1. Photon fundamentals.(a) Show that hc= 1240 eV∙nm.(b) Calculate
the energy of a photon with each of the following wavelengths: 650 nm
(red light); 450 nm (blue light); 0.1 nm (x-ray); 1 mm (typical for the
cosmic background radiation).(c) Calculate the number of photons emitted
in one second by a 1-milliwatt red He-Ne laser (λ = 633 nm). Get solution
2. Suppose that, in a photoelectric effect experiment of the type described above, light, with a wavelength of 400 nm results in a voltage reading of 0.8 V.(a) What is the work function for this photocathode?(b) What voltage reading would you expect to obtain if the wavelength were changed to 300 nm? What if the wavelength were changed to 500 nm? 600 nm? Get solution
3. Use the Einstein relation E = hf and the relation E = pc to show that the de Broglie relation A.3 holds for photons. Get solution
4. Use the relativistic definitions of energy and momentum to show that E = pc for any particle traveling at the speed of light. (For electromagnetic waves this relation can also be derived from Maxwell’s equations, but this is much harder.) Get solution
5. The electrons in a television picture tube are typically accelerated to an energy of 10,000 eV. Calculate the momentum of such an electron, and then use the de Broglie relation to calculate its wavelength. Get solution
6. In the experiment shown in Figure, the effective slit spacing was 6 μm and the distance from the “slits” to the detection screen was 16 cm. The spacing between the center of one bright line and the next (before magnification) was typically 100 nm. From these parameters, determine the wavelength of the electron beam. What voltage was used to accelerate the electrons? Figure: These images were produced using the beam of an electron microscope. A positively charged wire was placed in the path of the beam, causing the electrons to bend around either side and interfere as if they had passed through a double slit. The current in the electron beam increases from one image to the next, showing that lhe interference pattern is built up from the statistically distributed light flashes of individual electrons. From P. G. Merli, G. F. Missiroli, and G. Pozzi, American Journal of Physics 44, 306 (1976).... Get solution
7. The de Broglie relation applies to all “particles,” not just electrons and photons.(a) Calculate the wavelength of a neutron whose kinetic energy is 1 eV.(b) Estimate the wavelength of a pitched baseball. (Use any reasonable values for the mass and speed.) Explain why you don’t see baseballs diffracting around bats. Get solution
8. A definite-momentum wave function can be expressed by the formula Ψ (x) = A(cos kx +i sin kx), where A and k are constants.(a) How is the constant k related to the particle’s momentum? (Justify your answer.)(b) Show that, if a particle has such a wavefunction, you are equally likely to find it at any position x.(c) Explain why the constant A must be infinitesimal, if this formula is to be valid for all x.(d) Show that this wave function satisfies the differential equation dΨ/dx = ikΨ.(e) Often the function cosθ + i sinθ is written instead as eiθ. Treating the i as an ordinary constant. show that the function. A eikx obeys the same differential equation as in part (d). Get solution
9. The formula for a “properly constructed” wavepacket is...where A, a, and k0 are constants. (The exponential of an imaginary number is defined in Problem. In this problem, just assume that you can manipulate the i like any other constant.)(a) Compute and sketch |Ψ (x)|2 for this wavefunction.(b) Show that the constant A must, equal (2a/π)1/4 . (Hint: The probability of finding the particle somewhere between x = −∞ and x = ∞ must equal 1. See Section 1 of Appendix B for help with the integral.)(c) The standard deviation Δx can be computed as...and the average value of x2 is just the sum of all values of x2, weighted by their probabilities:...Use these formulas to show that for this wave packet, ...(d) The Fourier transform of a function Ψ(x) is defined as...Show that...for a properly constructed wave packet. Sketch this function.(e) Using formulas analogous to those in part (c), show that, for this wave function, ...(Hint: The standard deviation does not depend on k0, so you can simplify the calculation by setting k0 = 0 from the start.)(f) Compute Δpx for this wave function, and check whether the uncertainty principle is satisfied.Problem:The de Broglie relation applies to all “particles,” not just electrons and photons.(a) Calculate the wavelength of a neutron whose kinetic energy is 1 eV.(b) Estimate the wavelength of a pitched baseball. (Use any reasonable values for the mass and speed.) Explain why you don’t see baseballs diffracting around bats. Get solution
