1cc. If the width of the potential well is reduced to half its
original value, the energy of the n = 3 wave function willa) stay the
same.b) be reduced by a factor of 2.c) be reduced by a factor of 4.d) be
increased by a factor of 2.e) be increased by a factor of 4. Get solution
1mcq. The wavelength of an electron in an infinite potential well is α/2, where α is the width of the well. Which state is the electron in?a) n = 3b) n = 6c) n = 4d) n = 2 Get solution
2cc. If the width of the potential well in Example 37.2 is doubled, to 2.6 nm, and the depth remains the same, the number of bound states willa) stay the same.b) increase.c) decrease.Example 37.2Bound StatesProblemAn electron is trapped in a finite potential well of the kind shown in Figure 37.10 with a depth of U1 = 0.824 eV and a width of a = 1.30 nm. What wave numbers correspond to the possible bound states in this well?SolutionThe well depth is the same as in Solved Problem 37.1, so one solution for the wave number should correspond to the value 2π/(1.20a), the starting point in that problem. We also found two conditions for the exponential decay constant γ. Equation37.22 gave ..., and equation 37.21 gave .... Combining these results gives...We omitted the index 2 for the wave number ... from equation 37.22, because we want to search for all possible values of ... that satisfy equation (i). This equation usually does not have an algebraic solution, but we can solve it numerically quite straightforwardly. The numerical constants in this case are a = 1.30 nm and 2a2mU1/ħ2 = 36.5. Thus, we have to solve the equation...Figure 37.12 plots the two functions ... (in blue) and –y cot y (in red) and shows the positions where they intersect.As you can see, there are only two positions where the two functions have the same value. Therefore, the potential well of depth U1 = 0.824 eV and width a = 1.30 nm has only two bound states. Numerically, we find ... and .... This second value is nothing other than 2π/1.20, the value in Solved Problem 37.1. However, the value of ... is new information. Note that, unlike the case of the infinite potential well, ... for the finite potential well.In conclusion, Figure 37.13 shows the energies that correspond to the two bound states, overlaid on the shape of the potential energy function used for the potential well.Figure 37.10 Finite potential energy well....Figure 37.12 Finding the wave numbers for the bound states....Figure 37.13 Energies of the lowest two possible wave functions in the finite potential well.... Get solution
2mcq. For which of the following states will the particle never be found in the exact center of a square infinite potential well?a) the ground stateb) the first excited statec) the second excited stated) any of the abovee) none of the above Get solution
3cc. To increase the tunneling probability for the neutron in Example 37.3, it would be necessary toa) increase the barrier height and/or increase the barrier width.b) increase the barrier height and/or decrease the barrier width.c) decrease the barrier height and/or increase the barrier width.d) decrease the barrier height and/or decrease the barrier width.Example 37.3 is increased by 15%. By what factor does the neutron’s probability of tunneling through the barrier increase?Example 37.3 Neutron Tunneling...... Get solution
3mcq. The probability of finding an electron at a particular location in a hydrogen atom is directly proportional toa) its energy.b) its momentum.c) its wave function.d) the square of its wave function.e) the product of the position coordinate and the square of the wave function.f) none of the above. Get solution
4cc. In equation 37.39, the term ψa(x2) · ψb(x1) has a negative sign, and the term ψa(x1) · ψb(x2) has a positive sign. How important is this sign convention?a) These are the only signs possible for the two terms, because the second term is the exchange term and thus must have a negative sign.b) It does not matter which term gets which sign; all that matters is that they have opposite signs. If you multiply a wave function by an overall factor of –1, you still obtain a valid wave function.c) The sign of the exchange term is arbitrary; it could be positive or negative. But the first term must always be positive.d) Both terms can have either sign, and each of the four sign combinations (++,––,–+,+–) leads to a valid wave function.Equation 37.39... Get solution
