As an electron in a one-dimensional box of length 0.600 nm

Problem 1CQ The first-order x-ray diffraction of monochromatic x rays from a crystal occurs at angle . The crystal is then compressed, causing a slight reduction in its volume. Does increase, decrease, or stay the same? Explain.

Problem 1P X rays with a wavelength of 0.12 nm undergo first-order diffraction from a crystal at a 68° angle of incidence. What is the angle of second-order diffraction?

Problem 2CQ Explain the reasoning by which we claim that the stopping potential measures the maximum kinetic energy of the electrons in a photoelectric-effect experiment.

Problem 2P X rays with a wavelength of 0.20 nm undergo first-order diffraction from a crystal at a 54° angle of incidence. At what angle does first-order diffraction occur for x rays with a wavelength of 0.15 nm?

Problem 3CQ How does Einstein’s explanation account for each of these characteristics of the photoelectric effect? a. The photoelectric current is zero for frequencies below some threshold. b. The photoelectric current increases with increasing light intensity. c. The photoelectric current is independent of ?V for ?V ? 1V. d. The photoelectric current decreases slowly as ?V becomes more negative. e. The stopping potential is independent of the light intensity. Which of these cannot be explained by classical physics? Explain.

Problem 3P X rays diffract from a crystal in which the spacing between atomic planes is 0.175 nm. The second-order diffraction occurs at 45.0°. What is the angle of the first-order diffraction?

Problem 4CQ How would the graph of Figure 28.7a look if the emission of electrons from the cathode was due to the heating of the metal by light falling on it? Draw the graph and explain your reasoning. Assume that the light intensity remains constant as its frequency and wavelength are varied.

Problem 4P The spacing between atomic planes in a crystal is 0.110 nm. If 12.0 keV x rays are diffracted by this crystal, what are the angles of (a) first-order and (b) second-order diffraction?

Problem 5CQ Figure Q28.5 shows the typical photoelectric behavior of a metal as the anode-cathode potential difference ?V is varied. a. Why do the curves become horizontal for ?V ? 1V? Shouldn’t the current increase as the potential difference increases? Explain. b. Why doesn’t the current immediately drop to zero for ?V < 0 V? Shouldn’t ?V < 0 V prevent the electrons from reaching the anode? Explain. c. The current is zero for ?V < -2.0 V . Where do the electrons go? Are no electrons emitted if ?V < -2.0 V? Or if they are, why is there no current? Explain.

Problem 5P X rays with a wavelength of 0.085 nm diffract from a crystal in which the spacing between atomic planes is 0.18 nm. How many diffraction orders are observed?

Problem 6CQ In the photoelectric effect experiment, 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.

Problem 6P Which metals in Table 28.1 exhibit the photoelectric effect for (a) light with ?? = 400 nm and (b) light with ?? = 250 nm?

Problem 7CQ Metal surfaces on spacecraft in bright sunlight develop a net electric charge. Do they develop a negative or a positive charge? Explain.

Problem 7P Electrons are emitted when a metal is illuminated by light with a wavelength less than 388 nm but for no greater wavelength. What is the metal’s work function?

Problem 8CQ Metal 1 has a larger work function than metal 2. Both are illuminated with the same short-wavelength ultraviolet light. Do electrons from metal 1 have a higher speed, a lower speed, or the same speed as electrons from metal 2? Explain.

Problem 8P Electrons in a photoelectric-effect experiment emerge from a copper surface with a maximum kinetic energy of 1.10 eV. What is the wavelength of the light?

Problem 9CQ A gold cathode is illuminated with light of wavelength 250 nm. It is found that the current is zero when ?V = 1.0 V. Would the current change if a. The light intensity is doubled? b. The anode-cathode potential difference is increased to ?V = 5.5 V?

Problem 9P You need to design a photodetector that can respond to the entire range of visible light. What is the maximum possible work function of the cathode?

Problem 10CQ

Problem 10P A photoelectric-effect experiment finds a stopping potential of 1.93 V when light of 200 nm wavelength is used to illuminate the cathode. a. From what metal is the cathode made? b. What is the stopping potential if the intensity of the light is doubled?

Problem 11CQ When we say that a photon is a “quantum of light,” what does that mean? What is quantized?

Problem 11P Zinc has a work function of 4.3 eV. a. What is the longest wavelength of light that will release an electron from a zinc surface? b. A 4.7 eV photon strikes the surface and an electron is emitted. What is the maximum possible speed of the electron?

