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Concepts of Modern. Physics. Sixth Edition. Arthur Beiser. Boston Burt Ridge, II, Dubuque, IA Instructor's Solution Manual for Fundamentals of Physics. Problem Solutions. 1. If the speed of light were smaller than it is, would relativistic phenomena soundofheaven.info Beiser – Modern Physics. Concepts of Modern. Physics. Sixth Edition. Arthur Beiser. Boston Burr Ridge, IL . It can have a variety of solutions, including complex ones.
A single photon is emitted in this process. Differentiating the above expression for P r with respect to r and setting the derivative equal to zero. Female Branch: Equation 4. The best part?
Determine the distance of closest approach of 1. In this case, at the point of closest approach the proton will have no kinetic energy, and so the potential energy at closest approach will be the initial kinetic energy, taking the potential energy to be zero in the limit of very large separation.
Equating these energies, m. What is the shortest wavelength present in the Brackett series of spectral lines? In the Bohr model, the electron is in constant motion. How can such an electron have a negative amount of energy? For the system to be bound, the total energy, the sum of the positive kinetic energy and the total negative potential energy, must be negative.
For a classical particle subject to an inverse-square attractive force such as two oppositely charged particles or two uniform spheres subject to gravitational attraction in a circular orbit, the potential energy is twice the negative of the kinetic energy.
R Inha University Department of Physics 9. This quantity got its name because it first appeared in a theory by the German physicist Arnold Sommerfeld that tried to explain the fine structure in spectral lines multiple lines close together instead of single lines by assuming that elliptical as well as circular orbits are possible in the Bohr model.
Combining to find v 1 2 ,. A close cheek of the units is worthwhile; treating the units as algebraic quantities the units as given in the above calculation are.
Find the quantum number that characterizes the earth's orbit around the sun. The earth's mass is 6. Compare the uncertainty in the momentum of an electron confined to a region of linear dimension a o with the momentum of an electron in a ground-state Bohr orbit. What effect would you expect the rapid random motion of the atoms of an excited gas to have on the spectral lines they produce? A proton and an electron, both at rest initially, combine to form a hydrogen atom in the ground state.
A single photon is emitted in this process. What is its wavelength? The energy of the photon emitted is then -E l , and the wavelength is nm, In what part of the spectrum is this? A beam of electrons bombards a sample of hydrogen. Through what potential difference must the electrons have been accelerated if the first line of the Balmer series is to be emitted?
A potential difference of The longest wavelength in the Lyman series is Use the figures to find the longest wavelength of light that could ionize hydrogen. The energies are proportional to the reciprocals of the wavelengths, and so the wavelength of the photon needed to ionize hydrogen is nm. When an excited atom emits a photon, the linear momentum of the photon must be balanced by the recoil momentum of the atom.
As a result, some of the excitation energy of the atom goes into the kinetic energy of its recoil. Is the effect a major one? A nonrelativistic calculation is sufficient here. The fact that this mass change is too small to measure that is, the change is measured indirectly by measuring the energies of the emitted photons means that a nonrelativistic calculation should suffice.
Equation 4. As in Problem , a relativistic calculation is manageable; the result would be ,. In the above, the rest energy of the hydrogen atom is from the front endpapers. Find the wavelength of the photon emitted when the muonic atom drops to its ground state. In what part of the spectrum is this wavelength?
A mixture of ordinary hydrogen and tritium, a hydrogen isotope whose nucleus is approximately 3 times more massive than ordinary hydrogen, is excited and its spectrum observed. The difference between the wavelengths would then be The values of R and RT are proportional to the respective reduced masses, and their ratio is. Inserting numerical values, nm.
Find the wavelength of the photon emitted in this process if the electron is assumed to have had no kinetic energy when it combined with the nucleus. The scale is close, but not exact, and of course there are many more levels corresponding to higher n. The emitted photon's wavelength is nm. The Rutherford scattering formula fails to agree with the data at very small scattering angles. Can you think of a reason? To these nonpenetrating particles, the nucleus is either partially or completely screened by the atom's electron cloud, and the scattering analysis, based on a pointlike positively charged nucleus, is not applicable.