4mcq. Is the superposition of two wave functions, which are solutions to the time-independent Schrödinger equation for the same potential energy, also a solution to the Schrödinger equation?a) nob) yesc) depends on the value of the potential energyd) only if ... Get solution
5mcq. Let ... be the wave number of a particle moving in one dimension with velocity v. If the velocity of the particle is doubled, to 2v, then the magnitude of the wave number isa) ....b) 2 ....c) .../2.d) none of these. Get solution
6mcq. An electron is in a square infinite potential well of width a: U(x) = ∞, for x x > a. If the electron is in the first excited state, ψ(x) = A sin (2πx/a), at what position(s) is the probability function a maximum?a) 0b) a/4c) a/2d) 3a/4e) both a/4 and 3a/4 Get solution
7mcq. Which of the following statements is (are) true?a) The energy of electrons is always discrete.b) The energy of a bound electron is continuous.c) The energy of a free electron is discrete.d) The energy of an electron is discrete when it is bound to an ion. Get solution
8mcq. Which of the following statements is (are) true?a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced.b) In a one-dimensional infinite potential well, the energy levels are evenly spaced.c) The minimum total energy possible for a classical harmonic oscillator is zero.d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state (n = 0) of the one-dimensional quantum harmonic oscillator should also be zero.e) The n = 0 state of a one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty, ∆x∆p. Get solution
9mcq. Simple harmonic oscillation occurs when the potential energy function is equal to ..., where κ is a constant. What happens to the ground-state energy level if κ is increased?a) It increases.b) It remain the same.c) It decreases. Get solution
10mcq. A particle with energy E = 5 eV approaches a barrier of height U = 8 eV. Quantum mechanically there is a nonzero probability that the particle will tunnel through the barrier. If the barrier height is slowly decreased, the probability that the particle will deflect off the barrier willa) decrease.b) increase.c) not change. Get solution
11cq. Is the following statement true or false? The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer. Get solution
12cq. For a particle trapped in a square infinite potential well of length L, what happens to the probability that the particle is found between 0 and L/2 as the particle’s energy increases? Get solution
13cq. Think about what happens to wave functions for a particle in a square infinite potential well as the quantum number n approaches infinity. Does the probability distribution in that limit obey the correspondence principle? Explain. Get solution
14cq. Show by symmetry arguments that the expectation value of the momentum for an even-n state of a one-dimensional harmonic oscillator is zero. Get solution
15cq. Is it possible for the expectation value of the position of an electron to correspond to a position where the electron’s probability function, Π(x), is zero? If it is possible, give a specific example. Get solution
16cq. Sketch the two lowest-energy wave functions for an electron in an infinite potential well that is 20 nm wide and a finite potential well that is 1 eV deep and is also 20 nm wide. Using your sketches, can you determine whether the energy levels in the finite potential well are lower, the same, or higher than in the infinite potential well? Get solution
17cq. In the cores of white dwarf stars, carbon nuclei are thought to be locked into very ordered lattices because the temperature is very low, ~104 K. Consider a one-dimensional lattice in which the atoms are separated by 20 fm (1 fm = 1 · 10–15 m). Approximate the Coulomb potentials of the two outside atoms as following a quadratic relationship, and assume that the atoms’ vibrational motions are small. What energy state would the central carbon atom be in at this temperature? ... Get solution
18cq. For a square finite potential well, you have seen solutions for particle energies greater than and less than the well depth. Show that these solutions are equal outside the potential well if the particle energy is equal to the well depth. Explain your answer and the possible difficulty with it. Get solution
19cq. Consider the energies allowed for bound states of a half-harmonic oscillator, having the potential...Using simple arguments based on the characteristics of normalized wave functions, what are the energies allowed for bound states in this potential? Get solution
