Physics by Example 1st edition W. G. Rees - Solutions

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(a) Derive an expression for the total shift in photon wavelength after two successive Compton scatterings from electrons at rest. The photon is scattered by an angle θ1, in the first scattering and by θ2 in the second. (b) In general, is the total shift in wavelength produced by two
Nuclear fusion reactions at the center of the sun produce gamma-ray photons with energies of order 1 MeV (106eV). By contrast, what we see emanating from the sun's surface are visible light photons with wavelengths of order 500 nm. A simple model that explains this difference in wavelength is that
An x-ray photon is scattered from a free electron (mass m) at rest. The wavelength of the scattered photon is A', and the final speed of the struck electron is v. (a) What was the initial wavelength λ of the photon? Express your answer in terms of λ', v, and m. (b) Through what angle
A photon with wavelength 0.1100nm collides with a free electron that is initially at rest. After the collision the wavelength is 0.1132 nm. (a) What is the kinetic energy of the electron after the collision? What is its speed? (b) If the electron is suddenly stopped (for example, in a solid
(a) Calculate the maximum increase in photon wavelength that can occur during Compton scattering. (b) What is the energy (in electron volts) of the lowest-energy x-ray photon for which Compton scattering could result in doubling the original wavelength?
(a) Write the Planck distribution law in terms of the frequency f rather than the wavelength A, to obtain I(f) .(b) Show that where I(λ) is the Planck distribution formula of Eq. (38.32). You will need to use the following tabulated integral:(c) The result of (b) is I and has the form of the
An Ideal Blackbody a large cavity with a very small hole and maintained at a temperature T is a good approximation to an ideal radiator or blackbody. Radiation can pass into or out of the cavity only through the hole. The cavity is a perfect absorber, since any radiation incident on the hole
The motion of an oscillating flywheel is defined by the relation θ = θ0e−3πt cos4πt, where θ is expressed in radians and t in seconds. Knowing that θ0 = 0.5 rad, determine the angular coordinate, the angular velocity, and the angular acceleration of the flywheel when (a) t = 0, (b) t =
Consider a beam of monochromatic light with intensity I incident on a perfectly absorbing surface oriented perpendicular to the beam. Use the photon concept to show that the radiation pressure exerted by the light on the surface is given by I/c.
Consider Compton scattering of a photon by a moving electron. Before the collision the photon has wavelength ? and is moving in the +x-direction, and the electron is moving in the -x-direction with total energy E (including its rest energy mc2). The photon and electron collide head-on. After the
(a) An electron moves with a speed of 4.70 x 106m/s. What is its de Broglie wavelength? (b) A proton moves with the same speed. Determine its de Broglie wavelength.
For crystal diffraction experiments (discussed in Section 39.2), wavelengths on the order of 0.20 nm are often appropriate. Find the energy in electron volts for a particle with this wavelength if the particle is (a) A photon; (b) An electron; (c) An alpha particle (m = 6.64 x 10-27 kg).
An electron has a de Broglie wavelength of 2.80 X 10-10 m. Determine (a) The magnitude of its momentum and (b) Its kinetic energy (in joules and in electron volts).
Wavelength of an Alpha Particle an alpha particle (m = 6.64 x 10-27 kg) emitted in the radioactive decay of uranium-238 has an energy of 4.20MeV. What is its de Broglie wavelength?
In the Bohr model of the hydrogen atom, what is the de Broglie wavelength for the electron when it is in (a) The n = 1 level and (b) The n = 4 level? In each case, compare the de Broglie wavelength to the circumference 2πrn of the orbit.
(a) A non-relativistic free particle with mass m has kinetic energy K. Derive an expression for the de Broglie wavelength of the particle in terms of m and K. (b) What is the de Broglie wavelength of an 800-eV electron?
Why Don't We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable approximations for the necessary quantities. (b) Will the person in part (a) exhibit wave-like behavior when walking through the "single alit" of a doorway? Why?
What is the de Broglie wavelength for an electron with speed (a) v = 0.480c and (b) v = 0.960c?
(a) If a photon and an electron each have the same energy of 20.0eV, find the wavelength of each. (b) If a photon and an electron each have the same wavelength of 250 run, find the energy' of each.(c) You want to study an organic molecule that is about 250 nm long using either a photon or an
Hydrogen gas (H2) is at 0oC. The mass of a hydrogen atom is 1.67 x 10-27kg. (a) What is the average de Broglie wavelength of the hydrogen molecules? (b) How fast would an electron have to move to have the same de Broglie wavelength as the hydrogen? Do we need to consider relativity for this
Wavelength of a Bullet calculates the de Broglie wavelength of a 5.00-g bullet that is moving at 340m/s. Will the bullet exhibit wavelike properties?