10. Sketch a wave function for which the product (Δx)(Δpx) is much greater than h/4π. Explain how you would estimate Δx and Δpx for your wavefunction. Get solution
11. Consider the functions Ψ1(x) = sin(x) and Ψ2 (x) = sin(2x), where x can range from 0 to π. Write down formulas for three different nontrivial linear combinations of Ψ1 and Ψ2, and sketch each of your three functions. For simplicity, keep your functions real-valued. Get solution
12. Make a rough estimate of the minimum energy of a proton confined inside a box of width 10−15m (the size of an atomic nucleus). Get solution
13. For ultrarelativistic particles such as photons or high-energy electrons, the relation between energy and momentum is not E = p2/2m but rather E = pc. (This formula is valid for massless particles, and also for massive particles in the limit E ≫ mc2.)(a) Find a formula for the allowed energies of an ultrarelativistic particle confined to a one-dimensional box of length L .(b) Estimate the minimum energy of an electron con lined inside a box of width 10−15m. It was once thought that atomic nuclei might cont ain electrons; explain why this would be very unlikely.(c) A nucleon (proton or neutron) can be thought of as a bound state of three quarks that are approximately massless, held together by a very strong force that effectively confines them inside a box of width 10−15 m. Estimate the minimum energy of three such particles (assuming all three of them Lobe in the lowest-energy state), and divide by c2 to obtain an estimate of the nucleon mass. Get solution
14. Draw an energy level diagram for a. nonrelativistic particle confined inside a three-dimensional cube-shaped box, showing all states with energies below 15. (h2/8mL2 ) . Be sure to show each linearly independent state separately, to indicate the degeneracy of each energy level. Does the average number of states per unit energy increase or decrease as E increases? Get solution
15. A CO molecule can vibrate with a natural frequency of 6.4 × 1013 s−1.(a) What aret energies (in eV) of the five lowest vibrational slates of a CO molecule?(b) If a CO molecule is initially in its ground stale and you wish to excite it into its first vibrational level, what wavelength of light should you aim at it? Get solution
16. In this problem you will analyze the spectrum of molecular nitrogen shown in Figure. You may assume that all of the transit ions are correctly identified in the energy level diagram.(a) What is the approximate difference in energy between the upper and lower electronic states, neglecting any vibrational energy (aside from the zeropoint energies ...(b) Determine the approximate spacing in energy between the vibrational levels, for both the lower and upper electronic states.(c) Repeat part (b) using a different set of spectral lines, to verify that the diagram is consistent.(d) How can you tell from the spectrum that the vibrational levels (for either electronic state) are not quite evenly spaced? (This is an indication that the potential energy function is not exactly quadratic.)Figure: A port ion of the emission spectrum of molecular nitrogen, N2. The energy level diagram shows the transitions corresponding lo the various spectral lines. All or the lines shown are from transitions between the same pair of electronic states. In either electronic slate, however, the molecule can also have one or more “units” of vibrational energy; these numbers are labeled at left. The spectral lines are grouped according to the number of units of vibrational energy gained or lost. The splitting within each group of lines occurs because the vibrational levels are spaced farther apart in one electronic state than in the other. From Gordon M. Barrow, Introduction to Molecular Spectroscopy (McGraw-Hill, New York, 1962). Photo originally provided by J. A. Marquisee.... Get solution
17. A two-dimensional harmonic oscillator can be considered as a system of two independent one-dimensional oscillators. Consider an isotropic two-dimensional oscillator, for which the natural frequency is the same in both directions. Write a formula for the allowed energies of this system, and draw an energy level diagram showing the degeneracy of each level. Get solution
18. Repeat the previous problem for a three-dimensional isotropic oscillator. Find a formula for the number of degenerate states with any given energy. Get solution