20cq. Suppose ψ(x) is a properly normalized wave function describing the state of an electron. Consider a second wave function, ψnew(x) = eiϕ ψ(x), for some real number ϕ. How does the probability density associated with ψnew compare to that associated with ψ? Get solution
21cq. A particle of mass m is in the potential U(x) = U0 cosh(x/a), where U0 and a are constants. Show that the ground-state energy of the particle can be estimated as... Get solution
22cq. The time-independent Schrödinger equation for a nonrelativistic free particle of mass m is obtained from the energy relationship E = p2/ (2m) by replacing E and p with appropriate derivative operators, as suggested by the de Broglie relations. Using this procedure, derive a quantum wave equation for a relativistic particle of mass m, for which the energy relation is E2 – p2c2 = m2c4, without taking any square root of this relation. Get solution
23. A neutron has a kinetic energy of 10.0 MeV. What size object would have to be used to observe neutron diffraction effects? Is there anything in nature of this size that could serve as a target to demonstrate the wave nature of 10.0-MeV neutrons? Get solution
24. Given the complex function f(x) = (8 + 3i) + (7 – 2i)x of the real variable x, what is |f(x)|2? Get solution
26. Determine the three lowest energies of the wave function of a proton in a box of width 1.0 · 10–10 m. Get solution
27. What is the ratio of the energy difference between the ground state and the first excited state for a square infinite potential well of length L and that for a square infinite potential well of length 2L? That is, find (E2 – E1)L/(E2 – E1)2L. Get solution
29. Find the wave function for a particle in an infinite square well centered at the origin, with the walls at ±a/2. Get solution
30. In Example 37.1, we calculated the energy of the wave function with the lowest quantum number for an electron confined to a one-dimensional box of width 2.00 Å. However, atoms are three-dimensional entities with a typical diameter of 1.00 Å = 10–10 m. It would seem then that the better approximation would be an electron trapped in a three-dimensional infinite potential well (a potential cube with sides of 1.00 Å).a) Derive an expression for the wave function and the corresponding energies of an electron in a three-dimensional rectangular infinite potential well.b) Calculate the lowest energy allowed for the electron in this case.Example 37.1 Electron in a Box...... Get solution
31. An electron is confined to a potential well shaped as shown in Figure 37.10. The width of the well is 1.0·10–9 m, and U1 = 2.0 eV. Is the n = 3 state bound in this well?Figure 37.10 Finite potential energy well... Get solution
32. If a proton of kinetic energy 18.0 MeV encounters a rectangular potential energy barrier of height 29.8 MeV and width 1.00·10–15 m, what is the probability that the proton will tunnel through the barrier? Get solution
33. Suppose the kinetic energy of the neutron described in Example 37.3 is increased by 15%. By what factor does the neutron’s probability of tunneling through the barrier increase?Example 37.3 Neutron Tunneling...... Get solution
34. A beam of electrons moving in the positive x-direction encounters a potential barrier that is 2.51 eV high and 1.00 nm wide. Each electron has a kinetic energy of 2.50 eV, and the electrons arrive at the barrier at a rate of 1000. electrons/s. What is the rate IT (in electrons/s) at which electrons pass through the barrier, on average? What is the rate IR (in electrons/s) at which electrons bounce back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.... Get solution
35. Consider an electron approaching a potential barrier 2.00 nm wide and 7.00 eV high. What is the energy of the electron if it has a 10.0% probability of tunneling through this barrier? Get solution
36. Consider an electron in a three-dimensional box—with infinite potential walls—of dimensions 1.00 nm × 2.00 nm × 3.00 nm. Find the quantum numbers nx, ny, and nz and the energies (in eV) of the six lowest energy levels. Are any of these levels degenerate, that is, do any distinct quantum states have identical energies? Get solution
37. In a scanning tunneling microscope, the probability that an electron from the probe will tunnel through a 0.100-nm gap is 0.100%. Calculate the work function of the probe. Get solution
