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physics
modern physics

Modern Physics 6th Edition Paul A. Tipler, Ralph Llewellyn - Solutions

The wave function describing a state of an electron confined to move along the x axis is given at time zero byFind the probability of finding the electron in a region dx centered at (a) x = 0, (b) x = σ, and (c) x = 2σ. (d) Where is the electron most likely to be found? (x,0)
A radar transmitter used to measure the speed of pitched baseballs emits pulses of 2.0 cm wavelength that are 0.25 μs in duration.(a) What is the length of the wave packet produced? (b) To what frequency should the receiver be tuned?(c) What must be the minimum bandwidth of the receiver?
A certain standard tuning fork vibrates at 880 Hz. If the tuning fork is tapped, causing it to vibrate, then stopped a quarter of a second later, what is the approximate range of frequencies contained in the sound pulse that reached your ear?
If a phone line is capable of transmitting a range of frequencies Δf = 5000 Hz, what is the approximate duration of the shortest pulse that can be transmitted over the line?
(a) You are given the task of constructing a double-slit experiment for 5 eV electrons. If you want the first minimum of the diffraction pattern to occur at 5º, what must be the separation of the slits?(b) How far from the slits must the detector plane be located if the first minima on each side
A 100 g rigid sphere of radius 1 cm has a kinetic energy of 2 J and is confined to move in a force-free region between two rigid walls separated by 50 cm. (a) What is the probability of finding the center of the sphere exactly midway between the two walls? (b) What is the probability of
A particle moving in one dimension between rigid walls separated by a distance L has the wave function Ψ (x) = A sin(πx/L). Since the particle must always be located between the walls, what must be the value of A?
A ladybug 5 mm in diameter with a mass of 1.0 mg being viewed through a low-power magnifier with a calibrated reticule is observed to be stationary with an uncertainty of 10-2 mm. How fast might the ladybug actually be walking?
222Rn decays by the emission of an a particle with a lifetime of 3.823 days. The kinetic energy of the a particle is measured to be 5.490 MeV. What is the uncertainty in this energy? Describe in one sentence how the finite lifetime of the excited state of the radon nucleus translates into an energy
If the uncertainty in the position of a wave packet representing the state of a quantum-system particle is equal to its de Broglie wavelength, how does the uncertainty in momentum compare with the value of the momentum of the particle?
An excited state of a certain nucleus has a half-life of 0.85 ns. Taking this to be the uncertainty Δt for emission of a photon, calculate the uncertainty in the frequency Δf, using Equation 5-25. If λ = 0.01 nm, find Δf/f. ΔΕΔt – † 5-25
The lifetimes of so-called resonance particles cannot be measured directly but is computed from the energy width (or uncertainty) of the scattering cross section versus energy graph. For example, the scattering of a pion (p meson) and a proton can produce a short-lived Δ resonance particle with a
In order to locate a particle, for example, an electron, to within 5 x 10-12 m using electromagnetic waves (“light”), the wavelength must be at least this small. Calculate the momentum and energy of a photon with λ = 5 x 10-12 m. If the particle is an electron with Δx = 5 x 10-12m, what is
Laser pulses of femtosecond duration can be produced, but for such brief pulses it makes no sense to speak of the pulse’s color. To demonstrate this, compute the time duration of a laser pulse whose range of frequencies covers the entire visible spectrum (4.0 x 1014 Hz to 7.5 x1014 Hz).
Show that Equation 5-11 is satisfied by y = f (φ), where φ = x - vt for any function f. ay 1 a²y əx² var 5-11
Protons and neutrons in nuclei are bound to the nucleus by exchanging pions (π mesons) with each other. This is possible to do without violating energy conservation provided the pion is re-absorbed within a time consistent with the Heisenberg uncertainty relations. Consider the emission reaction p
Show that the relation Δps Δs > ћ can be written ΔL Δφ > ћ for a particle moving in a circle about the z axis, where ps is the linear momentum tangential to the circle, s is the arc length, and L is the angular momentum. How well can the angular position of the electron be
De Broglie developed Equation 5-2 initially for photons, assuming that they had a small but finite mass. His assumption was that RF waves with λ = 30 m traveled at a speed of at least 99 percent of that of visible light with λ = 500 nm. Beginning with the relativistic expression hf = ϒmc2,
A particle with a mass of 4 g is moving at 100 m/s. What size aperture would be needed in order to observe diffraction of this particle wave? Explain why no common object could pass through such an aperture.