19. Suppose that a hydrogen atom makes a transition from a high-n state to a low-n state, emitting a photon in the process. Calculate the energy and wavelength of the photon for each of the following transitions: 2 → 1, 3 → 2, 4 → 2, 5 → 2. Get solution
20. A very naive, but partially correct, way to understand quantization of angular momentum is as follows. Imagine that a particle is confined to travel around a circle of radius r. Its angular momentum about the center is then ±rp, where p is the magnitude of its linear momentum at any moment. Let s be a coordinate that labels the position of the particle around the circle, so that s ranges from 0 to 2πr. The wavefunction of this particle is a function of s. Now suppose that the wavefunction is sinusoidal, so that p is well defined. Using the fact that the wavefunction must undergo an integer number of complete oscillations over the entire circle, find the allowed values of p and the allowed values of the angular momentum. Get solution
21. Enumerate the quantum numbers (n, ...., and m) for all the independent states of a hydrogen atom with definite ..., and Lz, up to n = 3. Check that the number of independent states for level n is equal to n2 Get solution
22. In Section 6.2 I used the symbol € as an abbreviation for the constant .... This constant is ordinarily measured by microwave spectroscopy: bombarding' the molecule with microwaves and looking at what frequencies are absorbed. ... Get solution
23. Draw a cone diagram, as in Figure, showing the spin states of a particle with s = 1/2. Repeat for a particle with s = 3/2. Draw both diagrams on the same scale, and be as accurate as you can with magnitudes and directions.Figure: A particle with well-defined ... and Lz has completely undefined Lx and Ly, so we can visualize its angular momentum “vector” as a cone, smeared over all possible Lx and Ly values. Shown here are the allowed states for ℓ = 1 and ℓ = 2.... Get solution
24. According to equation A.30, each mode of a quantum field has a “zero-point” energy of ...even when no further units of energy are present. If the field is really a vibrating string or some other material object, this isn’t a problem because the total number of modes is finite: You can’t have a mode whose wavelength is shorter than half the atomic spacing (see Section 7.5). But for the electromagnetic field and other fundamental fields corresponding to elementary particles, there is no obvious limit on the number of modes, and the zero-point energies can add up to something embarassing.(a) Consider just the electromagnetic field inside a box of volume L3 . Use the methods of Chapter 7 to write down a formula for the total zero-point energy of all the modes of the field inside this box, in terms of a triple integral over the mode numbers in the x, y, and z directions.(b) There are good reasons to believe that most of our current laws of physics, including quantum field theory, break down at the very small length scale where quantum gravity becomes important. By dimensional analysis, you can guess that this length scale is of order ..., a quantity called the Planck length. Show that the Planck length indeed has units of length, and calculate it numerically. (c) Going back to your expression from part (a), cut off the integrals at a mode number corresponding to a wavelength of the Planck length. Then evaluate your expression to obtain an estimate of the energy per unit volume in empty space, due to the zero-point energy of the electromagnetic field. Express your answer in J /m3, then divide by c2 to obtain the equivalent mass density of empty space (in kg/m3). Compare to the average mass density of ordinary matter in the universe, which is roughly equivalent to one proton per cubic meter. [Comment: Since most physical effects depend only on differences in energy, and since the zero-point energy never A.6 Quantum Field Theory changes, a large energy density in “empty” space would be harmless as far as most of the laws of physics are concerned. The only exception, and it’s a big one, is gravity: Energy gravitates, so a large energy density in empty space would affect the expansion rate of the universe. The energy density of empty space is therefore known as the cosmological constant. From the observed expansion rate of the universe, cosmologists estimate that the actual cosmological constant cannot be any greater than 10−7 J/m3. The discrepancy between this observational bound and your calculated value is one of the greatest paradoxes in theoretical physics. (The obvious solution would be to find some negative contribution to the energy density coming from some other source. In fact, fermionic fields give a negative contribution to the cosmological constant, but nobody knows how to make this negative contribution cancel the positive contribution from bosonic fields to the required precision.*)] Get solution