Through what potential difference must electrons be accelerated so they will have? (a) The same wavelength as an x ray of wavelength 0.150 nm and (b) The same energy as the x ray in part (a)?
(a) Approximately how fast should an electron move so it has a wavelength that makes it useful to measure the distance between adjacent atoms in typical crystals (about 0.10 nm)? (b) What is the kinetic energy of the electron in part (a)? (c) What would be the energy of a photon of the same
A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit the diffracted beam shows its first diffraction minima at ± 11.5° from the original direction of the beam when viewed far from the slit. (a) Do we need to use
A beam of neutrons that all have the same energy scatters from the atoms that have a spacing of 0.0910 nm in the surface plane of a crystal. The m = 1 intensity maximum occurs when the angle θ in Fig. 39.3 is 28.6°. What is the kinetic energy (in electron volts) of each neutron in the beam?
A beam of 188-eV electrons is directed at normal incidence onto a crystal surface as shown in Fig. 39.4b. The m = 2 intensity maximum occurs at an angle θ = 60.6°. (a) What is the spacing between adjacent atoms on the surface? (b) At what other angle or angles is there an intensity
A beam of electrons is accelerated from rest through a potential difference of 0.100 kV and then passes through a thin slit the diffracted beam shows its first diffraction minima at ± 11.5° from the original direction of the beam when viewed far from the slit. (a) Do we need to use relativity
A pesky 1.5-mg mosquito is annoying you as you attempt to study physics in your room, which is 5.0 m wide and 2.5 m high. You decide to swat the bothersome insect as it flies toward you, but you need to estimate its speed to make a successful hit. (a) What is the maximum uncertainty in the
By extremely careful measurement, you determine the x-coordinate of a car's center of mass with an uncertainty of only 1.00µm. The car has a mass of 1200 kg. (a) What is the minimum uncertainty in the x-component of the velocity of the car's center of mass as prescribed by the Heisenberg
A 10.0-g marble is gently placed on a horizontal tabletop that is 1.75 m wide. (a) What is the maximum uncertainty in the horizontal position of the marble? (b) According to the Heisenberg uncertainty principle, what is the minimum uncertainty in the horizontal velocity of the marble? (c) In light
A scientist has devised a new method of isolating individual particles. He claims that this method enables him to detect simultaneously the position of a particle along an axis with a standard deviation of 0.12 nm and its momentum component along this axis with a standard deviation of 3.0 x l0-25kg
(a) The x-coordinate of an electron is measured with an uncertainty of 0.20 mm. What is the x-component of the electron's velocity, v" if the minimum percentage uncertainty in a simultaneous measurement of v. is 1.0%?(b) Repeat part (a) for a proton.
An atom in a meta-stable state has a lifetime of 5.2ms. What is the uncertainty in energy of the meta-stable state?
(a) The uncertainty in the y-component of a proton's position is 2.0 x 10-12m. What is the minimum uncertainty in a simultaneous measurement of the y-component of the proton's velocity? (b) The uncertainty in the z-component of an electron’s velocity is 0.250mn/s. What is the minimum uncertainty
The ψ (''psi'') particle has a rest energy of 3097MeV (1 MeV = 106eV). The ψ particle is unstable with a lifetime of 7.6 x 10-21s. Estimate the uncertainty in rest energy of the ψ particle. Express your answer in MeV and as a fraction of the rest energy of the particle.
Particle Lifetime the unstable W+ particle has a rest energy of 80.41GeV (1GeV = 109eV) and an uncertainty in rest energy of 2.06GeV. Estimate the lifetime of the W+ particle.
You want to study a biological specimen by means of a wavelength of 10.0 nm, and you have a choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a 10.0-nm-wavelength photon to the kinetic energy of a 10.0-nm-wavelength electron. (b) In view of
(a) In an electron microscope, what accelerating voltage is needed to produce electrons with wavelength 0.0600 nm? (b) If protons are used instead of electrons, what accelerating voltage is needed to produce protons with wavelength 0.0600 nm?
Consider a wave function given by ψ (x) = A sin kx, where k = 2π/λ and A is areal constant. (a) For what values of x is there the highest probability of finding the particle described by this wave function? Explain. (b) For which values of x is the probability zero? Explain.