Problem 12CQ An investigator is measuring the current in a photoelectric effect experiment. The cathode is illuminated by light of a single wavelength. What happens to the current if the intensity of the light is doubled while the wavelength is held constant?

Problem 12P Image intensifiers used in night vision devices create a bright image from dim light by letting the light first fall on a photocathode. Electrons emitted by the photoelectric effect are accelerated and then strike a phosphorescent screen, causing it to glow more brightly than the original scene. Recent devices are sensitive to wavelengths as long as 900 nm, in the infrared. a. If the threshold wavelength is 900 nm, what is the work function of the photocathode? b. If light of wavelength 700 nm strikes such a photocathode, what will be the maximum kinetic energy, in eV, of the emitted electrons?

Problem 13CQ An investigator is measuring the current in a photoelectric 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 while keeping the intensity constant?

Problem 13P Light with a wavelength of 350 nm shines on a metal surface, which emits electrons. The stopping potential is measured to be 1.25 V. a. What is the maximum speed of emitted electrons? b. Calculate the work function and identify the metal.

Problem 14CQ To have the best resolution, should an electron microscope use very fast electrons or very slow electrons? Explain.

Problem 14P When an ultraviolet photon is absorbed by a molecule of DNA, the photon’s energy can be converted into vibrational energy of the molecular bonds. Excessive vibration damages the molecule by causing the bonds to break. Ultraviolet light of wavelength less than 290 nm causes significant damage to DNA; ultraviolet light of longer wavelength causes minimal damage. What is the threshold photon energy, in eV, for DNA damage?

Problem 15CQ An electron and a proton are accelerated from rest through potential differences of the same magnitude. Afterward, which particle has the larger de Broglie wavelength? Explain.

Problem 15P The spacing between atoms in graphite is approximately 0.25 nm. What is the energy of an x-ray photon with this wavelength?

Problem 16CQ A neutron is shot straight up with an initial speed of 100 m/s. As it rises, does its de Broglie wavelength increase, decrease, or not change? Explain.

Problem 16P A firefly glows by the direct conversion of chemical energy to light. The light emitted by a firefly has peak intensity at a wavelength of 550 nm. a. What is the minimum chemical energy, in eV, required to generate each photon? b. One molecule of ATP provides 0.30 eV of energy when it is metabolized in a cell. What is the minimum number of ATP molecules that must be consumed in the reactions that lead to the emission of one photon of 550 nm light?

Problem 17CQ Double-slit interference of electrons occurs because: A. The electrons passing through the two slits repel each other. B. Electrons collide with each other behind the slits. C. Electrons collide with the edges of the slits. D. Each electron goes through both slits. E. The energy of the electrons is quantized. F. Only certain wavelengths of the electrons fit through the slits.

Problem 17P Your eyes have three different types of cones with maximum absorption at 437 nm, 533 nm, and 564 nm. What photon energies correspond to these wavelengths?

Problem 18CQ Can an electron with a de Broglie wavelength of 2 mm pass through a slit that is 1 mm wide? Explain.

Problem 18P What is the wavelength, in nm, of a photon with energy (a) 0.30 eV, (b) 3.0 eV, and (c) 30 eV? For each, is this wavelength visible light, ultraviolet, or infrared?

Problem 19CQ 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.

Problem 19P What is the ratio of me energy of a photon of light at the far red end of the visible spectrum (700 nm) to that of a photon at the far blue end of the visible spectrum (400 nm)?

Problem 20CQ 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” particle is moving to the right or to the left? If so, which is it? If not, why not?

Problem 20P The wavelengths of light emitted by a firefly span the visible spectrum but have maximum intensity near 550 nm. A typical flash lasts for 100 ms and has a power of 1.2 mW. If we assume that all of the light is emitted at the peak-intensity wavelength of 550 nm, how many photons are emitted in one flash?

Problem 21CQ

Problem 21P Station KAIM in Hawaii broadcasts on the AM dial at 870 kHz, with a maximum power of 50,000 W. At maximum power, how many photons does the transmitting antenna emit each second?

Problem 22CQ Imagine that the horizontal box of Figure 28.18 is instead oriented vertically. Also imagine the 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 uniformly spaced.