38. Consider a square potential, U(x) = 0 for x α, U(x) = –U0 for –α ≤ x ≤ α, where U0 is a positive constant, and U(x) = 0 for x > α. For E > 0, the solutions of the time-independent Schrodinger equation in the three regions will be the following:For ..., where ... and R is the amplitude of a reflected wave.For..., and ....For ... where T is the amplitude of the transmitted wave.Match ψ(x) and dψ(x)/dx at –α and α and find an expression for R. What is the condition for which R = 0 (that is, there is no reflected wave)? Get solution
39. a) Determine the wave function and the energy levels for the bound states of an electron in the symmetrical one-dimensional potential well of finite depth shown in the figure.b) If the penetration distance η into the classically forbidden region is defined as the distance at which the wave function decreases to 1/e of its value at the edge of the well, determine an expression for this penetration distance....c) The electrons in a typical GaAs-GaAlAs quantum-well laser diode are confined within a one-dimensional potential well like the one shown in the figure, of width 1 nm and depth 0.300 eV. Numerical solutions to the Schrodinger equation show that there is only one possible bound state for the electrons in this case, with energy of 0.125 eV. Calculate the penetration distance for these electrons. Get solution
40. An oxygen molecule has a vibrational mode that behaves approximately like a simple harmonic oscillator with frequency 2.99·1014 rad/s. Calculate the energy of the ground state and the first two excited states. Get solution
42. An experimental measurement of the energy levels of a hydrogen molecule, H2, shows that they are evenly spaced and separated by about 9·10–20 J. A reasonable model of one of the hydrogen atoms would then seem to be that of a particle in a simple harmonic oscillator potential. Assuming that the hydrogen atom is attached by a spring with a spring constant κ to the other atom in the molecule, what is the spring constant k? Get solution
43. Calculate the ground-state energy for an electron confined to a cubic potential well with sides equal to twice the Bohr radius (R = 0.0529 nm). Determine the spring constant that would give this same ground-state energy for a harmonic oscillator. Get solution
44. A particle in a harmonic oscillator potential has the initial wave function Ψ(x, 0) = A [ψ0(x) + ψ1(x)]. Normalize Ψ(x, 0). Get solution
45. The ground-state wave function for a harmonic oscillator is given by ....a) Determine the normalization constant, A2.b) Determine the probability that a quantum harmonic oscillator in the n = 0 state will be found in the classically forbidden region. Get solution
46. A particle is in a square infinite potential well of width L and is in the n = 3 state. What is the probability that, when observed, the particle is found to be in the rightmost 10.0% of the well? Get solution
47. An electron is confined between x = 0 and x = L. The wave function of the electron is ψ(x) = Asin(2πx/L). The wave function is zero for the regions x x > L.a) Determine the normalization constant, A.b) What is the probability of finding the electron in the region 0 ≤ x ≤ L/3? Get solution
48. Find the probability of finding an electron trapped in a one-dimensional infinite well of width 2.00 nm in the n = 2 state between 0.800 and 0.900 nm (assume that the left edge of the well is at x = 0 and the right edge is at x = 2.00 nm). Get solution
49. An electron is trapped in a one-dimensional infinite potential well that is L = 300. pm wide. What is the probability that the electron in the first excited state will be detected in an interval between x = 0.500 L and x = 0.750 L? Get solution
50. Find the uncertainty of x for the wave function ... Get solution
51. Write a wave function ... for a nonrelativistic free particle of mass m moving in three dimensions with momentum ..., including the correct time dependence as required by the Schrodinger equation. What is the probability density associated with this wave? Get solution
52. Suppose a quantum particle is in a stationary state with a wave function ... (x,t). The calculation of x, the expectation value of the particle’s position, is shown in the text. Calculate ... Get solution
53. Although quantum systems are frequently characterized by their stationary states, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass m in a one-dimensional potential well with infinite walls (a box) of width a is actually in a state with the wave function....where ... denotes the stationary state with quantum number n = 1 and ... denotes the state with n = 2. Calculate the probability density distribution for the position x of the particle in this state. Get solution
