1. Try to normalize the wave function .... Why can’t it be done over all space? Explain why this is not possible. Get solution
1q. Why can we use the nonrelativistic form of the kinetic energy in treating the structure of the hydrogen atom? Get solution
2. (a) In what direction does a wave of the form A sin(kx - ωt) move? (b) What about B sin(kx + ωt)? (c) Is ...a real number? Explain. (d) In what direction is the wave in (c) moving? Explain. Get solution
2q. How do you reconcile the fact that the probability density for the ground state of the quantum harmonic oscillator (Figure 6.10c) has its peak at the center and its minima at its ends, whereas the classical harmonic oscillator’s probability density (Figure 6.11) has a minimum at the center and peaks at each end? If you do this experiment with an actual mass and spring, what experimental result for its position distribution would you expect to obtain? Why? Figure 6.11 ... Figure 6.10c ... Get solution
3. Show directly that the trial wave function ψ (x, t) = ...satisfies Equation (6.1). ... Get solution
4. Normalize the wave function ...in the region x = 0 to a. Get solution
4q. In a given tunnel diode the pn junction (see Chapter 11) width is fixed. How can we change the time response of the tunnel diode most easily? Explain. Get solution
5. Normalize the wave function Are-r/α from r = 0 to q where α and A are constants. See Appendix 3 for useful integrals. Get solution
5q. A particle in a box has a first excited state that is 3 eV above its ground state. What does this tell you about the box? Get solution
6. Property 2 of the boundary conditions for wave functions specifies that ψ must be continuous in order to avoid discontinuous probability values. Why can’t we have discontinuous probabilities? Get solution
6q. Does the wavelength of a particle change after it tunnels through a barrier as shown in Figure 6.15? Explain. ... Get solution
7. Consider the wave function ...that we used in Example 6.4. (a) Does this wave function satisfy the boundary conditions of Section 6.1? (b) What does your analysis in part (a) imply about this wave function? (c) If the wave function is unacceptable as is, how could it be fixed? ... Get solution
7q. Can a particle be observed while it is tunneling through a barrier? What would its wavelength, momentum, and kinetic energy be while it tunnels through the barrier? Get solution
8. A set of measurements has given the following result for the measurement of x (in some units of length): 3.4, 3.9, 5.2, 4.7, 4.1, 3.8, 3.9, 4.7, 4.1, 4.5, 3.8, 4.5, 4.8, 3.9, and 4.4. Find the average value of x, called x or x>, and average value of x2, represented by x 2>. Show that the standard deviation of x, given by ... where xi is the individual measurement and N is the number of measurements, is also give.... Find the value of σ for the set of data given here. Get solution
8q. Is it easier for an electron or a proton of the same energy to tunnel through a given potential barrier? Explain. Get solution
9. If the potential V(x) for a one-dimensional system is independent of time, show that the expectation value for x is independent of time. Get solution
9q. Can a wave packet be formed from a superposition of wave functions of the type ei(kx-ωt) ? Can it be normalized? Get solution
10q. Given a particular potential V and wave function ψ, how could you prove that the given ψ is correct? Could you determine an appropriate energy E if the potential is independent of time? Get solution
11. A wave function has the value A sin x between x = 0 and π but zero elsewhere. Normalize the wave function and find the probability that the particle is (a) between x = 0 and x = π/4 and (b) between x = 0 and π/2. Get solution
11q. Compare the infinite square-well potential with the finite one. Where is the Schrödinger wave equation the same? Where is it different? Get solution
12. Find an equation for the difference between adjacent energy levels ...for the infi nite square-well potential. Calculate ΔE1, ΔE8, and ΔE800. Get solution
12q. Tunneling can occur for an electron trying to pass through a very thin tunnel diode. Can a baseball tunnel through a very thin window? Explain. Get solution
13. Determine the average value of ... inside the well for the infinite square-well potential for n = 1, 5, 10, and 100. Compare these averages with the classical probability of detecting the particle inside the box. Get solution
13q. For the three-dimensional cubical box, the ground state is given by n1 =n2 = n3 =1. Why is it not possible to have one ni = 1 and the other two equal to zero? Get solution
14. A particle in an infi nite square-well potential has ground-state energy 4.3 eV. (a) Calculate and sketch the energies of the next three levels, and (b) sketch the wave functions on top of the energy levels. Get solution