A certain ruby laser emits 1. What fraction of a beam of 7. Through what angle will it be scattered? The scattering angle is then.
Show that twice as many alpha particles are scattered by a foil through angles between 60 o and 90 o as are scattered through angles of 90 o or more. This suggests that we can treat a photon that passes near the sun in the same way as Rutherford treated an alpha particle that passes near a nucleus, with an attractive gravitational force replacing the repulsive electrical force. Adapt Eq. The mass and radius of the sun are respectively 2.
In fact, general relativity shows that this result is exactly half the actual deflection, a conclusion supported by observations made during solar clipses as mentioned in Sec.
Which of the wave functions in Fig. Why not? Figure c has discontinuous derivative in the shown interval. Figure d is finite everywhere in the shown interval. Figure f is discontinuous in the shown interval. As mentioned in Sec. Equation 5. Does this wave function meet all the above requirements? If not, could a linear superposition of such wave functions meet these requirements? What is the significance of such a superposition of wave functions?
Inha University Department of Physics A linear superposition of such waves could give a normalizable wave function, corresponding to a real particle. One of the possible wave functions of a particle in the potential well of Fig. As the potential energy increases with x, the particle's kinetic energy must decrease, and so the wavelength increases. The amplitude increases as the wavelength increases because a larger wavelength means a smaller momentum indicated as well by the lower kinetic energy , and the particle is more likely to be found where the momentum has a lower magnitude.
From either a table or repeated integration by parts, the indefinite integral is.
A particle is in a cubic box with infinitely hard walls whose edges are L long Fig. Before substitution into Equation 5. A beam of electrons is incident on a barrier 6. Use Eq. What bearing would you think the uncertainty principle has on the existence of the zero-point energy of a harmonic oscillator?
The uncertainty principle dictates that such a particle would have an infinite uncertainty in momentum, and hence an infinite uncertainty in energy. This contradiction implies that the zero-point energy of a harmonic oscillator cannot be zero. The other two integrals may be found from tables, from symbolic-manipulation programs, or by any of the methods outlined at the end of this chapter or in Special Integrals for Harmonic Oscillators, preceding the solutions for Section 5.
The integrals are. A pendulum with a 1. The period of the pendulum is 1. Would you expect the zero-point oscillations to be detectable? What is the corresponding quantum number? Find the transmitted and reflected currents. The transmitted current is T 1. The three quantum numbers needed to describe an atomic electron correspond to the variation in the radial direction, the variation in the azimuthal direction the variation along the circumference of the classical orbit , and the variation with the polar direction variation along the direction from the classical axis of rotation.
Chapter 6 Problem Solutions 1. Why is it natural that three quantum numbers are needed to describe an atomic electron apart from electron spin? The improper definite integral in u is known to have the value 2 and so the given function is normalized.
In Exercise 12 of Chap. Inha University Department of Physics 7. Compare the angular momentum of a ground-state electron in the Bohr model of the hydrogen atom with its value in the quantum theory. Under what circumstances, if any, is L z equal to L? Find the percentage difference between L and the maximum value of L z for an atomic electron in p, d , and f states.
Verify this. Differentiating the above expression for P r with respect to r and setting the derivative equal to zero,. Find the most probable value of r for a 3d electron in a hydrogen atom. How much more likely is the electron in a ground-state hydrogen atom to be at the distance a o from the nucleus than at the distance 2a o?
The ratio of the probabilities is then the ratio of the product r 2 R 10 r 2 evaluated at the different distances. The probability of finding an atomic electron whose radial wave function is R r outside a sphere of radius r o centered on the nucleus is a Calculate the probability of finding a 1s electron in a hydrogen atom at a distance greater than a o from the nucleus. According to classical physics, the electron therefore cannot ever exceed the distance 2a o from the nucleus.