Neutrons and protons in atomic nuclei are confined within a region whose diameter is about 10-15 m.(a) At any given instant, how fast might an individual proton or neutron be moving? (b) What is the approximate kinetic energy of a neutron that is localized to within such a region? (c)
A particle of mass m moves in a one-dimensional box of length L. (Take the potential energy of the particle in the box to be zero so that its total energy is its kinetic energy p2/2m.) Its energy is quantized by the standing-wave condition n(λ/2) = L, where λ is the de Broglie wavelength of the
(a) Suppose that a particle of mass m is constrained to move in a one-dimensional space between two infinitely high barriers located A apart. Using the uncertainty principle, find an expression for the zero-point (minimum) energy of the particle.(b) Using your result from (a), compute the minimum
A proton and a bullet each move with a speed of 500 m/s, measured with an uncertainty of 0.01 percent. If measurements of their respective positions are made simultaneous with the speed measurements, what is the minimum uncertainty possible in the position measurements?
An electron and a positron are moving toward each other with equal speeds of 3 x 106 m/s. The two particles annihilate each other and produce two photons of equal energy.(a) What were the de Broglie wavelengths of the electron and positron? Find the(b) Energy(c) Momentum(d) Wavelength of each
It is possible for some fundamental particles to “violate” conservation of energy by creating and quickly re-absorbing another particle. For example, a proton can emit a π+ according to p = n + π+, where the n represents a neutron. The π+ has a mass of 140 MeV/c2. The re-absorption must
Suppose that you drop BBs onto a bull’s-eye marked on the floor. According to the uncertainty principle, the BBs do not necessarily fall straight down from the release point to the center of the bull’s-eye but are affected by the initial conditions. (a) If the location of the release point
Using the first-order Doppler-shift formula f= f0 (1 + v/)c2, calculate the energy shift of a 1 eV photon emitted from an iron atom moving toward you with energy (3/2)kT at T = 300 K. Compare this Doppler line broadening with the natural line width calculated in Example 5-9. Repeat the
Calculate the order of magnitude of the shift in energy of a (a) 1 eV photon and(b) 1 MeV photon resulting from the recoil of an iron nucleus. Do this by first calculating the momentum of the photon and then by calculating p2/2m for the nucleus using that value of momentum. Compare with the
If a particle is deflected by 0.01º in each collision, about how many collisions would be necessary to produce an rms deflection of 10º? (Use the result from the one-dimensional random walk problem in statistics stating that the rms deflection equals the magnitude of the individual deflections
Consider the foil and a particle energy in Problem 4-6. Suppose that 1000 of those particles suffer a deflection of more than 25º. (a) How many of these are deflected by more than 45º? (b) How many are deflected between 25º and 45º? (c) How many are deflected between 75º and
The radius of the n = 1 orbit in the hydrogen atom is a0 = 0.053 nm. (a) Compute the radius of the n = 6 orbit. (b) Compute the radius of the n = 6 orbit in singly ionized helium (He+), which is hydrogenlike, that is, it has only a single electron outside the nucleus.
Calculate the three longest wavelengths in the Lyman series (nf = 1) in nm, and indicate their position on a horizontal linear scale. Indicate the series limit (shortest wavelength) on this scale. Are any of these lines in the visible spectrum?