15. We can approximate an electron moving in a nanowire (a small, thin wire) as a one-dimensional infinite square-well potential. Let the wire be 2.0 μm long. The nanowire is cooled to a temperature of 13 K, and we assume the electron’s average kinetic energy is that of gas molecules at this temperature (= 3kT/2). (a) What are the three lowest possible energy levels of the electrons? (b) What is the approximate quantum number of electrons moving in the wire? Get solution
16. An electron moves with a speed v = 1.25 × 10-4c inside a one-dimensional box (V = 0) of length 48.5 nm. The potential is infinite elsewhere. The particle may not escape the box. What approximate quantum number does the electron have? Get solution
17. For the infinite square-well potential, find the probability that a particle in its ground state is in each third of the one-dimensional box: 0 ≤ x ≤ L/3, L/3 ≤ x ≤ 2L/3, 2L/3 ≤ x ≤ L. Check to see that the sum of the probabilities is one. Get solution
18. Repeat the previous problem using the first excited state. For the infinite square-well potential, find the probability that a particle in its ground state is in each third of the one-dimensional box: 0 ≤ x ≤ L/3, L/3 ≤ x ≤ 2L/3, 2L/3 ≤ x ≤ L. Check to see that the sum of the probabilities is one. Get solution
19. Repeat Example 6.9 for an electron inside the nucleus. Assume nonrelativistic equations and find the transition energy for an electron. (See Example 6.9 for an interpretation of the result.) ... Get solution
20. What is the minimum energy of (a) a proton and (b) an α particle trapped in a one-dimensional region the size of a uranium nucleus (radius = 7.4 × 10-15 m)? Get solution
21. An electron is trapped in an infinite square-well potential of width 0.70 nm. If the electron is initially in the n = 4 state, what are the various photon energies that can be emitted as the electron jumps to the ground state? Get solution
22. Consider a finite square-well potential well of width 3.00 × 10-15 m that contains a particle of mass 1.88 GeV/c2. How deep does this potential well need to be to contain three energy levels? (This situation approximates a deuteron inside a nucleus.) Get solution
23. Compare the results of the infinite and finite square well potentials. (a) Are the wavelengths longer or shorter for the finite square well compared with the infinite well? (b) Use physical arguments to decide whether the energies (for a given quantum number n) are (i) larger or (ii) smaller for the finite square well than for the infinite square well? (c) Why will there be a finite number of bound energy states for the finite potential? Get solution
24. Apply the boundary conditions to the finite square well potential at x = 0 to find the relationships between the coefficients A, C, and D and the ratio C/D. Get solution
25. Apply the boundary conditions to the finite square well potential at x = L to find the relationship between the coefficients B, C, and D and the ratio C/D. Get solution
26. Find the energies of the second, third, fourth, and fifth levels for the three-dimensional cubical box. Which energy levels are degenerate? Get solution
27. Write the possible (un normalized) wave functions for each of the first four excited energy levels for the cubical box. Get solution
28. Find the normalization constant A for the ground state wave function for the cubical box in Equation (6.52). ... Get solution
29. Complete the derivation of Equation (6.49) by substituting the wave function given in Equation (6.47) into Equation (6.46). What is the origin of the three quantum numbers? ... ... ... Get solution
30. Find the normalization constant A [in Equation (6.47)] for the first excited state of a particle trapped in a cubical potential well with sides L. Does it matter which of the three degenerate excited states you consider? ... Get solution
31. A particle is trapped in a rectangular box having sides L, 2L, and 4L. Find the energy of the ground state and fi rst three excited states. Are any of these states degenerate? Get solution
33. What is the energy level difference between adjacent levels ΔEn = En - 1 - En for the simple harmonic oscillator? What are ΔE0, ΔE2, and ΔE20? How many possible energy levels are there? Get solution
34. The wave function for the first excited state c1 for the simple harmonic oscillator is .... Normalize the wave function to find the value of the constant A. Determine x>, x 2> and ... Get solution
35. A nitrogen atom of mass 2.32 × 10-26 kg oscillates in one dimension at a frequency of 1013 Hz. What are its effective force constant and quantized energy levels? Get solution
36. One possible solution for the wave function ψn for the simple harmonic oscillator is ... where A is a constant. What is the value of the energy level En? Get solution