54. In Chapter 40, you will see that a nuclear-fusion reaction between two protons (creating a deuteron, a positron, and a neutrino) releases 0.42 MeV of energy. Nuclear fusion is what causes the stars to shine, and if we can harness it, we can solve the world’s energy problems. Compare the energy released in this fusion reaction with what would be released by the annihilation of a proton and an antiproton. Get solution
55. Particle-antiparticle pairs are occasionally created out of empty space. Considering energy-time uncertainty, how long would such pairs be expected to exist at most if they consist ofa) an electron and a positron?b) a proton and an antiproton? Get solution
56. A positron and an electron annihilate, producing two 2.0-MeV gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron. Get solution
57. Calculate the energy of the first excited state of a proton in a one-dimensional infinite potential well of width α = 1.00 nm. Get solution
58. Electrons in a scanning tunneling microscope encounter a potential barrier that has a height of U = 4.0 eV above their total energy. By what factor does the tunneling current change if the tip moves a net distance of 0.10 nm farther from the surface? Get solution
59. An electron is confined in a three-dimensional cubic space of L3 with infinite potentials.a) Write down the normalized solution of the wave function for the ground state.b) How many energy states are available between the ground state and the second excited state? (Take the electron’s spin into account.) Get solution
60. A mass-and-spring harmonic oscillator used for classroom demonstrations has the angular frequency ω0 = 4.45 s–1. If this oscillator has a total (kinetic plus potential) energy E = 1.00 J, what is its corresponding quantum number, n? Get solution
61. The neutrons in a parallel beam, each having kinetic energy ... (which approximately corresponds to room temperature), are directed through two slits 0.50 mm apart. How far apart will the peaks of the interference pattern be on a screen 1.5 m away? Get solution
62. Find the ground-state energy (in eV) of an electron in a one-dimensional box, if the box is of length L = 0.100 nm. Get solution
63. An approximately one-dimensional potential well can be formed by surrounding a layer of GaAs with layers of AlxGa1–xAs. The GaAs layers can be fabricated in thicknesses that are integral multiples of the single-layer thickness, 0.28 nm. Some electrons in the GaAs layer behave as if they were trapped in a box. For simplicity, treat the box as a one-dimensional infinite potential well and ignore the interactions between the electrons and the Ga and As atoms (such interactions are often accounted for by replacing the actual electron mass with an effective electron mass). Calculate the energy of the ground state in this well for these cases:a) 2 GaAs layersb) 5 GaAs layers Get solution
64. Consider a water molecule in the vapor state in a room 4.00 m × 10.0 m × 10.0 m.a) What is the ground-state energy of this molecule, treating it as a simple particle in a box?b) Compare this energy to the average thermal energy of such a molecule, taking the temperature to be 300. K.c) What can you conclude from the two numbers you just calculated? Get solution
65. A neutron moves between rigid walls 8.4 fm apart. What is the energy of its n = 1 state? Get solution
66. A surface is examined using a scanning tunneling microscope (STM). For the range of the gap, L, between the tip of the probe and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as .... The tunneling current through the tip of the probe is proportional to the tunneling probability. In this situation, what is the ratio of the current when the tip is 0.400 nm above a surface feature to the current when the tip is 0.420 nm above the surface? Get solution
67. An electron is trapped in a one-dimensional infinite well of width 2.00 nm. It starts in the n = 4 state, and then goes into the n = 2 state, emitting radiation with energy corresponding to the energy difference between the two states. What is the wavelength of the radiation? Get solution
68. Two long, straight wires that lie along the same line have a separation at their tips of 2.00 nm. The potential energy of an electron in this gap is about 1.00 eV higher than it is in the conduction band of the two wires. Conduction-band electrons have enough energy to contribute to the current flowing in the wire. What is the probability that a conduction electron from one wire will be found in the other wire after arriving at the gap? Get solution