With the help of the wave functions listed in Table 6. The terms involving sines vanish, with the result of. The integrand is then an odd function of u when n and m are both even or both odd, and hence the integral is zero. If one of n or m is even and the other odd, the integrand is an even function of u and the integral is nonzero. Show that the magnetic moment of an electron in a Bohr orbit of radius r n is proportional to Find the minimum magnetic field needed for the Zeeman effect to be observed in a spectral line of nm wavelength when a spectrometer whose resolution is 0.
The orbital radius is proportional to n 2 See Equation 4. A beam of electrons enters a uniform 1. Find the possible angles between the z axis and the direction of the spin angular-momentum vector S.
The nuclei of ordinary helium atoms, , contain two protons and two neutrons each; the nuclei of another type of helium atom, , contain two protons and one neutron each. The properties of liquid and liquid are different because one type of helium atom obeys the exclusion principle but the other does not. Which is which, and why? Such atoms do not obey the exclusion principle.
In what way does the electron structure of an alkali metal atom differ from that of a halogen atom? From that of an inert gas atom? A halogen atom lacks one electron of having a closed outer shell: An inert gas atom has a closed outer shell. How many electrons can occupy an f subshell? Each state can have two electrons of opposite spins, for a total of 14 electrons. Repeated use of Equation 7. All are in group 1 of the periodic table.
Account for the decrease in ionization energy with increasing atomic number. The outermost electron in each of these atoms is further from the nucleus for higher atomic number, and hence has a successively lower binding energy.
Would you think that such an electron is relatively easy or relatively hard to detach from the atom? This outer electron is then relatively hard to detach. Why are Cl atoms more chemically active than Cl - ions?
In each of the following pairs of atoms, which would you expect to be larger in size? If the total number of electrons were odd, the net spin would be nonzero, and the anomalous Zeeman effect would be observable. Why is the normal Zeeman effect observed only in atoms with an even number of electrons? Use these wavelengths to calculate the effective magnetic field experienced by the outer electron in the sodium atom as a result of its orbital motion.
If , what values of l are possible? What must be true of the subshells of an atom which has a 1 S 0 ground state? There axe no other allowed states. This state has the lowest possible values of L and J , and is the only possible ground state. The lithium atom has one 2s electron outside a filled inner shell. The aluminum atom has two 3s electrons and one 3p electron outside filled inner shells.
Find the term symbol of its ground state. Answer the questions of Exercise 34 for an f electron in an atom whose total angular momentum is provided by this electron. How many substates are there for a given value of J?
What is the energy difference between different substates? Explain why the x-ray spectra of elements of nearby atomic numbers are qualitatively very similar, although the optical spectra of these elements may differ considerably. Optical spectra, however, depend upon the possible states of the outermost electrons, which, together with the transitions permitted for them, are different for atoms of different atomic number. The wavelength is In a triplet state, they are parallel Distinguish between singlet and triplet states in atoms with two outer electrons.
Inha University Department of Physics 1. The energy needed to detach the electron from a hydrogen atom is Why do you think the latter energy is greater? This means that the additional attractive force of the two protons exceeds the mutual repulsion of the electrons to increase the binding energy. At what temperature would the average kinetic energy of the molecules in a hydrogen sample be equal to their binding energy?
When a molecule rotates, inertia causes its bonds to stretch. This is why the earth bulges at the equator. What effects does this stretching have on the rotational spectrum of the molecule? In addition, the higher the quantum number J and hence the greater the angular momentum , the faster the rotation and the greater the distortion, so the spectral lines are no longer evenly spaced.
Quantitatively, the parameter I the moment of inertia of the molecule is a function of J, with I larger for higher J. Thus, all of the levels as given by Equation 8. It should be noted that if I depends on J, the algebraic steps that lead to Equation 8. Find the mass number of the unknown carbon isotope. For the different isotopes, the atomic separation, which depends on the charges of the atoms, will be essentially the same.
