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physics
light and optics
Fundamentals of Physics 9th Edition Jearl Walker, Halliday Resnick - Solutions
Show that the ground-state wave function and that of the first excited state of the harmonic oscillator are orthogonal; i.e., show that ∫ ψ0(x) ψ 1(x) dx = 0.
The wave function for the state n = 2 of the harmonic oscillator is ψ2(x) = A2(2ax2 – ½)e-ax2, where A2 is the normalization constant for this wave function. Show that the wave functions for the states n = 1 and n = 2 of the harmonic oscillator are orthogonal.
For the wave functions ψn(x) = √2/L sin(nπx/L) corresponding to a particle in an infinite square well potential from 0 to L, show that∫ ψn(x) ψm(x)dx = 0, that is, ψn and ψm are orthogonal.
Consider a particle in a one-dimensional box of length L that is centered at the origin.(a) What are the values of ψ1(0) and ψ2(0)?(b) What are the values of <x> for the states n = 1 and n = 2?(c) Evaluate <x2> for the states n = 1 and n = 2.
Eight identical non interacting fermions (such as neutrons) are confined to a two-dimensional square box of side length L. Determine the energies of the three lowest states.
A particle is confined to a two-dimensional box defined by the following boundary conditions: U(x,y) = 0 for –L/2 ≤ x ≤ L/2 and –3L/2 ≤ y v 3L/2; and U(x,y) = ∞ elsewhere.(a) Determine the energies of the lowest three bound states. Are any of these states degenerate?(b) Identify the
A particle moves in a potential given by U(x) = A|x|. Without attempting to solve the Schrödinger equation, sketch the wave function for(a) The ground-state energy of a particle inside this potential and(b) The first excited state for this potential.
The classical probability distribution function for a particle in a one-dimensional box of length L is P = 1/L.(a) Show that the classical expectation value of x2 for a particle in a one-dimensional box of length L centered at the origin (Problem 32) is L2/12.(b) Find the quantum
Show that Equations 36-27 and 36-28 imply that the transmission coefficient for particles of energy E incident on a step barrier U0Where r = k2/k1
Determine the normalization constant A2 in Problem 30.
Consider the time-independent one-dimensional Schrödinger equation when the potential function is symmetric about the origin, i.e., when U(x) = U(–x).(a) Show that if ψ(x) is a solution of the Schrödinger equation with energy E, then ψ(–x) is also a solution with the same energy E, and
In this problem you will derive the ground-state energy of the harmonic oscillator using the precise form of the uncertainty principle, Δx Δp ≥ h/2, where Δx and Δp are defined to be the standard deviations (Δx)2 = [(x – xav)2]av and (Δp)2 = [(p – pav)2]av
A particle of mass m near the earth’s surface at z = 0 can be described by the potential energyU = mgz, z > 0U = ∞, z < 0For some positive value of total energy E, indicate the classically allowed region on a sketch of U(z) versus z. Sketch also the classical kinetic energy versus z. The
(Multiple choice)(1) True or falseBoundary conditions on the wave function lead to energy quantization.
1. Give the wave functions for the lowest ten quantum states of the particle in Problem 19.2. What is the ground-state energy of ten noninteracting bosons in a one-dimensional box of length L?3. Give the wave functions for the lowest ten quantum states of the particle in Problem 21.
During the Apollo XI Moon landing, a retro reflecting panel was erected on the Moon’s surface. The speed of light can be found by measuring the time it takes a laser beam to travel from Earth, reflect from the panel, and return to Earth. If this interval is found to be 2.51 s, what is the
Figure shows the apparatus used by Armand H. L. Fizeau (1819€“1896) to measure the speed of light. The basic idea is to measure the total time it takes light to travel from some point to a distant mirror and back. If d is the distance between the light source and the mirror, and if the
In an experiment designed to measure the speed of light using the apparatus of Fizeau described in the preceding problem, the distance between light source and mirror was 11.45 km and the wheel had 720 notches. The experimentally determined value of c was 2.998 x 108 m/s. Calculate the minimum
Albert A. Michelson very carefully measured the speed of light using an alternative version of the technique developed by Fizeau. (See Problem 22.2.) Shows the approach Michelson used. Light was reflected from one face of a rotating eight-sided mirror towards a stationary mirror 35.0 km away. At
Figure shows an apparatus used to measure the distribution of the speeds of gas molecules. The device consists of two slotted rotating disks separated by a distance d, with the slots displaced by the angle θ. Suppose the speed of light is measured by sending a light beam toward the
The two mirrors in figure meet at a right angle. The beam of light in the vertical plane P strikes mirror 1 as shown.(a) Determine the distance the reflected light beam travels before striking mirror 2.(b) In what direction does the light beam travel after being reflected from mirror2?