69. Consider an electron that is confined to a one-dimensional infinite potential well of width a = 0.10 nm, and another electron that is confined to a three-dimensional (cubic) infinite potential well with sides of length a = 0.10 nm. Let the electron confined to the cube be in its ground state. Determine the difference in energy ground state of the two electrons and the excited state of the one-dimensional electron that minimizes the difference in energy with the three-dimensional electron. Get solution
70. An electron with an energy of 129 KeV is trapped in a potential well defined by an infinite potential at x U1 extending from x = 529.2 fm to x = 2116.8 fm (1 fm = 1 · 10–15 m), as shown in the figure. It is found that the electron can be detected beyond the barrier with a probability of 10%. Calculate the height of the potential barrier.... Get solution
71. Consider an electron that is confined to the xy-plane by a two-dimensional rectangular infinite potential well. The width of the well is w in the x-direction and 2w in the y-direction. What is the lowest energy that is shared by more than one distinct state, that is, two different states having the same energy? Get solution
73. A 6.31-MeV alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.7 MeV and whose width is 13.7 fm (1 fm = 1·10–15 m). What is the value of the decay constant, γ (in fm–1)? (Hint: A potentially useful value is ħc = 197.327 MeV fm.) Get solution
75. A 8.59-MeV alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.9 MeV. The tunneling probability is measured to be 1.042·10–18. What is the width of the barrier? (Hint: A potentially useful value is ħc = 197.327 MeV fm.) Get solution
74. An alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.7 MeV and whose width is 15.5 fm (1 fm = 1·10–15 m). The decay constant, γ, is measured to be 1.257 fm–1. What is the kinetic energy of the alpha particle? (Hint: A potentially useful value is ħc = 197.327 MeV fm.)... Get solution
76. An electron with a mass of 9.109·10–31 kg is trapped inside a one-dimensional infinite potential well of width 13.5 nm. What is the energy difference between the n = 5 and the n = 1 states? Get solution
77. A proton with a mass of 1.673·10–27 kg is trapped inside a one-dimensional infinite potential well of width 23.9 nm. What is the quantum number, n, of the state that has an energy difference of 1.08 ·10–3 meV with the n = 2 state? Get solution
78. A particle is trapped inside a one-dimensional infinite potential well of width 19.3 nm. The energy difference between the n = 2 and the n = 1 states is 2.639·10–25 J. What is the mass of the particle? Get solution
1mcq. The wavelength of an electron in an infinite potential well is α/2, where α is the width of the well. Which state is the electron in?a) n = 3b) n = 6c) n = 4d) n = 2 Get solution
2cc. If the width of the potential well in Example 37.2 is doubled, to 2.6 nm, and the depth remains the same, the number of bound states willa) stay the same.b) increase.c) decrease.Example 37.2Bound StatesProblemAn electron is trapped in a finite potential well of the kind shown in Figure 37.10 with a depth of U1 = 0.824 eV and a width of a = 1.30 nm. What wave numbers correspond to the possible bound states in this well?SolutionThe well depth is the same as in Solved Problem 37.1, so one solution for the wave number should correspond to the value 2π/(1.20a), the starting point in that problem. We also found two conditions for the exponential decay constant γ. Equation37.22 gave ..., and equation 37.21 gave .... Combining these results gives...We omitted the index 2 for the wave number ... from equation 37.22, because we want to search for all possible values of ... that satisfy equation (i). This equation usually does not have an algebraic solution, but we can solve it numerically quite straightforwardly. The numerical constants in this case are a = 1.30 nm and 2a2mU1/ħ2 = 36.5. Thus, we have to solve the equation...Figure 37.12 plots the two functions ... (in blue) and –y cot y (in red) and shows the positions where they intersect.As you can see, there are only two positions where the two functions have the same value. Therefore, the potential well of depth U1 = 0.824 eV and width a = 1.30 nm has only two bound states. Numerically, we find ... and .... This second value is nothing other than 2π/1.20, the value in Solved Problem 37.1. However, the value of ... is new information. Note that, unlike the case of the infinite potential well, ... for the finite potential well.In conclusion, Figure 37.13 shows the energies that correspond to the two bound states, overlaid on the shape of the potential energy function used for the potential well.Figure 37.10 Finite potential energy well....Figure 37.12 Finding the wave numbers for the bound states....Figure 37.13 Energies of the lowest two possible wave functions in the finite potential well.... Get solution