37. What would you expect for p> and p2> for the ground state of the simple harmonic oscillator? (Hint: Use symmetry and energy arguments.) Get solution
38. Show that the energy of a simple harmonic oscillator in the n = 1 state is ...directly into the Schrödinger equation. Get solution
39. An H2 molecule can be approximated by a simple harmonic oscillator with a force constant k = 1.1 × 103 N/m. Find (a) the energy levels and (b) the possible wavelengths of photons emitted when the H2 molecule decays from the third excited state eventually to the ground state. Get solution
40. The creation of elements in the early universe and in stars involves protons tunneling through nuclei. Find the probability of the proton tunneling through 12C when the temperature of the star containing the proton and carbon is 12,000 K. Get solution
41. Compare the wavelength of a particle when it passes a barrier of height (a) +V0 (see Figure 6.12) and (b) -V0 where E > |V0 | (see Figure 6.18). Calculate the momentum and kinetic energy for both cases. ... Get solution
42. (a) Calculate the transmission probability of an α particle of energy E = 5.0 MeV through a Coulomb barrier of a heavy nucleus that is approximated by a square barrier with V0 = 15 MeV and barrier width L = 1.3 × 10-14 m. Also, calculate the probability (b) by doubling the potential barrier height and (c) by using the original barrier height but doubling the barrier width. Compare all three probabilities. Get solution
43. Consider a particle of energy E trapped inside the potential well shown in the accompanying figure. Make an approximate sketch of possible wave functions inside and outside the potential well. Explain your sketch. ... Get solution
44. When a particle of energy E approaches a potential barrier of height V0, where E>> V0, show that the reflection coefficient is about ... Get solution
45. Let 12.0-eV electrons approach a potential barrier of height 4.2 eV. (a) For what barrier thickness is there no reflection? (b) For what barrier thickness is the reflection a maximum? Get solution
46. A 1.0-eV electron has a 2.0 × 10-4 probability of tunneling through a 2.5-eV potential barrier. What is the probability of a 1.0-eV proton tunneling through the same barrier? Get solution
47. An electron is attempting to tunnel through a square barrier potential. (a) Draw a potential function with zero potential on either side of a square-top potential similar to Figure 6.12. Draw the wave function before, after, and inside the barrier. (b) Let the barrier be twice as wide and repeat part (a). (c) Let the barrier be about twice as tall as in (a) and repeat (a). Do not perform calculations; make estimates only. Get solution
48. Use the approximate Equation (6.73) to estimate the probability of (a) a 1.4-eV electron tunneling through a 6.4-eV-high barrier of width 2.8 nm, and (b) a 4.4- MeV α particle tunneling through a uranium nucleus where the potential barrier is 19.2 MeV and 7.4 fm wide. (c) Discuss whether the approximation was valid for these two cases. Explain. ... Get solution
49g. Check to see whether the simple linear combination of sine and cosine functions satisfies the time-independent Schrödinger equation for a free particle (V = 0). ... Get solution
50g. (a) Check to see whether the simple linear combination of sine and cosine functions ... where A and B are real numbers, satisfies the time dependent Schrödinger equation for a free particle (V = 0). (b) Repeat for the modified version ... Get solution
51g. A particle of mass m is trapped in a three-dimensional rectangular potential well with sides of length L, ...and 2L. Inside the box V = 0, outside V = ∞. Assume that ... inside the well. Substitute this wave function into the Schrödinger equation and apply appropriate boundary conditions to find the allowed energy levels. Find the energy of the ground state and fi rst four excited levels. Which of these levels are degenerate? Get solution
52g. For a region where the potential V = 0, the wave function is given by .... Calculate the energy of this system. Get solution
53g. Consider the semi-infinite-well potential in which V =∞ for x = 0, V = 0 for 0? x? L, and V = V0 for x≥L. (a) Show that possible wave functions are A sin kx inside the well and Be-kx for x ? L, where k = ...and k = .... (b) Show that the application of the boundary conditions gives k tan (kL) = -k Get solution
55g. Prove that there are a limited number of bound solutions for the semi-infinite well. Get solution
56g. Use the semi-infinite-well potential to model a deuteron, a nucleus consisting of a neutron and a proton. Let the well width be 3.5 × 10-15 m and V0 - E = 2.2 MeV. Determine the energy E. How many excited states are there, and what are their energies? Get solution
