency b. The threshold wavelength c. The maximum electron ejection speed if the light has a wavelength of 220 nm d. The stopping potential if the wavelength is 220 nm

In a photoelectric-effect experiment, the maximum kinetic energy of electrons is 2.8 eV. When the wavelength of the light is increased by 50%, the maximum energy decreases to I. I eV. What are (a) the work function of the cathode and (b) the initial wavelength?

In a photoelectric-effect experiment, the stopping potential at a wavelength of 400 nm is 25.7% of the stopping potential at a wavelength of 300 nm. Of what metal is the cathode made?

Light of constant intensity but varying wavelength was used to illuminate the cathode in a photoelectric-e ffect experiment. The graph of Figure P28.54 shows how the stopping potential depended on the frequency of the light. What is the work function, in eV, of the cathode?

What is the de Broglie wavelength of a red blood cell BKJ with a mass of 1.00 X IO- l I g that is moving with a speed of 0.400 emfs? Do we need to be concerned wi th the wave nature of the blood cells when we describe the now of blood in the body?

Suppose you need to image the structure of a virus with a diameter of 50 nm. For a sharp image, the wavelength of the probing wave must be 5.0 nm or less. We have seen that, for imaging such small objects, this short wavelength is obtained by using an e lectron beam in an electron microscope. Why don't we simply use sho1t-wavelength electromagnetic waves? There's a problem with this approach: As the wavelength gets shorter, the energy of a photon of light gets greater and could damage or destroy the object being studied. Let's compare the energy of a photon and an electron that can provide the same resolution. a. For light of wavelength 5.0 nm, what is the ene rgy (in eV) of a single photon? In what part of the electromagnetic spectrum is this? b. For an electron with a de Broglie wavelength of 5.0 nm, what is the kinetic energy (in e V)?

Gamma rays are photons with very high energy. a. What is the wavelength of a gamma-ray photon with energy 625 keV? b. How many visible-light photons with a wavelength of 500 nm would you need Lo match the energy of this one gamma-ray photon?

A red laser with a wavelength of 650 nm and a blue laser with a wavelength of 450 nm emit laser beams with the same light power. What is the ratio of the red laser's photon emission rate (photons per second) to the blue laser's photon emission rate?

A typical incandescent lightbulb e mits approximately 3 X 1018 visible-light photons per second. Your eye, when it is INT fully dark adapted, can bare ly see the light from an incandescent lightbulb 10 km away. How many photons per second are incident at the image point on your retina? The diameter of a dark-adapted pupil is 6 mm.

The intensity of sunlight hitting the surface of the ea11h on a cloudy day is about 0.50 kW/m2 Assuming your pupil can close down to a diameter of 2.0 mm and that the average wavelength of visible light is 550 nm, how many photons per second of visible light enter your eye if you look up at the sky on a cloudy day?

A red LED (light emitting diode) is connected to a battery; it carries a current. As electrons move through the diode, they j ump between states, emitting photons in the process. Assume that each electron that travels through the diode causes the emission of a single 630 nm photon. What current is necessary to produce 5.0 mW of emitted light?

A ruby laser emits an intense pulse of light that lasts a mere I 0 ns. The I ight has a wavelength of 690 nm, and each pulse has an energy of 500 mJ. a. How many p hotons are emitted in each pulse? b. What is the rate of photon emi ssion, in photons per second, during the 10 ns that the laser is "on"?

In a laser range-fi nding experi ment, a pulse of laser light is fired toward an a1Tay of renecting mirrors left on the moon by Apollo astronauts. By measuring the time it takes for the pulse to travel to the moon, renect off the mirrors, and return to earth, scientists can measure the distance to the moon to an accuracy of a few centimeters. The light pulses are 100 ps long, and the laser wavelength is 532 nm. When the pulse reaches the moon, it has an intensity of300 W/m2 How many photons strike a single 4.0-cm-diameter mirror from one laser pulse?

The wavelength of the radiation in a microwave oven is 12 cm. How many photons are absorbed by 200 g of water as it's heated from 20C to 90C?

Exposure to a sufficient quantity of ultraviolet light will redden the skin, producing e1ytli ema-a sunburn. The amount of exposure necessary to produce this reddening depends on the wavelength. For a 1.0 cm2 patch of skin, 3.7 mJ of ultraviolet light at a wavelength of 254 nm will produce reddening; at 300 nm wavelength, 13 mJ are required. a. What is the photon energy corresponding to each of these wavelengths? b. How many total photons does each of these exposures correspond to? c. Explain why there is a di fference in the number of photons needed to provoke a response in the two cases.

A silicon solar cell behaves like a bauery with a 0.50 V termiINT nal voltage. Suppose that 1.0 W of light of wavelength 600 nm falls on a solar cell and that 50% of the photons give their energy to charge carriers, creating a current. What is the solar cell's e fficiency-that is, what percentage of the energy incident on the cell is converted to electric energy?

What is the kinetic energy in eV of an electron whose de Broglie wavelength is I 0% of the wavelength of a photon of the same energy?