The ratio of the moments of inertia will then be the ratio of the reduced masses. The rotational spectrum of HCI contains the following wavelengths: A least-squares fit from a spreadsheet program gives 0. From Equation 8. A Hg 35 Cl Molecule emits a 4. Find the interatomic distance in this molecule. This is an example of Bohr's correspondence principle.
Show that a similar correspondence holds for a diatomic molecule rotating about its center of mass. The hydrogen isotope deuterium has an atomic mass approximately twice that of ordinary hydrogen. Does H 2 or HD have the greater zero-point energy?
How does this affect the binding energies of the two molecules? HD has the greater reduced mass, and hence the smaller frequency of vibration v o and the smaller zero- point energy. HD is the more tightly bound, and has the greater binding energy since its zero-point energy contributes less energy to the splitting of the molecule.
Plot the potential energy of this molecule versus internuclear distance in the vicinity of 0. The levels are shown below, where the vertical scale is in units of 10 J and the horizontal scale is in units of 10 m. The lowest vibrational states of the 23 Na 35 Cl molecule are 0.
Find the approximate force constant of this molecule. Solving Equation 8. Is it likely that an HCl molecule will be vibrating in its first excited vibrational state at room temperature? Atomic masses are given in the Appendix. It's important to note that in the above calculations, the symbol "k " has been used for both a spring constant and Boltzmann's constant, quantities that are not interchangeable.
Find the ratio between the numbers of atoms in each state in sodium vapor at l K. The moment of inertia of the H 2 molecule is 4. If so, at what temperature does this occur? Find and v rms for an assembly of two molecules, one with a speed of 1. At what temperature will the average molecular kinetic energy in gaseous hydrogen equal the binding energy of a hydrogen atom?
Find the width due to the Doppler effect of the How many independent standing waves with wavelengths between 95 and How many with wavelengths between Similarly, the number of waves between A thermograph measures the rate at which each small portion of a persons skin emits infrared radiation. To verify that a small difference in skin temperature means a significant difference in radiation rate, find the percentage difference between the total radiation from skin at 34 o and at 35 o C.
At what rate would solar energy arrive at the earth if the solar surface had a temperature 10 percent lower than it is? Using 1. An object is at a temperature of o C. At what temperature would it radiate energy twice as fast? At what rate does radiation escape from a hole l0 cm 2 in area in the wall of a furnace whose interior is at o C?
Find the surface area of a blackbody that radiates kW when its temperature is o C. If the blackbody is a sphere, what is its radius? The brightest part of the spectrum of the star Sirius is located at a wavelength of about nm. What is the surface temperature of Sirius? A gas cloud in our galaxy emits radiation at a rate of 1. If the cloud is spherical and radiates like a blackbody, find its surface temperature and its diameter.
Find the specific heat at constant volume of 1. The median energy is that energy for which there are many occupied states below the median as there are above.
The Fermi energy in silver is 5. This is the reason for the symmetry of the curves in Fig. The density of zinc is 7. The electronic structure of zinc is given in Table 7. Calculate the Fermi energy in zinc. Thus, there are two free electrons per atom.
Are we justified in considering the electron energy distribution as continuous in a metal? The number of states per electronvolt is and the distribution may certainly be considered to be continuous.
To do this, use Eq 9. The Fermi-Dirac distribution function for the free electrons in a metal cannot be approximated by the Maxwell-Boltzmann function at STP for energies in the neighborhood of k T. As calculated in Sec. Note that Eq. In this problem, the time t is the time that observer A measures as the time that B's clock takes to record a time change of to. For this problem. If one of the characteristic wavelengths of the light the galaxy emits is nm. A galaxy in the constellation Ursa Major is receding from the earth at The classical and relativistic frequencies.