57g. Consider as a model of a hydrogen atom a particle trapped in a one-dimensional, infinite potential well of width 2a0 (the ground-state hydrogen atom’s diameter). Find the electron’s ground-state energy and comment on the result. Get solution
58g. (a) Repeat the preceding problem using a cubical infinite potential well, with each side of the cube equal to 2a0. Get solution
59g. In the lab you make a simple harmonic oscillator with a 0.15-kg mass attached to a 12-N/m spring. (a) If the oscillation amplitude is 0.10 m, what is the corresponding quantum number n for the quantum harmonic oscillator? (b) What would be the amplitude of the quantum ground state for this oscillator? (c) What is the energy of a photon emitted when this oscillator makes a transition between adjacent energy levels? Comment on each of your results. Get solution
60g. In gravity-free space, a 2.0-mg dust grain is confined to move back and forth between rigid walls 1.0 mm apart. (a) What is the speed of the dust grain if it is in the quantum ground state? (b) If it is actually moving at a speed of 0.25 mm/s, what is the quantum number associated with its quantum state? Get solution
61g. The wave function for the n _ 2 state of a simple harmonic oscillator is .... (a) Show that its energy level is ... by substituting the wave function into the Schrödinger equation. (b) Find x> and x 2>. Get solution
62g. A particle is trapped inside an infinite square-well potential between x = 0 and x = L. Its wave function is a superposition of the ground state and first excited state. The wave function is given by ... Show that the wave function is normalized. Get solution
63g. The Morse potential is a good approximation for a real potential to describe diatomic molecules. It is given by ... where D is the molecular dissociation energy, and re is the equilibrium distance between the atoms. For small vibrations, r - re is small, and V(r) can be expanded in a Taylor series to reduce to a simple harmonic potential. Find the lowest term of V(r) in this expansion and show that it is quadratic in (r - re). Get solution
64g. Show that the vibrational energy levels Ev for the Morse potential of the previous problem are given by ... Where ... and n is the vibrational quantum number, mr is the reduced mass, and Ev ?? D. Find the three lowest energy levels for KCl where D = 4.42 eV, and a = 7.8 nm-1. Get solution
65g. Consider a particle of mass m trapped inside a two dimensional square box of sides L aligned along the x and y axes. Show that the wave function and energy levels are given by ... Plot the first six energy levels and give their quantum numbers. Get solution
66g. Make a sketch for each of the following situations for both the infinite square-well and finite square-well potentials in one dimension: (a) the four lowest energy levels and (b) the probability densities for the four lowest states. (c) Discuss the differences between the two potentials and why they occur. Get solution
67g. Two nanowires are separated by 1.3 nm as measured by STM. Inside the wires the potential energy is zero, but between the wires the potential energy is greater than the electron’s energy by only 0.9 eV. Estimate the probability that the electron passes from one wire to the other. Get solution
68g. The WKB approximation is useful to obtain solutions to the one-dimensional time-independent Schrödinger equation in cases where E ? V(x) and the potential V(x) changes slowly and gradually with x. In this case the wavelength λ (x) varies with x because of the V(x) dependence on x. (a) Argue that we can write the wavelength as ... for a particle of mass m in a potential V(x). (b) By considering the number of oscillations that can be fi t into a distance dx, show that the following equation is valid, where n is an integer and represents the number of standing waves that fi t inside the potential well. ... where n is an integer (6.74) This is the WKB approximation. (Hint: the equation might be helpful) ... Get solution
69g. Use the WKB approximation of Equation (6.74) in the previous problem with the potential shown in the accompanying figure. V(x) = ∞ for x = 0, and V(x) = Ax for x ? 0. (a) Find the quantized energy values, and (b) sketch the wave functions on top of the V(x) function for the three lowest states. ... Get solution
70g. In the special topic box in Section 6.7, the extreme sensitivity of scanning tunneling microscopes (STM) was described. It was reported that a change in the tunneling gap of only 0.4 nm between STM sample and probe can change the tunneling current by a factor of 104. Check the plausibility of this statement by using the “wide-barrier” approximation in Equation (6.70) to find the difference between the electron energy and barrier height that would produce such a situation. Report your answer in eV and comment in the result. Get solution