Compute | ψ |2 for ψ = ψ sin wt, where ψ is time independent and w is a real constant. Is this a wave function for a stationary state? Why or why not?
Normalization of the Wave Function Consider a particle moving in one dimension which we shall call the x-axis. (a) What does it mean for the wave function of this particle to be normalized? (b) Is the wave function ψ (x) = eax, where a is a positive real number, normalized? Could this be a
A particle is described by a wave function ψ (x) = Ae–ax2, where A and a are real, positive constants. If the value of a is increased, what effect does this have on (a) The particle's uncertainty in position and (b) The particle's uncertainty in momentum? Explain your answers.
Consider the complex-valued function f (x, y) = (x - iy)/ (x + iy). Calculate | f |2.
Particle A is described by the wave function ψ (x, y, z). Particle B is described by the wave function ψ (x, y, z) eiφ, where Φ is a real constant. How does the probability of finding particle A within a volume dV around a certain point in space compare with the probability of
A particle moving in one dimension (the x-axis) is described by the wave function where b = 2.00 m-1,A > 0, and the +x-axis points toward the right(a) Determine A so that the wave function is norma1ized.(b) Sketch the graph of the wave function.(c) Find the probability of finding this particle
Linear Combinations of Wave Functions Let ψ1 and ψ2 be two solutions of Eq. (39.18) with the same energy E. Show that ψ = Bψ1 + Cψ2 is also a solution with energy E, for any values of the constants B and C.
Let ψ1 and ψ2 be two solutions of Eq. (39.18) with energies E1 and E2, respectively, where E1 ≠ E2 Is ψ = Aψ1 + Bψ2, where A and B are nonzero constants, a solution to Eq. (39.18)? Explain your answer.
A beam of 40-eV electrons traveling in the +x-direction passes through a slit that is parallel to the y-axis and 5.0µwide. The diffraction pattern is recorded on a screen 2.5 m from the slit. (a) What is the de Broglie wavelength of the electrons? (b) How much time does it take the electrons to
(a) What is the energy of a photon that has wavelength 0.10µm? (b) Through approximately what potential difference must electrons be accelerated so that they will exhibit wave nature in passing through a pinhole 0.10µm in diameter? What is the speed of these electrons? (c) If protons rather than
Electrons go through a single slit 150 nm wide and strike a screen 24.0cm away. You find that at angles of ±20.0o from the center of the diffraction pattern, no electrons hit the screen hut electrons bit at all points closer to the center. (a) How fast were these electrons moving when they went
A beam of electrons is accelerated from rest and then passes through a pair of identical thin slits that are 1.25 nm apart. You observe that the first double-slit interference dark fringe occurs at ±18.0° from the original direction of the beam when viewed on a distant screen. (a) Are these
A beam of protons and a beam of alpha particles (of mass 6.64 x 10-27 kg and charge +2e) are accelerated from rest through the same potential difference and pass through identical circular holes in a very thin, opaque film. When viewed far from the hole, the diffracted proton beam forms its first
An electron beam and a photon beam pass through identical slits. On a distant screen, the first dark fringe occurs at the same angle for both of the beams. The electron speeds are much slower than that of light. (a) Express the energy of a photon in terms of the kinetic energy K of one of the
Coherent light is passed through two narrow slits whose separation is 40.0µm. The second-order bright fringe in the interference pattern is located at an angle of 0.0300 rad. If electrons are used instead of light, what must the kinetic energy (in electron volts) of the electrons be if they are to
What is the de Broglie wavelength of a red blood cell, with mass 1.00 x 10-11 g, that is moving with a speed of 0.400cm/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?
High-speed electrons are used to probe the interior structure of the atomic nucleus. For such electrons the expression λ = h/p still holds, but we must use the relativistic expression for momentum, p = mv/√1 – v2/c2. (a) Show that the speed of an electron that has de Broglie
(a) What is the de Broglie wavelength of an electron accelerated from rest through a potential increase of 125 V? (b) What is the de Broglie wavelength of an alpha particle (q = + 2e, m = 6.64 x 10-27 kg) accelerated from rest through a potential drop of 125 V?
Suppose that the uncertainty of position of an electron is equal to the radius of the n = 1 Bohr orbit for hydrogen. Calculate the simultaneous minimum uncertainty of the corresponding momentum component and compare this with the magnitude of the momentum of the electron in the n = 1 Bohr orbit.
(a) A particle with mass m has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle?(b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.