Electrons wi th a speed of 2.0 X !06 mis pass through a doubleINT slit apparatus. Interference fringes are detected with a fringe spacing of 1.5 mm. a. What will the fringe spacing be if the electrons are replaced by neutrons with the same speed? b. What speed must neutrons have to produce interference fringes with a fringe spacing of 1.5 mm?

Electrons pass through a 200-nm-diameter circular aperture, forming a diffraction pattern on a detector 1.2 m behind the aperture. What is the kinetic energy of the electrons if the width of the central maximum is 0.37 mm?

The electron interference pattern of Figure 28. 14 was made by shooting electrons with 50 keV of kinetic energy through two slits spaced 1.0 ,m apart. The fonges were recorded on a detector 1.0 m behind the slits. a. What was the speed of the electrons? (The speed is large enough to justify using relativity, but for simplicity do this as a nonrelativistic calculation.) b. Figure 28. 14 is gre atly magnified. What was the actual spacing on the detector between adjacent bright fringes?

lt is stated in the text that special relativity must be used to INT calculate the de Broglie wavelength of electrons in an electron microscope. Let us discover how much of an effect relativity has. Consider an electron accelerated through a potential difference of 1.00 x 105 V. a. Using the Newtonian (nonrelativistic) expressions for kinetic energy and momentum, what is the electron's de Broglie wavelength? b. The de Broglie wavelength is A= hip, but the momentum of a relativistic particle is not mv. Using the relativistic expressions for kinetic energy and momentum, what is the electron's de Broglie wavelength?

An electron confined to a one-dimensional box of length 0. 70 nm jumps from then = 2 level to the ground state. Whal is the wavelength (in nm) of the emitted photon?

a. What is the minimum energy of a 2.7 g Ping-Pong ball in a IO-cm-long box? b. What speed co1Tesponds to this kinetic energy?

The color of dyes results from the preferential absorption of certain wavelengths of light. Certain dye molecules consist of symmetric pairs of rings joined at the center by a chain of carbon atoms, as shown in Figure P28.74. Electrons of the bonds along the chain of carbon atoms are shared among the atoms in the chain, but are repelled by the nitrogencontaining rings at the end of the chain. These electrons are thus free to move along the chain but not beyond its ends. They look very much like a particle in a one-dimensional box. For the molecule shown, the effective length of the "box" is 0.85 nm. Assuming that the electrons start in the lowest energy state, what are the three longest wavelengths this molecule will absorb?

What is the length of a box in which the difference between an electron's first and second allowed energies is 1.0 x I 0-19 J?

Two adjacent allowed energies of an electron in a onedimensional box are 2.0 eV and 4.5 eV. What is the length of the box?

An electron confined to a box has an energy of 1.28 eV. Another electron confined to an identical box has an energy of 2.88 eV. What is the smallest possible length for those boxes?

Consider a small virus having a diameter of I 0 nm. The atoms of the intracellular fluid are confined within this "box." INT Suppose we model the virus as a one-dimensional box of length 10 nm. What is the ground-state energy (in eV) of a sodium ion confined in such a box?

It can be shown that the allowed energies of a particle of mass 111 in a two-dimensional square box of side L are The energy depends on two quantum numbers, n and /, both of which must have an integer value I , 2, 3, .... a. What is the minimum energy for a particle in a two-dimensional square box of side L? b. What are the five lowest allowed energies? Give your values as multiples of 111;0

An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n = 2 to 11 = I. What is the length of the box?

A proton confined in a one-dimensional box emits a 2.0 MeV gamma-ray photon in a quantum jump from 11 = 2 to n = I. What is the length of the box?

As an electron in a one-dimensional box of length 0.600 nm jumps between two energy levels, a photon of energy 8.36 eV is emitted. What are the quantum numbers of the two levels?

Magnetic resonance is used in imaging; it is also a useful tool for analyzing chemical samples. Magnets for magnetic resonance experiments are often characterized by the proton resonance frequency they create. What is the field strength of an 800 MHz magnet?

The electron has a magnetic moment, so you can do magne tic resonance measurements on substances with unpaired electron spins. The e lectron has a magnetic moment ,= 9.3 x I 0- 24 Jff. A sample is placed in a solenoid of length 15 cm with 1200 turns of wire ca1Tying a current of 3.5 A. A probe coil provides radio waves to " flip" the spins. What is the necessary frequency for the probe coi I?

When the x-ray photon scatters from the electron, A. Its speed increases. B. Its speed decreases. C. Its speed stays the same.

When the x-ray photon scatters from the electron, A. Its wavelength increases. B. Its wavelength decreases. C. Its wavelength stays the same

When the electron is struck by the x-ray photon, A. Its de Broglie wavelength increases. B. Its de Broglie wavelength decreases. C. Its de Broglie wavelength stays the same .

X-ray diffraction can also change the direction of a beam of x rays. Which statement offers the best comparison between Compton scattering and x-ray diffraction? A. X-ray diffraction changes the wavelength of x rays; Compton scattering does not. B. Compton scattering changes the speed of x rays; x-ray diffraction does not. C. X-ray diffraction relies on the particle nature of the x rays; Compton scattering relies on the wave nature. D. X-ray diffraction relies on the wave nature of the x rays; Compton scattering relies on the particle nature.