1q. Why can we use the nonrelativistic form of the kinetic energy in treating the structure of the hydrogen atom? Get solution
2. (a) In what direction does a wave of the form A sin(kx - ωt) move? (b) What about B sin(kx + ωt)? (c) Is ...a real number? Explain. (d) In what direction is the wave in (c) moving? Explain. Get solution
2q. How do you reconcile the fact that the probability density for the ground state of the quantum harmonic oscillator (Figure 6.10c) has its peak at the center and its minima at its ends, whereas the classical harmonic oscillator’s probability density (Figure 6.11) has a minimum at the center and peaks at each end? If you do this experiment with an actual mass and spring, what experimental result for its position distribution would you expect to obtain? Why? Figure 6.11 ... Figure 6.10c ... Get solution
3. Show directly that the trial wave function ψ (x, t) = ...satisfies Equation (6.1). ... Get solution
4. Normalize the wave function ...in the region x = 0 to a. Get solution
4q. In a given tunnel diode the pn junction (see Chapter 11) width is fixed. How can we change the time response of the tunnel diode most easily? Explain. Get solution
5. Normalize the wave function Are-r/α from r = 0 to q where α and A are constants. See Appendix 3 for useful integrals. Get solution
5q. A particle in a box has a first excited state that is 3 eV above its ground state. What does this tell you about the box? Get solution
6. Property 2 of the boundary conditions for wave functions specifies that ψ must be continuous in order to avoid discontinuous probability values. Why can’t we have discontinuous probabilities? Get solution
6q. Does the wavelength of a particle change after it tunnels through a barrier as shown in Figure 6.15? Explain. ... Get solution
7. Consider the wave function ...that we used in Example 6.4. (a) Does this wave function satisfy the boundary conditions of Section 6.1? (b) What does your analysis in part (a) imply about this wave function? (c) If the wave function is unacceptable as is, how could it be fixed? ... Get solution
7q. Can a particle be observed while it is tunneling through a barrier? What would its wavelength, momentum, and kinetic energy be while it tunnels through the barrier? Get solution
8. A set of measurements has given the following result for the measurement of x (in some units of length): 3.4, 3.9, 5.2, 4.7, 4.1, 3.8, 3.9, 4.7, 4.1, 4.5, 3.8, 4.5, 4.8, 3.9, and 4.4. Find the average value of x, called x or x>, and average value of x2, represented by x 2>. Show that the standard deviation of x, given by ... where xi is the individual measurement and N is the number of measurements, is also give.... Find the value of σ for the set of data given here. Get solution
8q. Is it easier for an electron or a proton of the same energy to tunnel through a given potential barrier? Explain. Get solution
9. If the potential V(x) for a one-dimensional system is independent of time, show that the expectation value for x is independent of time. Get solution
9q. Can a wave packet be formed from a superposition of wave functions of the type ei(kx-ωt) ? Can it be normalized? Get solution
10q. Given a particular potential V and wave function ψ, how could you prove that the given ψ is correct? Could you determine an appropriate energy E if the potential is independent of time? Get solution
11. A wave function has the value A sin x between x = 0 and π but zero elsewhere. Normalize the wave function and find the probability that the particle is (a) between x = 0 and x = π/4 and (b) between x = 0 and π/2. Get solution
11q. Compare the infinite square-well potential with the finite one. Where is the Schrödinger wave equation the same? Where is it different? Get solution
12. Find an equation for the difference between adjacent energy levels ...for the infi nite square-well potential. Calculate ΔE1, ΔE8, and ΔE800. Get solution
12q. Tunneling can occur for an electron trying to pass through a very thin tunnel diode. Can a baseball tunnel through a very thin window? Explain. Get solution
13. Determine the average value of ... inside the well for the infinite square-well potential for n = 1, 5, 10, and 100. Compare these averages with the classical probability of detecting the particle inside the box. Get solution
13q. For the three-dimensional cubical box, the ground state is given by n1 =n2 = n3 =1. Why is it not possible to have one ni = 1 and the other two equal to zero? Get solution
14. A particle in an infi nite square-well potential has ground-state energy 4.3 eV. (a) Calculate and sketch the energies of the next three levels, and (b) sketch the wave functions on top of the energy levels. Get solution