Proton Energy in a Nucleus the radii of atomic nuclei are of the order of 5.0 x l0-l5 m. (a) Estimate the minimum uncertainty in the momentum of a proton if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the
Electron Energy in a Nucleus the radii of atomic nuclei are of the order of 5.0 x 10-15 m. (a) Estimate the minimum uncertainty in the momentum of an electron if it is confined within a nucleus. (b) Take this uncertainty in momentum to be an estimate of the magnitude of the momentum. Use the
In a TV picture tube the accelerating voltage is 15.0kV and the electron beam passes through an aperture 0.50 mm in diameter to a screen 0.300 m away. (a) Calculate the uncertainty in the component of the electron's velocity perpendicular to the line between aperture and screen. (b) What is the
The neutral pion (π0) is an unstable particle produced in high-energy particle collisions. Its mass is about 264 times that of the electron, and it exists for an average lifetime of 8.4 x 10-17 s before decaying into two gamma-ray photons. Using the relationship E = mc2 between rest mass and
Quantum Effects in Daily Life a 1.25-mg insect flies through a 4.00-mm-diameter hole in an ordinary window screen. The thickness of the screen is 0.500 mm. (a) What should be the approximate wavelength and speed of the insect for her to show wave behavior as she goes through the hole? (b) At the
Doorway Diffraction if your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take
Doorway Diffraction if your wavelength were 1.0 m, you would undergo considerable diffraction in moving through a doorway. (a) What must your speed be for you to have this wavelength? (Assume that your mass is 60.0 kg.) (b) At the speed calculated in part (a), how many years would it take you to
You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic? How do you know? (b) What accelerating potential is needed?(c) What is the kinetic energy of the
For x rays with wavelength 0.0300 nm, the m = 1 intensity maximum for a crystal occurs when the angle θ in Fig. 36.23c is 35.8°. At what angle θ does the m = 1 maximum occur when a beam of 4.50-keV electrons is used instead? Assume that the electrons also scatter from the atoms in the
You intend to use an electron microscope to study the structure of some crystals. For accurate resolution, you want the electron wavelength to be 1.00 nm. (a) Are these electrons relativistic? How do you know? (b) What accelerating potential is needed?(c) What is the kinetic energy of the electrons
Zero-Point Energy Consider a particle with mass m moving in a potential U = ½ kx2, as in a mass-spring system. The total energy of the particle is E = p2/2m + ½ kx2. Assume that p and x are approximately related by the Heisenberg uncertainty principle, px ≈ h. (a) Calculate the minimum
A particle with mass m moves in a potential U(x) = A |x|, where A is a positive constant. In a simplified picture, quarks have a potential energy of interaction of approximately this form, where x represents the separation between a pair of quarks. Because U(x) → ∞ as x → ∞,
The discussion in Section 39.5 shows that the wave function ψ = ψe-iwt is a stationary state, where ψ is time independent and w is a real (not complex) constant. Consider the wave function ψ = ψ1e-iwt + ψ2ε-iw2t, where ψ1 and ψ2 are different time
The Time-Dependent Schrodinger Equation (39.18) is the time-independent Schrodinger equation in one dimension. The time-dependent Schrodinger equation is If ?(x) is a solution to Eq. (39.18) with energy E, show that the time-dependent function ? (x, t) = ? (x)e-iwt is a solution to the
Time-Dependent Wave Function for a Free Particle One example of a time-dependent wave function is that of a free particle [one for which U(x) = 0 for all x] of energy E and x-component of momentum p. From the de Broglie relationships (see Section 39.1), such a particle has associated with it a
(a) What is the uncertainty in the ball's horizontal momentum, in a direction perpendicular to that in which it is being thrown, if the student throwing the ball knows that it is located within a cube with volume 125cm3 at the time she throws it? (b) By what horizontal distance could the ball miss
A particle is described by the normalized wave function ψ(x, y, z) = Axe-ax2 e -βy2 e-yz2', where A, a, β, and γ are all real, positive constants. The probability that the particle will be found in the infinitesimal volume dx dy dz centered at the point (x0, y0, z0) is | ψ
A particle is described by the normalized wave function ψ(x, y, z) = Ae –a(x2+y2+z2), where A and a are real, positive constants. (a) Determine the probability of finding the particle at a distance between r and r + dr from the origin. b) For what value of r does the probability in part
Consider the wave packet defined byLet B(k) = e €“ a2k2 (a) The function B(k) has its maximum value at k = 0. Let kh, be the value of k at which B(k) bas fallen to half its maximum value, and define the width of B(k) as wk = kh. In terms of α, what is wk?(b) Use integral tables
(a) Using the integral in Problem 39.68, determine the wave function ψ (x) for a function B(k) given by This represents an equal combination of all wave numbers between 0 and k0. Thus ψ (x) represents a particle with average wave number k0/2, with a total spread or uncertainty in wave
The wave nature of particles results in the quantum mechanical situation that a particle confined in a box can assume only wavelengths that result in standing waves in the box, with nodes at the box walls.(a) Show that an electron confined in a one dimensional box of length L will have energy
You have entered a contest in which the contestants drop a marble with mass 20.0 g from the roof of a building onto a small target 25.0 m below. From uncertainty considerations, what is the typical distance by which you will miss the target, given that you aim with the highest possible precision?