2. Suppose that, in a photoelectric effect experiment of the type described above, light, with a wavelength of 400 nm results in a voltage reading of 0.8 V.(a) What is the work function for this photocathode?(b) What voltage reading would you expect to obtain if the wavelength were changed to 300 nm? What if the wavelength were changed to 500 nm? 600 nm? Get solution
3. Use the Einstein relation E = hf and the relation E = pc to show that the de Broglie relation A.3 holds for photons. Get solution
4. Use the relativistic definitions of energy and momentum to show that E = pc for any particle traveling at the speed of light. (For electromagnetic waves this relation can also be derived from Maxwell’s equations, but this is much harder.) Get solution
5. The electrons in a television picture tube are typically accelerated to an energy of 10,000 eV. Calculate the momentum of such an electron, and then use the de Broglie relation to calculate its wavelength. Get solution
6. In the experiment shown in Figure, the effective slit spacing was 6 μm and the distance from the “slits” to the detection screen was 16 cm. The spacing between the center of one bright line and the next (before magnification) was typically 100 nm. From these parameters, determine the wavelength of the electron beam. What voltage was used to accelerate the electrons? Figure: These images were produced using the beam of an electron microscope. A positively charged wire was placed in the path of the beam, causing the electrons to bend around either side and interfere as if they had passed through a double slit. The current in the electron beam increases from one image to the next, showing that lhe interference pattern is built up from the statistically distributed light flashes of individual electrons. From P. G. Merli, G. F. Missiroli, and G. Pozzi, American Journal of Physics 44, 306 (1976).... Get solution
7. The de Broglie relation applies to all “particles,” not just electrons and photons.(a) Calculate the wavelength of a neutron whose kinetic energy is 1 eV.(b) Estimate the wavelength of a pitched baseball. (Use any reasonable values for the mass and speed.) Explain why you don’t see baseballs diffracting around bats. Get solution
8. A definite-momentum wave function can be expressed by the formula Ψ (x) = A(cos kx +i sin kx), where A and k are constants.(a) How is the constant k related to the particle’s momentum? (Justify your answer.)(b) Show that, if a particle has such a wavefunction, you are equally likely to find it at any position x.(c) Explain why the constant A must be infinitesimal, if this formula is to be valid for all x.(d) Show that this wave function satisfies the differential equation dΨ/dx = ikΨ.(e) Often the function cosθ + i sinθ is written instead as eiθ. Treating the i as an ordinary constant. show that the function. A eikx obeys the same differential equation as in part (d). Get solution
9. The formula for a “properly constructed” wavepacket is...where A, a, and k0 are constants. (The exponential of an imaginary number is defined in Problem. In this problem, just assume that you can manipulate the i like any other constant.)(a) Compute and sketch |Ψ (x)|2 for this wavefunction.(b) Show that the constant A must, equal (2a/π)1/4 . (Hint: The probability of finding the particle somewhere between x = −∞ and x = ∞ must equal 1. See Section 1 of Appendix B for help with the integral.)(c) The standard deviation Δx can be computed as...and the average value of x2 is just the sum of all values of x2, weighted by their probabilities:...Use these formulas to show that for this wave packet, ...(d) The Fourier transform of a function Ψ(x) is defined as...Show that...for a properly constructed wave packet. Sketch this function.(e) Using formulas analogous to those in part (c), show that, for this wave function, ...(Hint: The standard deviation does not depend on k0, so you can simplify the calculation by setting k0 = 0 from the start.)(f) Compute Δpx for this wave function, and check whether the uncertainty principle is satisfied.Problem:The de Broglie relation applies to all “particles,” not just electrons and photons.(a) Calculate the wavelength of a neutron whose kinetic energy is 1 eV.(b) Estimate the wavelength of a pitched baseball. (Use any reasonable values for the mass and speed.) Explain why you don’t see baseballs diffracting around bats. Get solution