An atom in an excited state will on the average undergo a transition to a state of lower energy in about 10-8 seconds. If the electron in a doubly ionized lithium atom (Li+2, which is hydrogenlike) is placed in the n = 4 state, about how many revolutions around the nucleus does it make before
It is possible for a muon to be captured by a proton to form a muonic atom. A muon is identical to an electron except for its mass, which is 105.7 MeV/c2. (a) Calculate the radius of the first Bohr orbit of a muonic atom. (b) Calculate the magnitude of the lowest energy state. (c)
In the lithium atom (Z = 3) two electrons are in the n = 1 orbit and the third is in the n = 2 orbit. (Only two are allowed in the n = 1 orbit because of the exclusion principle, which will be discussed in Chapter 7.) The interaction of the inner electrons with the outer one can be approximated by
Draw to careful scale an energy-level diagram for hydrogen for levels with n = 1, 2, 3, 4, ∞. Show the following on the diagram: (a) The limit of the Lyman series, (b) The Hβ line, (c) The transition between the state whose binding energy (= energy needed to remove the electron
A hydrogen atom at rest in the laboratory emits the Lyman a radiation. (a) Compute the recoil kinetic energy of the atom. (b) What fraction of the excitation energy of the n = 2 state is carried by the recoiling atom?
(a) Draw accurately to scale and label completely a partial energy-level diagram for C5+ including at minimum the energy levels for n = 1, 2, 3, 4, 5, and ∞. (b) Compute the wavelength of the spectral line resulting from the n = 3 to the n = 2 transition, the C5+Ha line. (c) In what
The electron-positron pair that was discussed in Chapter 2 can form a hydrogenlike system called positronium. Calculate(a) The energies of the three lowest states and (b) The wavelength of the Lyman a and β lines. (Detection of those lines is a “signature” of positronium formation.)
(a) Compute the energy of an electron in the n = 1 (K shell) of tungsten, using Z - 1 for the effective nuclear charge. (b) The experimental result for this energy is 69.5 keV. Assume that the effective nuclear charge is Z - σ, where s is called the screening constant, and calculate s from
Suppose that, in a Franck-Hertz experiment, electrons of energy up to 13.0 eV can be produced in the tube. If the tube contained atomic hydrogen,(a) what is the shortest-wavelength spectral line that could be emitted from the tube? (b) List all of the hydrogen lines that can be emitted by this
If we could somehow fill a Franck-Hertz tube with positronium, what cathode-grid voltage would be needed to reach the second current decrease in the positronium equivalent of Figure 4-23? Figure 4-23 1 350 300 250 200 150 100 50 0 0 00 8 5 Vo. V 10 15
The Li2+ ion is essentially identical to the H atom in Bohr’s theory, aside from the effect of the different nuclear charges and masses. (a) What transitions in Li2+ will yield emission lines whose wavelengths are very nearly equal to the first two lines of the Lyman series in
If electric charge did not exist and electrons were bound to protons by the gravitational force to form hydrogen, derive the corresponding expressions for a0 and En and compute the energy and frequency of the Ha line and the limit of the Balmer series. Compare these with the corresponding
(a) Write a computer program for your personal computer or programmable calculator that will provide you with the spectral series of H-like atoms. Inputs to be included are ni, nf, Z, and the nuclear mass M. Outputs are to be the wavelengths and frequencies of the first six lines and the series
In this problem you are to obtain the Bohr results for the energy levels in hydrogen without using the quantization condition of Equation 4-17. In order to relate Equation 4-14to the Balmer-Ritz formula, assume that the radii of allowed orbits are given by rn = n2r0, where n is an integer and r0 is
Listed in the table are the La x-ray wavelengths for several elements. Construct a Moseley plot from these data. Compare the slope with the appropriate one in Figure 4-19. Determine and interpret the intercept on your graph, using a suitably modified version of Equation 4-35.Figure 4-19Equation
A small shot of negligible radius hits a stationary smooth, hard sphere of radius R, making an angle β with the normal to the sphere, as shown in Figure 4-25. It is reflected at an equal angle to the normal. The scattering angle is θ = 180º - 2β, as shown.(a) Show by the geometry of the figure
Figure 3-15b shows the Ka and Kβ characteristic x rays emitted by a molybdenum (Mo) target in an x-ray tube whose accelerating potential is 35 kV. The wavelengths are Ka = 0.071 nm and Kβ = 0.063 nm.(a) Compute the corresponding energies of these photons. (b) Suppose we wish to prepare a