Ground-Level Billiards (a) Find the lowest energy level for a particle in a box if the particle is a billiard ball (m = 0.20kg) and the box has a width of 1.5 m, the size of a billiard table. (Assume that the billiard ball slides without friction rather than rolls. That is, ignore the rotational
A proton is in a box of width L. What must the width of the box be for the ground-level energy to be 5.0 MeV, a typical value for the energy with which the particles in a nucleus are bound? Compare your result to the size of a nucleus--that is, on the order of 10-14 m.
Find the width L of a one-dimensional box that would correspond to the absolute value of the ground state of a hydrogen atom.
(b) For a box with the width calculated in part (a), what is the ground-state energy? How does this correspond to the ground-state energy of a hydrogen atom? (c) Do you think a one-dimensional box is a good model for a hydrogen atom? Explain.
A certain atom requires 3.0eV of energy to excite an electron from the ground level to the first excited level. Model the atom as an electron in a box and find the width L of the box.
(a) For which values of x, if any, in the range from 0 to L is the probability of finding the particle zero? (b) For which values of x is the probability highest? (c) In parts (a) and (b) are your answers consistent with Fig. 40.8? Explain.
Repeat Exercise 40.6 for the particle in the first excited level.
(a) Show that ψ = A sin kx is a solution to Eq. (40.3) if k = √2mE/h. (b) Explain why this is an acceptable wave function for a particle in a box with rigid walls at x = 0 and x = L only if k is an integer multiple of π/L.
(a) Repeat Exercise 40.8 for ψ = A cos kx. (b) Explain why this cannot be an acceptable wave function for a particle in a box with rigid walls at x = 0 and x = L no matter what the value of k.
(a) Find the excitation energy from the ground level to the third excited level for an electron confined to a box that has a width of 0.125 nm. (b) The electron makes a transition from the n = 1 to n = 4 level by absorbing a photon. Calculate the wavelength of this photon.
An electron is in a box of width 3.0 x l0-10 m. What are the de Broglie wavelength and the magnitude of the momentum of the electron if it is in? (a) The n = 1 level; (b) The n = 2 level;(c) The n = 3 level? In each case how does the wavelength compare to the width of the box?
Show that the time-dependent wave function given by Eq. (40.15) is a solution to the one-dimensional Schrodinger equation, Eq. (40.1).
(a) Show that ψ = A sin kx, where k is a constant, is not a solution of Eq. (40.1) for U = U0 and E < U0. 9 (b) Is this ψ a solution for E > U0?
An electron is moving past the square well shown in Fig. 40.6. The electron has energy E = 3U0. What is the ratio of the de Broglie wavelength of the electron in the region x > L to the wavelength for 0 < x < L?
An electron is bound in a square well of depth U0 = 6E∞. What is the width of the well if its ground-state energy is 2.00eV?
An electron is bound in a square well of width 1.50 nm and depth U0 = 6E∞. If the electron is initially in the ground level and absorbs a photon, what maximum wavelength can the photon have and still liberate the electron from the well?
Calculate d2 ψ/dx2 for the wave function of Eq. (40.17), and show that the function is a solution of Eq. (40.16).
Calculate d2 ψ/dx2 for the wave function of Eq. (40.19), and show that for any values of C and D it is a solution of Eq. (40.1) for x < 0 and x < L when U = U0 and E < U0.
A proton is bound in a square well of width 4.0 fm = 4.0 x 10-15 m. The depth of the well is six times the ground-level energy E= of the corresponding infinite well. If the proton makes a transition from the level with energy E, to the level with energy E, by absorbing a photon, find the wavelength

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Physics by Example 1st edition W. G. Rees - Solutions

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