The best part? As a Chegg Study subscriber, you can view available interactive solutions manuals for each of your classes for one low monthly price. Why buy extra books when you can get all the homework help you need in one place? You bet! Just post a question you need help with, and one of our experts will provide a custom solution.
You can also find solutions immediately by searching the millions of fully answered study questions in our archive. You can download our homework help app on iOS or Android to access solutions manuals on your mobile device. Is this a good idea? Actually, when the effects of length contraction are included discussed in Section 1.
Two observers, A on earth and B in a spacecraft whose speed is 2. To B, does A's watch seem to run fast, run slow, or keep the same time as his own watch? In Equation 1. In this problem, the time t is the time that observer A measures as the time that B's clock takes to record a time change of to. How fast must a spacecraft travel relative to the earth for each day on the spacecraft to correspond to 2 d on the earth?
A certain particle has a lifetime of 1. How far does it go before decaying if its speed is 0. If one of the characteristic wavelengths of the light the galaxy emits is nm, what is the corresponding wavelength measured by astronomers on the earth? For this problem, v 1. A spacecraft receding from the earth emits radio waves at a constant frequency of Hz. If the receiver on earth can measure frequencies to the nearest hertz, at what spacecraft speed can the difference between the relativistic and classical Doppler effects be detected?
For the classical effect, assume the earth is stationary. Show that this formula includes Eqs. An astronaut whose height on the earth is exactly 6 ft is lying parallel to the axis of a spacecraft moving at 0.
What is his height as measured by an observer in the same spacecraft? By an observer on the earth? From Equation 1.
How much time does a meter stick moving at 0. The meter stick is parallel to its direction of motion. A spacecraft antenna is at an angle of 10o relative to the axis of the spacecraft. If the spacecraft moves away from the earth at a speed of 0. To an observer on the earth, the length in the direction of the spacecraft's axis will be contracted as described by Equation 1. A woman leaves the earth in a spacecraft that makes a round trip to the nearest star, 4 light- years distant, at a speed of 0.
All definitions are arbitrary, but some are more useful than others. Dynamite liberates about 5. What fraction of its total energy content is this? At what speed does the kinetic energy of a particle equal its rest energy? An electron has a kinetic energy of 0. Find its speed according to classical and relativistic mechanics. A particle has a kinetic energy 20 times its rest energy. Find the speed of the particle in terms of c.
How much work in MeV must be done to increase the speed of an electron from 1. Figure 1. A burst of electromagnetic radiation of energy Eo is emitted by one end of the box. If the CM of the box is to remain in its original place, the radiation must have transferred mass from one end to the other.
In its own frame of reference, a proton takes 5 min to cross the Milky Way galaxy, which is about light-years in diameter. The result of Problem does not give an answer accurate to three significant figures.
The value of the speed may be substituted into Equation 1. An observer detects two explosions, one that occurs near her at a certain time and another that occurs 2. Another observer finds that the two explosions occur at the, same place. What time interval separates the explosions to the second observer? Inserting this into Equation 1. An equally valid method, and a good cheek, is to note that when the relative speed of the observers 5.
Algebraically and numerically, the different methods give the same result. Take the direction of the ship's motion assumed parallel to its axis to be the positive x-direction, so that in the frame of the fixed stars the unprimed frame , the signal arrives at an angle 0 with respect to the positive x-direction. A man on the moon sees two spacecraft, A and B, coming toward him from opposite directions at the respective speeds of 0.
For the speed with which he is approaching B? For the speed with which he is approaching A? The relative velocities will have opposite directions, but the relative speeds will be the same.
If Planck's constant were smaller than it is, would quantum phenomena be more or less conspicuous than they are now? That is, quantum phenomena would be less conspicuous than they are now. Find the energy of a nm photon.
How many photons per second does it emit? Light from the sun arrives at the earth, an average of 1. Assume that sunlight is monochromatic with a frequency of 5. Using the result from part a , 4.
The maximum wavelength for photoelectric emission in tungsten is nm. What wavelength of light must be used in order for electrons with a maximum energy of 1.