Problem 22P At 510 nm, the wavelength of maximum sensitivity of the human eye, the dark-adapted eye can sense a 100-ms-long flash of light of total energy . (Weaker flashes of light may be detected, but not reliably.) If 60% of the incident light is lost to reflection and absorption by tissues of the eye, how many photons reach the retina from this flash?

Problem 23CQ Figure Q28.23 shows a standing 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 particle moving faster and at which end is it moving slower?

Problem 23P 550 nm is the average wavelength of visible light. a. What is the energy of a photon with a wavelength of 550 nm? b. A typical incandescent light bulb emits about 1 J of visible light energy every second. Estimate the number of visible photons emitted per second.

Problem 24CQ 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”?

Problem 24P Dinoflagellates are single celled creatures that float in the world’s oceans; many types are bioluminescent. When disturbed by motion in the water, a typical bioluminescent dinoflagellate emits 100,000,000 photons in a 0.10-s-long flash of light of wavelength 460 nm. What is the power of the flash in watts?

Problem 25CQ Science fiction 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 final location where they reassemble. How do the principles of quantum mechanics pose problems for this futuristic means of transportation?

Problem 26MCQ A light sensor is based on a photodiode that requires a minimum photon energy of 1.7 eV to create 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

Problem 25P A circuit employs a silicon solar cell to detect flashes of light lasting 0.25 s. The smallest current the circuit can detect reliably is 0.42 ?A. Assuming that all photons reaching the solar cell give their energy to a charge carrier, what is the minimum power of a flash of light of wavelength 550 nm that can be detected?

Problem 26P Estimate your de Broglie wavelength while walking at a speed of 1 m/s.

Problem 27MCQ 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.

Problem 27P a. What is the de Broglie wavelength of a 200 g baseball with a speed of 30 m/s? b. What is the speed of a 200 g baseball with a de Broglie wavelength of 0.20 nm?

Problem 28MCQ 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. Neither A nor B.

Problem 28P a. What is the speed of an electron with a de Broglie wavelength of 0.20 nm? b. What is the speed of a proton with a de Broglie wavelength of 0.20 nm?

Problem 29MCQ In the photoelectric effect, electrons are never emitted from a metal if the frequency of the incoming light is below 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 field 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.

Problem 29P What is the kinetic energy, in eV, of an electron with a de Broglie wavelength of 1.0 nm?

Problem 30MCQ 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 B. 20,000 C. 200,000 D. 2,000,000

Problem 30P A paramecium is covered with motile hairs called cilia that propel it at a speed of 1 mm/s. If the paramecium has a volume of and a density equal to that of water, what is its de Broglie wavelength when in motion? What fraction of the paramecium’s 150 mm length does this wavelength represent?

Problem 31MCQ 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 100 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.

Problem 31P The diameter of an atomic nucleus is about 10 fm . What is the kinetic energy, in MeV, of a proton with a de Broglie wavelength of 10 fm?

Problem 32MCQ An electron is accelerated through a 5000 V potential difference, strikes a metal target, and causes an x ray to be emitted. What is the (approximate) minimum wavelength of the emitted x ray? A. 0.25 nm B. 1.0 nm C. 2.5 nm D. 4.0 nm

Problem 33MCQ How many photons does a 5.0 mW helium-neon laser (?? = 633 nm) emit in 1 second?

Problem 32P Rubidium atoms are cooled to 0.10 ?K in an atom trap. What is their de Broglie wavelength? How many times larger is this than the 0.25 nm diameter of the atoms?

Problem 33P Through what potential difference must an electron be accelerated from rest to have a de Broglie wavelength of 500 nm?

Problem 34MCQ You shoot a beam of electrons through a double slit to make an interference pattern. After noting the properties of the pattern, you then double the speed of the electrons. What effect would this have? A. The fringes would get closer together. B. The fringes would get farther apart. C. The positions of the fringes would not change.

Problem 34P What is the length of a box in which the minimum energy of an electron is ?

Problem 35MCQ Photon P in Figure Q28.35 moves an electron from energy level n = 1 to energy level n = 3. The electron jumps down to n = 2, emitting photon Q, and then jumps down to n = 1, emitting photon R. The spacing between energy levels is drawn to scale. What is the correct relationship among the wavelengths of the photons?

Problem 35P What is the length of a one-dimensional box in which an electron in the n = 1 state has the same energy as a photon with a wavelength of 600 nm?

Problem 36P An electron confined in a one-dimensional box is observed, at different times, to have energies of 12 eV, 27 eV, and 48 eV. What is the length of the box?