An underwater scuba diver sees the Sun at an apparent angle of 45.0° from the vertical. What is actual direction of the Sun?
Light is incident normal to a 1.00-cm layer of water that lies on top of a flat Lucite® plate with a thickness of 0.500 cm. How much more time is required for light to pass through this double layer than is required to traverse the same distance in air (nLucite = 1.59)?
A laser beam is incident at an angle of 30.0° to the vertical onto a solution of corn syrup in water. If the beam is refracted to 19.24° to the vertical, (a) What is the index of refraction of the syrup solution? Suppose the light is red, with wavelength 632.8 nm in a vacuum. Find its (b)
Light containing wavelengths of 400 nm, 500 nm, and 650 nm is incident from air on a block of crown glass at an angle of 25.0°. (a) Are all colors refracted alike, or is one color bent more than the others? (b) Calculate the angle of refraction in each case to verify your answer.
Light of wavelength λ0 in a vacuum has a wavelength of 438 nm in water and a wavelength of 390 nm in benzene.(a) What is the wavelength λ0? (b) Using only the given wavelengths, determine the ratio of the index of refraction of benzene to that of water.
Light of wavelength 436 nm in air enters a fishbowl filled with water, then exits through the crown-glass wall of the container. Find the wavelengths of the light (a) In the water and (b) In the glass.
A ray of light is incident on the surface of a block of clear ice at an angle of 40.0° with the normal. Part of the light is reflected and part is refracted. Find the angle between the reflected and refracted light.
The laws of refraction and reflection are the same for sound as for light. The speed of sound is 340 m/s in air and 1,510 m/s in water. If a sound wave traveling in air approaches a plane water surface at an angle of incidence of 12.0°, what is the angle of refraction?
The light emitted by a helium–neon laser has a wavelength of 632.8 nm in air. As the light travels from air into zircon, find (a) Its speed in zircon, (b) Its wavelength in zircon, and (c) Its frequency.
A flashlight on the bottom of a 4.00-m-deep swimming pool sends a ray upward and at an angle so that the ray strikes the surface of the water 2.00 m from the point directly above the flashlight. What angle (in air) does the emerging ray make with the water’s surface?
How many times will the incident beam shown in figure be reflected by each of the parallelmirrors?
A ray of light strikes a flat 2.00-cm-thick block of glass (n = 1.50) at an angle of 30.0° with the normal (Figure). Trace the light beam through the glass, and find the angles of incidence and refraction at each surface.
When the light ray in Problem 18 passes through the glass block, it is shifted laterally by a distance d (Figure). Find the value ofd.
Find the time required for the light to pass through the glass block described in Problem 19.
The light beam shown in figure makes an angle of 20.0° with the normal line NN€™ in the linseed oil. Determine the angles θ and θ€™. (The refractive index for linseed oil is1.48.)
A submarine is 300 m horizontally out from the shore and 100 m beneath the surface of the water. A laser beam is sent from the sub so that it strikes the surface of the water at a point 210 m from the shore. If the beam just strikes the top of a building standing directly at the water’s edge,
Two light pulses are emitted simultaneously from a source. The pulses take parallel paths to a detector 6.20 m away, but one moves through air and the other through a block of ice. Determine the difference in the pulses’ times of arrival at the detector.
A narrow beam of ultrasonic waves reflects off the liver tumor in figure. If the speed of the wave is 10.0% less in the liver than in the surrounding medium, determine the depth of thetumor.
A beam of light both reflects and refracts at the surface between air and glass, as shown in Figure. If the index of refraction of the glass is ng, find the angle of incidence, 1, in the air that would result in the reflected ray and the refracted ray being perpendicular to eachother.
Three sheets of plastic have unknown indices of refraction. Sheet 1 is placed on top of sheet 2, and a laser beam is directed onto the sheets from above so that it strikes the interface at an angle of 26.5° with the normal. The refracted beam in sheet 2 makes an angle of 31.7° with the normal.
An opaque cylindrical tank with an open top has a diameter of 3.00 m and is completely filled with water. When the afternoon Sun reaches an angle of 28.0° above the horizon, sunlight ceases to illuminate the bottom of the tank. How deep is the tank?