What is the maximum wavelength of light that will cause photoelectrons to be emitted from sodium? What will the maximum kinetic energy of the photoelectrons be if nm light falls on a sodium surface? From Equation 2. A metal surface illuminated by 8. Show that it is impossible for a photon to give up all its energy and momentum to a free electron. This is the reason why the photoelectric effect can take place only when photons strike bound electrons.
An easier alternative is to consider the interaction in the frame where the electron is at rest after absorbing the photon. In this frame, the final energy is the rest energy of the electron, mec2, but before the interaction, the electron would have been moving to conserve momentum , and hence would have had more energy than after the interaction, and the photon would have had positive energy, so energy could not be conserved. Electrons are accelerated in television tubes through potential differences of about 10 kV.
Find the highest frequency of the electromagnetic waves emitted when these electrons strike the screen of the tube. What kind of waves are these? The distance between adjacent atomic planes in calcite CaCO3 is 0.
Find the smallest angle of Bragg scattering for 0. What is the frequency of an x-ray photon whose momentum is 1. In See. Show that this assumption is reasonable by calculating the Compton wavelength of a Na atom and comparing it with the typical x-ray wavelength of 0.
A beam of x-rays is scattered by a target. At 45o from the beam direction the scattered x-rays have a wavelength of 2. What is the wavelength of the x-rays in the direct beam? An x-ray photon of initial frequency 3. Find its new frequency. At what scattering angle will incident keV x-rays leave a target with an energy of 90 keV? A photon whose energy equals the rest energy of the electron undergoes a Compton collision with an electron. If the electron moves off at an angle of 40o with the original photon direction, what is the energy of the scattered photon?
Consider the expression for the recoil angle as given preceding the solution to Problem A positron collides head on with an electron and both are annihilated. Each particle had a kinetic energy of 1. The wavelength of each photon will be hc 1. Show that, regardless of its initial energy, a photon cannot undergo Compton scattering through an angle of more than 60o and still be able to produce an electron-positron pair.
Start by expressing the Compton wavelength of the electron in terms of the maximum photon wavelength needed for pair production. The linear absorption coefficient for 1-MeV gamma rays in lead is 78 m The linear absorption coefficients for 2. What thickness of water would give the same shielding for such gamma rays as 10 mm of lead? What thickness of copper is needed to reduce the intensity of the beam in Exercise 48 by half.
The sun's mass is 2. Find the approximate gravitational red shift in light of wavelength nm emitted by the sun.
As discussed in Chap. These photons constitute gamma rays. When a nucleus emits a photon, it recoils in the opposite direction. The mass of a Fe atom is 9. By how much is the photon energy reduced from the full By how much is the photon energy reduced in this situation if the ex- cited Fe nucleus is part of a 1.
Such a source was used in the experiment described in See. What is the original frequency and the change in frequency of a This approximation gives the previous result. Of course, a relativistic calculation is correct here, but it is interesting to see what a classical calculation produces. That is, its total energy must be nonnegative. A photon and a particle have the same wavelength. Can anything be said about how their linear momenta compare? About how the photon's energy compares with the particle's total energy?
Problem Solutions 3. Find the de Broglie wavelength of a 1. By what percentage will a nonrelativistle calculation of the de Broglie wavelength of a keV electron be in error? Problem Solutions keeping extra figures in the intermediate calculations.
The atomic spacing in rock salt, NaCl, is 0. Find the kinetic energy in eV of a neutron with a de Broglie wavelength of 0. Is a relativistic calculation needed? Such neutrons can be used to study crystal structure. This energy is much less than the neutron's rest energy, and so the nonrelativistic calculation is completely valid.
Problem Solutions 9. Green light has a wavelength of about nm. Through what potential difference must an electron be accelerated to have this wavelength? Note that the kinetic energy is very small compared to the electron rest energy, so the nonrelativistic calculation is valid.