Problem 37P The nucleus of a typical atom is 5.0 fm in diameter. A very simple model of the nucleus is a one dimensional 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.

Problem 38P The allowed energies of a quantum system are 1.0 eV, 2.0 eV, 4.0 eV, and 7.0 eV. What wavelengths appear in the system’s emission spectrum?

Problem 40P The allowed energies of a quantum system are 0.0 eV, 4.0 eV, and 6.0 eV. a. Draw the system’s energy-level diagram. Label each level with the energy and the quantum number. b. What wavelengths appear in the system’s emission spectrum?

Problem 39P Figure P28.41 is an energy-level diagram for a quantum system. What wavelengths appear in the system’s emission spectrum?

Problem 41P The allowed energies of a quantum system are 0.0 eV, 1.5 eV, 3.0 eV, and 6.0 eV. How many different wavelengths appear in the emission spectrum?

Problem 42P The speed of an electron is known to be between . Estimate the smallest possible uncertainty in its position.

Problem 43P What is the smallest 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 m/s?

Problem 44P A spherical virus has a diameter of 50 nm. It is contained inside a long, narrow cell of length . What uncertainty does this imply for the velocity of the virus along the length of the cell? Assume the virus has a density equal to that of water.

Problem 45P A thin solid barrier in the xy-plane has a 10-mm-diameter circular hole. An electron traveling in the z-direction with passes through the hole. Afterward, is still zero? If not, within what range is likely to be?

Problem 46P A proton is confined within an atomic nucleus of diameter . Estimate the smallest range of speeds you might find for a proton in the nucleus.

Problem 47GP X rays with a wavelength of 0.0700 nm diffract from a crystal. Two adjacent angles of x-ray diffraction are 45.6° and 21.0°. What is the distance in nm between the atomic planes responsible for the diffraction?

Problem 48GP Potassium and gold cathodes are used in a photoelectric effect 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

Problem 49GP 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 1.1 eV. What are (a) the work function of the cathode and (b) the initial wavelength?

Problem 50GP 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?

Problem 51GP Light of constant intensity but varying wavelength was used to illuminate the cathode in a photoelectric-effect 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?

Problem 52GP What is the de Broglie wavelength of a red blood cell with a mass of that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

Problem 53GP 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 electron beam in an electron microscope. Why don’t we simply use short-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 energy (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 eV)?

Problem 54GP 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 to match the energy of this one gamma-ray photon?

Problem 55GP 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?

Problem 56GP A typical incandescent light bulb emits approximately visible-light photons per second. Your eye, when it is fully dark adapted, can barely see the light from an incandescent light bulb 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.

Problem 57GP The intensity of sunlight hitting the surface of the earth on a cloudy day is about . 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?

Problem 58GP A red LED (light emitting diode) is connected to a battery; it carries a current. As electrons move through the diode, they jump 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?

Problem 59GP A ruby laser emits an intense pulse of light that lasts a mere 10 ns. The light has a wavelength of 690 nm, and each pulse has an energy of 500 mJ. a. How many photons are emitted in each pulse? b. What is the rate of photon emission, in photons per second, during the 10 ns that the laser is “on”?

Problem 60GP The human body emits thermal electromagnetic radiation, as we’ve seen. Assuming that all radiation is emitted at the wavelength of peak intensity, for a skin temperature of 33°C and a surface area of 1.8 m2, how many photons per second does the body emit?

Problem 61GP 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 20°C to 90°C?

Problem 62GP Exposure to a sufficient quantity of ultraviolet light will redden the skin, producing erythema—a sunburn. The amount of exposure necessary to produce this reddening depends on the wavelength. For a 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 difference in the number of photons needed to provoke a response in the two cases.

Problem 63GP A silicon solar cell behaves like a battery with a 0.50 V terminal 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 efficiency—that is, what percentage of the energy incident on the cell is converted to electric energy?

Problem 64GP Electrons with a speed of pass through a double 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?

Problem 65GP Electrons pass through a 1.0-?m-wide slit with a speed of 1.5 × 106 m/s. How wide is the electron diffraction pattern on a detector 1.0 m behind the slit?

Problem 66GP 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 fringes 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 greatly magnified. What was the actual spacing on the detector between adjacent bright fringes?

Problem 67GP It is stated in the text that special relativity must be used to 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 . a. Using the Newtonian (non relativistic) expressions for kinetic energy and momentum, what is the electron’s de Broglie wavelength? b. The de Broglie wavelength is ??= h/p, 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?