A cylindrical cistern, constructed below ground level, is 3.0 m in diameter and 2.0 m deep and is filled to the brim with a liquid whose index of refraction is 1.5. A small object rests on the bottom of the cistern at its center. How far from the edge of the cistern can a girl whose eyes are 1.2 m
The index of refraction for red light in water is 1.331, and that for blue light is 1.340. If a ray of white light enters the water at an angle of incidence of 83.00°, what are the underwater angles of refraction for the blue and red components of the light?
A certain kind of glass has an index of refraction of 1.650 for blue light of wavelength 430 nm and an index of 1.615 for red light of wavelength 680 nm. If a beam containing these two colors is incident at an angle of 30.00° on a piece of this glass, what is the angle between the two beams inside
A ray of light strikes the midpoint of one face of an equiangular (60°–60°–60°) glass prism (n = 1.5) at an angle of incidence of 30°. (a) Trace the path of the light ray through the glass, and find the angles of incidence and refraction at each surface. (b) If a small fraction of light is
The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular dispersion of visible light passing through an equilateral prism of apex angle 60.0° if the angle of incidence is 50.0°? (SeeFigure)
Calculate the critical angles for the following materials when surrounded by air: (a) Zircon, (b) Fluorite, and(c) Ice. Assume that λ = 589 nm.
For light of wavelength 589 nm, calculate the critical angle for the following materials surrounded by air:(a) Diamond and (b) Flint glass.
Repeat Problem 34, but this time suppose that the materials are surrounded by water.
A beam of light is incident from air on the surface of a liquid. If the angle of incidence is 30.0° and the angle of refraction is 22.0°, find the critical angle for the liquid when surrounded by air.
A plastic light pipe has an index of refraction of 1.53. For total internal reflection, what is the minimum angle of incidence if the pipe is in? (a) Air? (b) Water?
Determine the maximum angle θ for which the light rays incident on the end of the light pipe in figure are subject to total internal reflection along the walls of the pipe. Assume that the light pipe has an index of refraction of 1.36 and that the outside medium isair.
Consider a common mirage formed by superheated air just above a roadway. A truck driver whose eyes are 2.00 m above the road, where n = 1.000 3, looks forward. She has the illusion of seeing a patch of water ahead on the road, where her line of sight makes an angle of 1.20° below the horizontal.
A jewel thief hides a diamond by placing it on the bottom of a public swimming pool. He places a circular raft on the surface of the water directly above and centered over the diamond, as shown in figure. If the surface of the water is calm and the pool is 2.00 m deep, find the minimum diameter of
A room contains air in which the speed of sound is 343 m/s. The walls of the room are made of concrete, in which the speed of sound is 1 850 m/s. (a) Find the critical angle for total internal reflection of sound at the concrete–air boundary. (b) In which medium must the sound be traveling in
Three adjacent faces (that all share a corner) of a plastic cube of index of refraction n are painted black. A clear spot at the painted corner serves as a source of diverging rays when light comes through it. Show that a ray from this corner to the center of a clear face is totally reflected if n
The light beam in figure strikes surface 2 at the critical angle. Determine the angle of incidence,θi.
(a) Consider a horizontal interface between air above and glass with an index of 1.55 below. Draw a light ray incident from the air at an angle of incidence of 30.0°. Determine the angles of the reflected and refracted rays, and show them on the diagram. (b) Suppose instead that the light ray is
A layer of ice having parallel sides floats on water. If light is incident on the upper surface of the ice at an angle of incidence of 30.0°, what is the angle of refraction in the water?
A light ray of wavelength 589 nm is incident at an angle θ on the top surface of a block of polystyrene surrounded by air, as shown in figure.(a) Find the maximum value of θ for which the refracted ray will undergo total internal reflection at the left vertical face of the
Figure shows the path of a beam of light through several layers with different indices of refraction.(a) If θ1 = 30.0°, what is the angle θ2 of the emerging beam?(b) What must the incident angle θ1 be in order to have total internal reflection at the
The walls of a prison cell are perpendicular to the four cardinal compass directions. On the first day of spring, light from the rising Sun enters a rectangular window in the eastern wall. The light traverses 2.37 m horizontally to shine perpendicularly on the wall opposite the window. A prisoner
As shown in figure, a light ray is incident normal to on one face of a 30°€“60°€“90° block of dense flint glass (a prism) that is immersed in water.(a) Determine the exit angle θ4 of the ray.(b) A substance is dissolved in the water to increase the index
A narrow beam of light is incident from air onto a glass surface with index of refraction 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence.