2mcq. For which of the following states will the particle never be found in the exact center of a square infinite potential well?a) the ground stateb) the first excited statec) the second excited stated) any of the abovee) none of the above Get solution
3cc. To increase the tunneling probability for the neutron in Example 37.3, it would be necessary toa) increase the barrier height and/or increase the barrier width.b) increase the barrier height and/or decrease the barrier width.c) decrease the barrier height and/or increase the barrier width.d) decrease the barrier height and/or decrease the barrier width.Example 37.3 is increased by 15%. By what factor does the neutron’s probability of tunneling through the barrier increase?Example 37.3 Neutron Tunneling...... Get solution
3mcq. The probability of finding an electron at a particular location in a hydrogen atom is directly proportional toa) its energy.b) its momentum.c) its wave function.d) the square of its wave function.e) the product of the position coordinate and the square of the wave function.f) none of the above. Get solution
4cc. In equation 37.39, the term ψa(x2) · ψb(x1) has a negative sign, and the term ψa(x1) · ψb(x2) has a positive sign. How important is this sign convention?a) These are the only signs possible for the two terms, because the second term is the exchange term and thus must have a negative sign.b) It does not matter which term gets which sign; all that matters is that they have opposite signs. If you multiply a wave function by an overall factor of –1, you still obtain a valid wave function.c) The sign of the exchange term is arbitrary; it could be positive or negative. But the first term must always be positive.d) Both terms can have either sign, and each of the four sign combinations (++,––,–+,+–) leads to a valid wave function.Equation 37.39... Get solution
4mcq. Is the superposition of two wave functions, which are solutions to the time-independent Schrödinger equation for the same potential energy, also a solution to the Schrödinger equation?a) nob) yesc) depends on the value of the potential energyd) only if ... Get solution
5mcq. Let ... be the wave number of a particle moving in one dimension with velocity v. If the velocity of the particle is doubled, to 2v, then the magnitude of the wave number isa) ....b) 2 ....c) .../2.d) none of these. Get solution
6mcq. An electron is in a square infinite potential well of width a: U(x) = ∞, for x x > a. If the electron is in the first excited state, ψ(x) = A sin (2πx/a), at what position(s) is the probability function a maximum?a) 0b) a/4c) a/2d) 3a/4e) both a/4 and 3a/4 Get solution
7mcq. Which of the following statements is (are) true?a) The energy of electrons is always discrete.b) The energy of a bound electron is continuous.c) The energy of a free electron is discrete.d) The energy of an electron is discrete when it is bound to an ion. Get solution
8mcq. Which of the following statements is (are) true?a) In a one-dimensional quantum harmonic oscillator, the energy levels are evenly spaced.b) In a one-dimensional infinite potential well, the energy levels are evenly spaced.c) The minimum total energy possible for a classical harmonic oscillator is zero.d) The correspondence principle states that because the minimum possible total energy for the classical simple harmonic oscillator is zero, the expected value for the fundamental state (n = 0) of the one-dimensional quantum harmonic oscillator should also be zero.e) The n = 0 state of a one-dimensional quantum harmonic oscillator is the state with the minimum possible uncertainty, ∆x∆p. Get solution
9mcq. Simple harmonic oscillation occurs when the potential energy function is equal to ..., where κ is a constant. What happens to the ground-state energy level if κ is increased?a) It increases.b) It remain the same.c) It decreases. Get solution
10mcq. A particle with energy E = 5 eV approaches a barrier of height U = 8 eV. Quantum mechanically there is a nonzero probability that the particle will tunnel through the barrier. If the barrier height is slowly decreased, the probability that the particle will deflect off the barrier willa) decrease.b) increase.c) not change. Get solution