10. Sketch a wave function for which the product (Δx)(Δpx) is much greater than h/4π. Explain how you would estimate Δx and Δpx for your wavefunction. Get solution
11. Consider the functions Ψ1(x) = sin(x) and Ψ2 (x) = sin(2x), where x can range from 0 to π. Write down formulas for three different nontrivial linear combinations of Ψ1 and Ψ2, and sketch each of your three functions. For simplicity, keep your functions real-valued. Get solution
12. Make a rough estimate of the minimum energy of a proton confined inside a box of width 10−15m (the size of an atomic nucleus). Get solution
13. For ultrarelativistic particles such as photons or high-energy electrons, the relation between energy and momentum is not E = p2/2m but rather E = pc. (This formula is valid for massless particles, and also for massive particles in the limit E ≫ mc2.)(a) Find a formula for the allowed energies of an ultrarelativistic particle confined to a one-dimensional box of length L .(b) Estimate the minimum energy of an electron con lined inside a box of width 10−15m. It was once thought that atomic nuclei might cont ain electrons; explain why this would be very unlikely.(c) A nucleon (proton or neutron) can be thought of as a bound state of three quarks that are approximately massless, held together by a very strong force that effectively confines them inside a box of width 10−15 m. Estimate the minimum energy of three such particles (assuming all three of them Lobe in the lowest-energy state), and divide by c2 to obtain an estimate of the nucleon mass. Get solution
14. Draw an energy level diagram for a. nonrelativistic particle confined inside a three-dimensional cube-shaped box, showing all states with energies below 15. (h2/8mL2 ) . Be sure to show each linearly independent state separately, to indicate the degeneracy of each energy level. Does the average number of states per unit energy increase or decrease as E increases? Get solution
15. A CO molecule can vibrate with a natural frequency of 6.4 × 1013 s−1.(a) What aret energies (in eV) of the five lowest vibrational slates of a CO molecule?(b) If a CO molecule is initially in its ground stale and you wish to excite it into its first vibrational level, what wavelength of light should you aim at it? Get solution
16. In this problem you will analyze the spectrum of molecular nitrogen shown in Figure. You may assume that all of the transit ions are correctly identified in the energy level diagram.(a) What is the approximate difference in energy between the upper and lower electronic states, neglecting any vibrational energy (aside from the zeropoint energies ...(b) Determine the approximate spacing in energy between the vibrational levels, for both the lower and upper electronic states.(c) Repeat part (b) using a different set of spectral lines, to verify that the diagram is consistent.(d) How can you tell from the spectrum that the vibrational levels (for either electronic state) are not quite evenly spaced? (This is an indication that the potential energy function is not exactly quadratic.)Figure: A port ion of the emission spectrum of molecular nitrogen, N2. The energy level diagram shows the transitions corresponding lo the various spectral lines. All or the lines shown are from transitions between the same pair of electronic states. In either electronic slate, however, the molecule can also have one or more “units” of vibrational energy; these numbers are labeled at left. The spectral lines are grouped according to the number of units of vibrational energy gained or lost. The splitting within each group of lines occurs because the vibrational levels are spaced farther apart in one electronic state than in the other. From Gordon M. Barrow, Introduction to Molecular Spectroscopy (McGraw-Hill, New York, 1962). Photo originally provided by J. A. Marquisee.... Get solution
17. A two-dimensional harmonic oscillator can be considered as a system of two independent one-dimensional oscillators. Consider an isotropic two-dimensional oscillator, for which the natural frequency is the same in both directions. Write a formula for the allowed energies of this system, and draw an energy level diagram showing the degeneracy of each level. Get solution
18. Repeat the previous problem for a three-dimensional isotropic oscillator. Find a formula for the number of degenerate states with any given energy. Get solution
19. Suppose that a hydrogen atom makes a transition from a high-n state to a low-n state, emitting a photon in the process. Calculate the energy and wavelength of the photon for each of the following transitions: 2 → 1, 3 → 2, 4 → 2, 5 → 2. Get solution