One technique for measuring the angle of a prism is shown in figure. A parallel beam of light is directed onto the apex of the prism so that the beam reflects from opposite faces of the prism. Show that the angular separation of the two reflected beams is given by B =2A.
An optical fiber with index of refraction n and diameter d is surrounded by air. Light is sent into the fiber along its axis, as shown in figure.(a) Find the smallest outside radius R permitted for a bend in the fiber if no light is to escape.(b) Does the result for part (a) predict reasonable
A piece of wire is bent through an angle θ. The bent wire is partially submerged in benzene (index of refraction = 1.50), so that, to a person looking along the dry part, the wire appears to be straight and makes an angle of 30.0° with the horizontal. Determine the value of θ.
A light ray traveling in air is incident on one face of a right-angle prism with index of refraction n = 1.50, as shown in figure and the ray follows the path shown in the figure. Assuming that θ = 60.0° and the base of the prism is mirrored, determine the angle Ñ„ made
A transparent cylinder of radius R = 2.00 m has a mirrored surface on its right half, as shown in figure. A light ray traveling in air is incident on the left side of the cylinder. The incident light ray and the exiting light ray are parallel, and d = 2.00 m. Determine the index of refraction of
A laser beam strikes one end of a slab of material, as in figure. The index of refraction of the slab is 1.48. Determine the number of internal reflections of the beam before it emerges from the opposite end of theslab.
For this problem, refer to figure. For various angles of incidence, it can be shown that the deviation angle δ is a minimum when the ray passes through the glass so that the interior ray is parallel to the base of the prism. A measurement of this minimum angle of deviation enables one
A hiker stands on a mountain peak near sunset and observes a rainbow caused by water droplets in the air about 8.00 km away. The valley is 2.00 km below the mountain peak and entirely flat. What fraction of the complete circular arc of the rainbow is visible to the hiker?
A light ray incident on a prism is refracted at the first surface, as shown in figure. Let Ñ„ represent the apex angle of the prism and n its index of refraction. Find, in terms of n and Ñ„, the smallest allowed value of the angle of incidence at the first surface for which
Students allow a narrow beam of laser light to strike a water surface. They arrange to measure the angle of refraction for selected angles of incidence and record the data shown in the following table:Use the data to verify Snell€™s law of refraction by plotting the sine of the angle of
A light ray enters a rectangular block of plastic at an angle θ1 = 45.0° and emerges at an angle θ2 = 76.0°, as shown in figure.(a) Determine the index of refraction of the plastic.(b) If the light ray enters the plastic at a point L = 50.0 cm from the bottom edge,
A. H. P fund€™s method for measuring the index of refraction of glass is illustrated in figure. One face of a slab of thickness t is painted white, and a small hole scraped clear at point P serves as a source of diverging rays when the slab is illuminated from below. Ray PBB€™
What are some reasons that most ceilings are made of white textured material?
How is it possible that a complete circle of a rainbow can sometimes be seen from an airplane?
Why does the arc of a rainbow appear with red on top and violet on the bottom?
Under what conditions is a mirage formed? On a hot day, what are we seeing when we observe a mirage of a water puddle on the road?
A type of mirage called a pingo is often observed in Alaska. Pingos occur when the light from a small hill passes to an observer by a path that takes the light over a body of water warmer than the air. What is seen is the hill and an inverted image directly below it. Explain how these mirages are
Is it possible to have total internal reflection for light incident from air on water? Explain.
1. Explain why an oar partially submearged in water appears to be bent.2. Suppose you are told that only two colors of light (X and Y) are sent through a glass prism and that X is bent more than Y. Which color travels more slowly in the prism?
If a beam of light with a given cross section enters a new medium, the cross section of the refracted beam is different from that of the incident beam. Is it larger or smaller, or is there no definite direction to the change?
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
Spherical mirrors. Object O stands on the central axis of a spherical mirror. For this situation, each problem in Table 34-3 gives object distance Ps (centimeters), the type of mirror, and then the distance (centimeters, without proper sign) between the focal point and the mirror. Find(a) The
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
22 More mirrors. Object O stands on the central axis of a spherical or plane mirror. For this situation, each problem in Table 34-4 refers to(a) The type of mirror,(b) The focal distance f,(c) The radius of curvature r,(d) The object distance p,(e) The image distance i, and(f) The lateral
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