15. We can approximate an electron moving in a nanowire (a small, thin wire) as a one-dimensional infinite square-well potential. Let the wire be 2.0 μm long. The nanowire is cooled to a temperature of 13 K, and we assume the electron’s average kinetic energy is that of gas molecules at this temperature (= 3kT/2). (a) What are the three lowest possible energy levels of the electrons? (b) What is the approximate quantum number of electrons moving in the wire? Get solution
16. An electron moves with a speed v = 1.25 × 10-4c inside a one-dimensional box (V = 0) of length 48.5 nm. The potential is infinite elsewhere. The particle may not escape the box. What approximate quantum number does the electron have? Get solution
17. For the infinite square-well potential, find the probability that a particle in its ground state is in each third of the one-dimensional box: 0 ≤ x ≤ L/3, L/3 ≤ x ≤ 2L/3, 2L/3 ≤ x ≤ L. Check to see that the sum of the probabilities is one. Get solution
18. Repeat the previous problem using the first excited state. For the infinite square-well potential, find the probability that a particle in its ground state is in each third of the one-dimensional box: 0 ≤ x ≤ L/3, L/3 ≤ x ≤ 2L/3, 2L/3 ≤ x ≤ L. Check to see that the sum of the probabilities is one. Get solution
19. Repeat Example 6.9 for an electron inside the nucleus. Assume nonrelativistic equations and find the transition energy for an electron. (See Example 6.9 for an interpretation of the result.) ... Get solution
20. What is the minimum energy of (a) a proton and (b) an α particle trapped in a one-dimensional region the size of a uranium nucleus (radius = 7.4 × 10-15 m)? Get solution
21. An electron is trapped in an infinite square-well potential of width 0.70 nm. If the electron is initially in the n = 4 state, what are the various photon energies that can be emitted as the electron jumps to the ground state? Get solution
22. Consider a finite square-well potential well of width 3.00 × 10-15 m that contains a particle of mass 1.88 GeV/c2. How deep does this potential well need to be to contain three energy levels? (This situation approximates a deuteron inside a nucleus.) Get solution
23. Compare the results of the infinite and finite square well potentials. (a) Are the wavelengths longer or shorter for the finite square well compared with the infinite well? (b) Use physical arguments to decide whether the energies (for a given quantum number n) are (i) larger or (ii) smaller for the finite square well than for the infinite square well? (c) Why will there be a finite number of bound energy states for the finite potential? Get solution
24. Apply the boundary conditions to the finite square well potential at x = 0 to find the relationships between the coefficients A, C, and D and the ratio C/D. Get solution
25. Apply the boundary conditions to the finite square well potential at x = L to find the relationship between the coefficients B, C, and D and the ratio C/D. Get solution
26. Find the energies of the second, third, fourth, and fifth levels for the three-dimensional cubical box. Which energy levels are degenerate? Get solution
27. Write the possible (un normalized) wave functions for each of the first four excited energy levels for the cubical box. Get solution
28. Find the normalization constant A for the ground state wave function for the cubical box in Equation (6.52). ... Get solution
29. Complete the derivation of Equation (6.49) by substituting the wave function given in Equation (6.47) into Equation (6.46). What is the origin of the three quantum numbers? ... ... ... Get solution
30. Find the normalization constant A [in Equation (6.47)] for the first excited state of a particle trapped in a cubical potential well with sides L. Does it matter which of the three degenerate excited states you consider? ... Get solution
31. A particle is trapped in a rectangular box having sides L, 2L, and 4L. Find the energy of the ground state and fi rst three excited states. Are any of these states degenerate? Get solution
33. What is the energy level difference between adjacent levels ΔEn = En - 1 - En for the simple harmonic oscillator? What are ΔE0, ΔE2, and ΔE20? How many possible energy levels are there? Get solution
34. The wave function for the first excited state c1 for the simple harmonic oscillator is .... Normalize the wave function to find the value of the constant A. Determine x>, x 2> and ... Get solution
35. A nitrogen atom of mass 2.32 × 10-26 kg oscillates in one dimension at a frequency of 1013 Hz. What are its effective force constant and quantized energy levels? Get solution
36. One possible solution for the wave function ψn for the simple harmonic oscillator is ... where A is a constant. What is the value of the energy level En? Get solution