20. A very naive, but partially correct, way to understand quantization of angular momentum is as follows. Imagine that a particle is confined to travel around a circle of radius r. Its angular momentum about the center is then ±rp, where p is the magnitude of its linear momentum at any moment. Let s be a coordinate that labels the position of the particle around the circle, so that s ranges from 0 to 2πr. The wavefunction of this particle is a function of s. Now suppose that the wavefunction is sinusoidal, so that p is well defined. Using the fact that the wavefunction must undergo an integer number of complete oscillations over the entire circle, find the allowed values of p and the allowed values of the angular momentum. Get solution
21. Enumerate the quantum numbers (n, ...., and m) for all the independent states of a hydrogen atom with definite ..., and Lz, up to n = 3. Check that the number of independent states for level n is equal to n2 Get solution
22. In Section 6.2 I used the symbol € as an abbreviation for the constant .... This constant is ordinarily measured by microwave spectroscopy: bombarding' the molecule with microwaves and looking at what frequencies are absorbed. ... Get solution
23. Draw a cone diagram, as in Figure, showing the spin states of a particle with s = 1/2. Repeat for a particle with s = 3/2. Draw both diagrams on the same scale, and be as accurate as you can with magnitudes and directions.Figure: A particle with well-defined ... and Lz has completely undefined Lx and Ly, so we can visualize its angular momentum “vector” as a cone, smeared over all possible Lx and Ly values. Shown here are the allowed states for ℓ = 1 and ℓ = 2.... Get solution
24. According to equation A.30, each mode of a quantum field has a “zero-point” energy of ...even when no further units of energy are present. If the field is really a vibrating string or some other material object, this isn’t a problem because the total number of modes is finite: You can’t have a mode whose wavelength is shorter than half the atomic spacing (see Section 7.5). But for the electromagnetic field and other fundamental fields corresponding to elementary particles, there is no obvious limit on the number of modes, and the zero-point energies can add up to something embarassing.(a) Consider just the electromagnetic field inside a box of volume L3 . Use the methods of Chapter 7 to write down a formula for the total zero-point energy of all the modes of the field inside this box, in terms of a triple integral over the mode numbers in the x, y, and z directions.(b) There are good reasons to believe that most of our current laws of physics, including quantum field theory, break down at the very small length scale where quantum gravity becomes important. By dimensional analysis, you can guess that this length scale is of order ..., a quantity called the Planck length. Show that the Planck length indeed has units of length, and calculate it numerically. (c) Going back to your expression from part (a), cut off the integrals at a mode number corresponding to a wavelength of the Planck length. Then evaluate your expression to obtain an estimate of the energy per unit volume in empty space, due to the zero-point energy of the electromagnetic field. Express your answer in J /m3, then divide by c2 to obtain the equivalent mass density of empty space (in kg/m3). Compare to the average mass density of ordinary matter in the universe, which is roughly equivalent to one proton per cubic meter. [Comment: Since most physical effects depend only on differences in energy, and since the zero-point energy never A.6 Quantum Field Theory changes, a large energy density in “empty” space would be harmless as far as most of the laws of physics are concerned. The only exception, and it’s a big one, is gravity: Energy gravitates, so a large energy density in empty space would affect the expansion rate of the universe. The energy density of empty space is therefore known as the cosmological constant. From the observed expansion rate of the universe, cosmologists estimate that the actual cosmological constant cannot be any greater than 10−7 J/m3. The discrepancy between this observational bound and your calculated value is one of the greatest paradoxes in theoretical physics. (The obvious solution would be to find some negative contribution to the energy density coming from some other source. In fact, fermionic fields give a negative contribution to the cosmological constant, but nobody knows how to make this negative contribution cancel the positive contribution from bosonic fields to the required precision.*)] Get solution
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