The Ka, La, and Ma x rays are emitted in the n = 2 → n = 1, n = 3 → n = 2, and n = 4 → n = 3 transitions respectively. For calcium (Z = 20) the energies of these transitions are 3.69 keV, 0.341 keV, and 0.024 keV, respectively. Suppose that energetic photons impinging on a calcium surface
In an a scattering experiment, the area of the a particle detector is 0.50 cm2. The detector is located 10 cm from a 1.0-μm-thick silver foil. The incident beam carries a current of 1.0 nA, and the energy of each a particle is 6.0 MeV. How many a particles will be counted per second by the
(a) The current i due to a charge q moving in a circle with frequency frev is q frev. Find the current due to the electron in the first Bohr orbit. (b) The magnetic moment of a current loop is iA, where A is the area of the loop. Find the magnetic moment of the electron in the first Bohr orbit
Geiger and Marsden used a particles with 7.7 MeV kinetic energy and found that when they were scattered from thin gold foil, the number observed to be scattered at all angles agreed with Rutherford’s formula. Use this fact to compute an upper limit on the radius of the gold nucleus.
Three isotopes of hydrogen occur in nature; ordinary hydrogen, deuterium, and tritium. Their nuclei consist of, respectively, 1 proton, 1 proton and 1 neutron (deuteron), and 1 proton and 2 neutrons (triton). The masses of the three nuclei are given in Table 11-1.(a) Use Equation 4-26 to determine
A Rydberg hydrogen atom is in the n = 45 energy state. (a) What is the energy difference (in eV) between this state and the n = 46 level? (b) What is the ionization energy of the atom in the n = 45 level? (c) What are the frequency and wavelength of a photon emitted in the n = 46 →
Electrons in the Franck-Hertz tube can also have elastic collisions with the Hg atoms. If such a collision is a head-on, what fraction of its initial kinetic energy will an electron lose, assuming the Hg atom to be at rest? If the collision is not head-on, will the fractional loss be greater or
Show that a small change in the reduced mass of the electron produces a small change in a spectral line given by Δλ/λ = Δμ/μ. Use this to calculate the difference Δλ in the Balmer red line λ = 656.3 nm between hydrogen and deuterium, which has a nucleus with twice the mass of hydrogen.
Consider the Franck-Hertz experiment with Hg vapor in the tube and the voltage between the cathode and the grid equal to 4.0 V, that is, not enough to for the electrons to excite the Hg atom’s first excited state. Therefore, the electron–Hg atom collisions are elastic.(a) If the kinetic energy
A particular radiating cavity has the maximum of its spectral distribution of radiated power at a wavelength of 27.0 μm (in the infrared region of the spectrum). The temperature is then changed so that the total power radiated by the cavity doubles.(a) Compute the new temperature.(b) At what
Equation 3-3 suggests how a velocity selector for particles or mixtures of different particles all having the same charge can be made. Suppose you wish to make a velocity selector that allows undeflected passage for electrons whose kinetic energy is 5.0 x 104 eV. The electric field available to you
On drop #16, Millikan measured the following total charges, among others, at different times:What value of the fundamental quantized charge e do these numbers imply? 25.41 X 10-19 C 20.64 × 10-19 C 17.47 x 10-19 C 19.06 x 10-19 C 12.70 X 10-19 C 14.29 X 10-19 C
If electrons have kinetic energy of 2000 eV, find (a) Their speed,(b) The time needed to traverse a distance of 5cm between plates D and E in Figure 3-1, and (c) The vertical component of their velocity after passing between the plates if the electric field is 3.33 x 103 V/m.Figure 3-1 в
One variation of the Millikan oil-drop apparatus arranges the electric field horizontal, rather than vertical, giving charged droplets acceleration in the horizontal direction. The result is that the droplet falls in a straight line that makes an angle θ with the vertical.Show thatwhere v't is the
A beam of charged particles consisting of protons, electrons, deuterons, and singly ionized helium atoms and H2 molecules all pass through a velocity selector, all emerging with speeds of 2.5 x 106 m/s. The beam then enters a region of uniform magnetic field B = 0.40 T directed perpendicular to
Consider Thomson’s experiment with the electric field turned “off.” If the electrons enter a region of uniform magnetic field B and length ℓ, show that the electrons are deflected through an angle θ ≈ eℓB/mu for small values of ℓ. (Assume that the electrons are moving at
Show that the electric field needed to make the rise time of the oil drop equal to its field-free fall time is 8 = 2mg/q.