37. What would you expect for p> and p2> for the ground state of the simple harmonic oscillator? (Hint: Use symmetry and energy arguments.) Get solution
38. Show that the energy of a simple harmonic oscillator in the n = 1 state is ...directly into the Schrödinger equation. Get solution
39. An H2 molecule can be approximated by a simple harmonic oscillator with a force constant k = 1.1 × 103 N/m. Find (a) the energy levels and (b) the possible wavelengths of photons emitted when the H2 molecule decays from the third excited state eventually to the ground state. Get solution
40. The creation of elements in the early universe and in stars involves protons tunneling through nuclei. Find the probability of the proton tunneling through 12C when the temperature of the star containing the proton and carbon is 12,000 K. Get solution
41. Compare the wavelength of a particle when it passes a barrier of height (a) +V0 (see Figure 6.12) and (b) -V0 where E > |V0 | (see Figure 6.18). Calculate the momentum and kinetic energy for both cases. ... Get solution
42. (a) Calculate the transmission probability of an α particle of energy E = 5.0 MeV through a Coulomb barrier of a heavy nucleus that is approximated by a square barrier with V0 = 15 MeV and barrier width L = 1.3 × 10-14 m. Also, calculate the probability (b) by doubling the potential barrier height and (c) by using the original barrier height but doubling the barrier width. Compare all three probabilities. Get solution
43. Consider a particle of energy E trapped inside the potential well shown in the accompanying figure. Make an approximate sketch of possible wave functions inside and outside the potential well. Explain your sketch. ... Get solution
44. When a particle of energy E approaches a potential barrier of height V0, where E>> V0, show that the reflection coefficient is about ... Get solution
45. Let 12.0-eV electrons approach a potential barrier of height 4.2 eV. (a) For what barrier thickness is there no reflection? (b) For what barrier thickness is the reflection a maximum? Get solution
46. A 1.0-eV electron has a 2.0 × 10-4 probability of tunneling through a 2.5-eV potential barrier. What is the probability of a 1.0-eV proton tunneling through the same barrier? Get solution
47. An electron is attempting to tunnel through a square barrier potential. (a) Draw a potential function with zero potential on either side of a square-top potential similar to Figure 6.12. Draw the wave function before, after, and inside the barrier. (b) Let the barrier be twice as wide and repeat part (a). (c) Let the barrier be about twice as tall as in (a) and repeat (a). Do not perform calculations; make estimates only. Get solution
48. Use the approximate Equation (6.73) to estimate the probability of (a) a 1.4-eV electron tunneling through a 6.4-eV-high barrier of width 2.8 nm, and (b) a 4.4- MeV α particle tunneling through a uranium nucleus where the potential barrier is 19.2 MeV and 7.4 fm wide. (c) Discuss whether the approximation was valid for these two cases. Explain. ... Get solution
49g. Check to see whether the simple linear combination of sine and cosine functions satisfies the time-independent Schrödinger equation for a free particle (V = 0). ... Get solution
50g. (a) Check to see whether the simple linear combination of sine and cosine functions ... where A and B are real numbers, satisfies the time dependent Schrödinger equation for a free particle (V = 0). (b) Repeat for the modified version ... Get solution
51g. A particle of mass m is trapped in a three-dimensional rectangular potential well with sides of length L, ...and 2L. Inside the box V = 0, outside V = ∞. Assume that ... inside the well. Substitute this wave function into the Schrödinger equation and apply appropriate boundary conditions to find the allowed energy levels. Find the energy of the ground state and fi rst four excited levels. Which of these levels are degenerate? Get solution
52g. For a region where the potential V = 0, the wave function is given by .... Calculate the energy of this system. Get solution
53g. Consider the semi-infinite-well potential in which V =∞ for x = 0, V = 0 for 0? x? L, and V = V0 for x≥L. (a) Show that possible wave functions are A sin kx inside the well and Be-kx for x ? L, where k = ...and k = .... (b) Show that the application of the boundary conditions gives k tan (kL) = -k Get solution
55g. Prove that there are a limited number of bound solutions for the semi-infinite well. Get solution
56g. Use the semi-infinite-well potential to model a deuteron, a nucleus consisting of a neutron and a proton. Let the well width be 3.5 × 10-15 m and V0 - E = 2.2 MeV. Determine the energy E. How many excited states are there, and what are their energies? Get solution