Problem 68GP An electron confined to a one-dimensional box of length 0.70 nm jumps from the n = 2 level to the ground state. What is the wavelength (in nm) of the emitted photon?

Problem 69GP a. What is the minimum energy of a 2.7 g Ping-Pong ball in a 10-cm-long box? b. What speed corresponds to this kinetic energy?

Problem 70GP 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 nitrogen containing 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?

Problem 71GP What is the length of a box in which the difference between an electron’s first and second allowed energies is ?

Problem 72GP Two adjacent allowed energies of an electron in a one dimensional box are 2.0 eV and 4.5 eV. What is the length of the box?

Problem 73GP 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?

Problem 74GP Consider a small virus having a diameter of 10 nm. The atoms of the intracellular fluid are confined within this “box.” 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?

Problem 75GP It can be shown that the allowed energies of a particle of mass m in a two-dimensional square box of side L are The energy depends on two quantum numbers, n and l, both of which must have an integer value 1, 2, 3,c. 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 .

Problem 76GP An electron confined in a one-dimensional box emits a 200 nm photon in a quantum jump from n = 2 to n = 1. What is the length of the box?

Problem 77GP A proton confined in a one-dimensional box emits a 2.0 MeV gamma-ray photon in a quantum jump from n = 2 to n = 1. What is the length of the box?

Problem 78GP 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?

Problem 79GP 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?

Problem 80GP The electron has a magnetic moment, so you can do magnetic resonance measurements on substances with unpaired electron spins. The electron has a magnetic moment . A sample is placed in a solenoid of length 15 cm with 1200 turns of wire carrying a current of 3.5 A. A probe coil provides radio waves to “flip” the spins. What is the necessary frequency for the probe coil?

Problem 81PP Compton Scattering Further support for the photon model of electromagnetic waves comes from Compton scattering, in which x rays scatter from electrons, changing direction and frequency in the process. Classical electromagnetic wave theory cannot explain the change in frequency of the x rays on scattering, but the photon model can. Suppose an x-ray photon is moving to the right. It has a collision with a slow-moving electron, as in Figure P28.85. The photon transfers energy and momentum to the electron, which recoils at a high speed. The x-ray photon loses energy, and the photon energy formula E = hf tells us that its frequency must decrease. The collision looks very much like the collision between two particles. When the x-ray photon scatters from the electron, A. Its speed increases. B. Its speed decreases. C. Its speed stays the same.

Problem 82PP Compton Scattering Further support for the photon model of electromagnetic waves comes from Compton scattering, in which x rays scatter from electrons, changing direction and frequency in the process. Classical electromagnetic wave theory cannot explain the change in frequency of the x rays on scattering, but the photon model can. Suppose an x-ray photon is moving to the right. It has a collision with a slow-moving electron, as in Figure P28.85. The photon transfers energy and momentum to the electron, which recoils at a high speed. The x-ray photon loses energy, and the photon energy formula E = hf tells us that its frequency must decrease. The collision looks very much like the collision between two particles. When the x-ray photon scatters from the electron, A. Its wavelength increases. B. Its wavelength decreases. C. Its wavelength stays the same.

Problem 83PP Compton Scattering Further support for the photon model of electromagnetic waves comes from Compton scattering, in which x rays scatter from electrons, changing direction and frequency in the process. Classical electromagnetic wave theory cannot explain the change in frequency of the x rays on scattering, but the photon model can. Suppose an x-ray photon is moving to the right. It has a collision with a slow-moving electron, as in Figure P28.85. The photon transfers energy and momentum to the electron, which recoils at a high speed. The x-ray photon loses energy, and the photon energy formula E = hf tells us that its frequency must decrease. The collision looks very much like the collision between two particles. 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.

Problem 84PP Compton Scattering Further support for the photon model of electromagnetic waves comes from Compton scattering, in which x rays scatter from electrons, changing direction and frequency in the process. Classical electromagnetic wave theory cannot explain the change in frequency of the x rays on scattering, but the photon model can. Suppose an x-ray photon is moving to the right. It has a collision with a slow-moving electron, as in Figure P28.85. The photon transfers energy and momentum to the electron, which recoils at a high speed. The x-ray photon loses energy, and the photon energy formula E = hf tells us that its frequency must decrease. The collision looks very much like the collision between two particles. 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.