A cosmic-ray proton approaches Earth vertically at the equator, where the horizontal component of Earth’s magnetic field is 3.5 x 10-5 T. If the proton is moving at 3.0 x106 m/s, what is the ratio of the magnetic force to the gravitational force on the proton?
In J. J. Thomson’s first method (see Problem 3-46), the heat capacity of the beam stopper was about 5 x10-3 cal/°C and the temperature increase was about 2°C. How many 2000 eV electrons struck the beam stopper?
Use the result of Example 3-4 and Equations 3-4 and 3-6 to express Stefan’s constant in terms of h, c, and k. Using the known values of these constants, calculate Stefan’s constant. R = GT 3-4
Show that Planck’s law, Equation 3-18, expressed in terms of the frequency f, is u(f) 8πf² hf c³ ehf/kT - 1 C
A charged oil droplet falls 5.0 mm in 20.0 s at terminal speed in the absence of an electric field. The specific gravity of air is 1.35 x10-3 and that of oil is 0.75. The viscosity of air is 1.80 x10-5 N . s/m2.(a) What are the mass and radius of the drop? (b) If the droplet carries two units
Find λm for blackbody radiation at (a) T = 3 K, (b) T = 300 K, and (c) T = 3000 K.
Use Planck’s law, Equation 3-18, to derive the constant in Wien’s law, Equation 3-5. XT = constant = m 2.898 x 10³ m. K 3-5
As noted in the chapter, the cosmic microwave background radiation fits the Planck equation for a blackbody at 2.7 K. (a) What is the wavelength at the maximum intensity of the spectrum of the background radiation? (b) What is the frequency of the radiation at the maximum? (c) What
Find the temperature of a blackbody if its spectrum has its peak at (a) λm = 700 nm (visible), (b) λm = 3 cm (microwave region), and (c) λm = 3 m (FM radio waves).
If the absolute temperature of a blackbody is doubled, by what factor is the total emitted power increased?
A certain very bright star has an effective surface temperature of 20,000 K.(a) Assuming that it radiates as a blackbody, what is the wavelength at which u(λ) is maximum? (b) In what part of the electromagnetic spectrum does the maximum lie?
The energy reaching Earth from the Sun at the top of the atmosphere is 1.36 x 103 W/m2, called the solar constant. Assuming that Earth radiates like a blackbody at uniform temperature, what do you conclude is the equilibrium temperature of Earth?
A 40 W incandescent bulb radiates from a tungsten filament operating at 3300 K. Assuming that the bulb radiates like a blackbody, (a) what are the frequency fm and the wavelength λm at the maximum of the spectral distribution? (b) If fm is a good approximation of the average frequency of
Use Compton’s equation (Equation 3-25) to compute the value of Δλ in Figure 3-17d. To what percent shift in the wavelength does this correspond?Figure 3-17(d) λε - λι h muc (1 - cost) 3-25
Using apparatus similar to that in Figure 3-8, the photoelectric effect data below were measured.(a) From a graph of Ek,max versus f, find a value for Planck’s constant. (b) By what percentage (+ or -) does the value found in (a) differ from the accepted value?(c) Based on the graph plotted
The wavelengths of visible light range from about 380 nm to about 750 nm.(a) What is the range of photon energies (in eV) in visible light? (b) A typical FM radio station’s broadcast frequency is about 100 MHz. What is the energy of an FM photon of that frequency?