57g. Consider as a model of a hydrogen atom a particle trapped in a one-dimensional, infinite potential well of width 2a0 (the ground-state hydrogen atom’s diameter). Find the electron’s ground-state energy and comment on the result. Get solution
58g. (a) Repeat the preceding problem using a cubical infinite potential well, with each side of the cube equal to 2a0. Get solution
59g. In the lab you make a simple harmonic oscillator with a 0.15-kg mass attached to a 12-N/m spring. (a) If the oscillation amplitude is 0.10 m, what is the corresponding quantum number n for the quantum harmonic oscillator? (b) What would be the amplitude of the quantum ground state for this oscillator? (c) What is the energy of a photon emitted when this oscillator makes a transition between adjacent energy levels? Comment on each of your results. Get solution
60g. In gravity-free space, a 2.0-mg dust grain is confined to move back and forth between rigid walls 1.0 mm apart. (a) What is the speed of the dust grain if it is in the quantum ground state? (b) If it is actually moving at a speed of 0.25 mm/s, what is the quantum number associated with its quantum state? Get solution
61g. The wave function for the n _ 2 state of a simple harmonic oscillator is .... (a) Show that its energy level is ... by substituting the wave function into the Schrödinger equation. (b) Find x> and x 2>. Get solution
62g. A particle is trapped inside an infinite square-well potential between x = 0 and x = L. Its wave function is a superposition of the ground state and first excited state. The wave function is given by ... Show that the wave function is normalized. Get solution
63g. The Morse potential is a good approximation for a real potential to describe diatomic molecules. It is given by ... where D is the molecular dissociation energy, and re is the equilibrium distance between the atoms. For small vibrations, r - re is small, and V(r) can be expanded in a Taylor series to reduce to a simple harmonic potential. Find the lowest term of V(r) in this expansion and show that it is quadratic in (r - re). Get solution
64g. Show that the vibrational energy levels Ev for the Morse potential of the previous problem are given by ... Where ... and n is the vibrational quantum number, mr is the reduced mass, and Ev ?? D. Find the three lowest energy levels for KCl where D = 4.42 eV, and a = 7.8 nm-1. Get solution
65g. Consider a particle of mass m trapped inside a two dimensional square box of sides L aligned along the x and y axes. Show that the wave function and energy levels are given by ... Plot the first six energy levels and give their quantum numbers. Get solution
66g. Make a sketch for each of the following situations for both the infinite square-well and finite square-well potentials in one dimension: (a) the four lowest energy levels and (b) the probability densities for the four lowest states. (c) Discuss the differences between the two potentials and why they occur. Get solution
67g. Two nanowires are separated by 1.3 nm as measured by STM. Inside the wires the potential energy is zero, but between the wires the potential energy is greater than the electron’s energy by only 0.9 eV. Estimate the probability that the electron passes from one wire to the other. Get solution
68g. The WKB approximation is useful to obtain solutions to the one-dimensional time-independent Schrödinger equation in cases where E ? V(x) and the potential V(x) changes slowly and gradually with x. In this case the wavelength λ (x) varies with x because of the V(x) dependence on x. (a) Argue that we can write the wavelength as ... for a particle of mass m in a potential V(x). (b) By considering the number of oscillations that can be fi t into a distance dx, show that the following equation is valid, where n is an integer and represents the number of standing waves that fi t inside the potential well. ... where n is an integer (6.74) This is the WKB approximation. (Hint: the equation might be helpful) ... Get solution
69g. Use the WKB approximation of Equation (6.74) in the previous problem with the potential shown in the accompanying figure. V(x) = ∞ for x = 0, and V(x) = Ax for x ? 0. (a) Find the quantized energy values, and (b) sketch the wave functions on top of the V(x) function for the three lowest states. ... Get solution
70g. In the special topic box in Section 6.7, the extreme sensitivity of scanning tunneling microscopes (STM) was described. It was reported that a change in the tunneling gap of only 0.4 nm between STM sample and probe can change the tunneling current by a factor of 104. Check the plausibility of this statement by using the “wide-barrier” approximation in Equation (6.70) to find the difference between the electron energy and barrier height that would produce such a situation. Report your answer in eV and comment in the result. Get solution