The orbiting space shuttle moved around Earth well above 99 percent of the atmosphere, yet it still accumulated an electric charge on its skin due, in part, to the loss of electrons caused by the photoelectric effect with sunlight. Suppose the skin of the shuttle were coated with Ni, which has a
(a) If 5 percent of the power of a 100 W bulb is radiated in the visible spectrum, how many visible photons are radiated per second? (b) If the bulb is a point source radiating equally in all directions, what is the flux of photons (number per unit time per unit area) at a distance of 2 m?
A Compton scattering experiment yielded the data in the table below.(a) Using Equation 3-25 as a guide, construct an appropriate graph that enables you to obtain a value for the Compton wavelength of the electron. (b) By what percent (+ or -) does your result differ from the accepted value?
The work function of molybdenum is 4.22 eV. (a) What is the threshold frequency for the photoelectric effect in molybdenum? (b) Will yellow light of wavelength 560 nm cause ejection of photoelectrons from molybdenum? Prove your answer.
The NaCl molecule has a bond energy of 4.26 eV; that is, this energy must be supplied in order to dissociate the molecule into neutral Na and Cl atoms.(a) What are the minimum frequency and maximum wavelength of the photon necessary to dissociate the molecule? (b) In what part of the
Under optimum conditions, the eye will perceive a flash if about 60 photons arrive at the cornea. How much energy is this in joules if the wavelength of the light is 550 nm?
The longest wavelength of light that will cause emission of electrons from cesium is 653 nm. (a) Compute the work function for cesium. (b) If light of 300 nm (ultraviolet) were to shine on cesium, what would be the energy of the ejected electrons?
x-ray tubes currently used by dentists often have accelerating voltages of 80 kV. What is the minimum wavelength of the x rays they produce?
Compton’s equation (Equation 3-25) indicates that a graph of λ2 versus (1- cos θ) should be a straight line whose slope h/mc allows a determination of h. Given that the wavelength of λ1 in Figure 3-17 is 0.0711 nm, compute λ2 for each scattering angle in the figure and graph the results
In the Compton scattering of a photon with energy E1 from an electron at rest, show that the energy of the scattered photon E2 is given by E₂ = E₁ (E₁/mc²) (1 cos ) + 1
Find the momentum of a photon in eV/c and in kg·m/s if the wavelength is (a) 400 nm, (b) 1 Å = 0.1 nm, (c) 3 cm, and (d) 2 nm.
Gamma rays emitted by radioactive nuclei also exhibit measurable Compton scattering. Suppose a 0.511 MeV photon from a positron-electron annihilation scatters at 110° from a free electron. What are the energies of the scattered photon and the recoiling electron? Relative to the initial direction
Assuming that the difference between Thomson’s calculated e/m in his second experiment (see Figure 3-19) and the currently accepted value was due entirely to his neglecting the horizontal component of Earth’s magnetic field outside the deflection plates, what value for that component does the
Compute Δλ for photons scattered at 120° from (a) Free protons, (b) Free electrons, and (c) N2 molecules in air.
Millikan’s data for the photoelectric effect in lithium are shown in the table.(a) Graph the data and determine the work function for lithium. (b) Find the value of Planck’s constant directly from the graph in (a).(c) The work function for lead is 4.14 eV.Which, if any, of the wavelengths
When light of wavelength 450 nm is incident on potassium, photoelectrons with stopping potential of 0.52 V are emitted. If the wavelength of the incident light is changed to 300 nm, the stopping potential is 1.90 V. Using only these numbers together with the values of the speed of light and the
Show that the maximum kinetic energy Ek, called the Compton edge, that a recoiling electron can carry away from a Compton scattering event is given by Ex = 2E² hf 1+ mc²/2hf 2E + mc²
In his first e/m experiment Thomson determined the speed of electrons accelerated through a potential ΔV by collecting them in an insulated beam stopper and measuring both the total collected charge Q and the temperature rise ΔT of the beam stopper.(a) Show that with those measurements, he could

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