11cq. Is the following statement true or false? The larger the amplitude of a Schrödinger wave function, the larger its kinetic energy. Explain your answer. Get solution
12cq. For a particle trapped in a square infinite potential well of length L, what happens to the probability that the particle is found between 0 and L/2 as the particle’s energy increases? Get solution
13cq. Think about what happens to wave functions for a particle in a square infinite potential well as the quantum number n approaches infinity. Does the probability distribution in that limit obey the correspondence principle? Explain. Get solution
14cq. Show by symmetry arguments that the expectation value of the momentum for an even-n state of a one-dimensional harmonic oscillator is zero. Get solution
15cq. Is it possible for the expectation value of the position of an electron to correspond to a position where the electron’s probability function, Π(x), is zero? If it is possible, give a specific example. Get solution
16cq. Sketch the two lowest-energy wave functions for an electron in an infinite potential well that is 20 nm wide and a finite potential well that is 1 eV deep and is also 20 nm wide. Using your sketches, can you determine whether the energy levels in the finite potential well are lower, the same, or higher than in the infinite potential well? Get solution
17cq. In the cores of white dwarf stars, carbon nuclei are thought to be locked into very ordered lattices because the temperature is very low, ~104 K. Consider a one-dimensional lattice in which the atoms are separated by 20 fm (1 fm = 1 · 10–15 m). Approximate the Coulomb potentials of the two outside atoms as following a quadratic relationship, and assume that the atoms’ vibrational motions are small. What energy state would the central carbon atom be in at this temperature? ... Get solution
18cq. For a square finite potential well, you have seen solutions for particle energies greater than and less than the well depth. Show that these solutions are equal outside the potential well if the particle energy is equal to the well depth. Explain your answer and the possible difficulty with it. Get solution
19cq. Consider the energies allowed for bound states of a half-harmonic oscillator, having the potential...Using simple arguments based on the characteristics of normalized wave functions, what are the energies allowed for bound states in this potential? Get solution
20cq. Suppose ψ(x) is a properly normalized wave function describing the state of an electron. Consider a second wave function, ψnew(x) = eiϕ ψ(x), for some real number ϕ. How does the probability density associated with ψnew compare to that associated with ψ? Get solution
21cq. A particle of mass m is in the potential U(x) = U0 cosh(x/a), where U0 and a are constants. Show that the ground-state energy of the particle can be estimated as... Get solution
22cq. The time-independent Schrödinger equation for a nonrelativistic free particle of mass m is obtained from the energy relationship E = p2/ (2m) by replacing E and p with appropriate derivative operators, as suggested by the de Broglie relations. Using this procedure, derive a quantum wave equation for a relativistic particle of mass m, for which the energy relation is E2 – p2c2 = m2c4, without taking any square root of this relation. Get solution
23. A neutron has a kinetic energy of 10.0 MeV. What size object would have to be used to observe neutron diffraction effects? Is there anything in nature of this size that could serve as a target to demonstrate the wave nature of 10.0-MeV neutrons? Get solution
24. Given the complex function f(x) = (8 + 3i) + (7 – 2i)x of the real variable x, what is |f(x)|2? Get solution
26. Determine the three lowest energies of the wave function of a proton in a box of width 1.0 · 10–10 m. Get solution
27. What is the ratio of the energy difference between the ground state and the first excited state for a square infinite potential well of length L and that for a square infinite potential well of length 2L? That is, find (E2 – E1)L/(E2 – E1)2L. Get solution
29. Find the wave function for a particle in an infinite square well centered at the origin, with the walls at ±a/2. Get solution
30. In Example 37.1, we calculated the energy of the wave function with the lowest quantum number for an electron confined to a one-dimensional box of width 2.00 Å. However, atoms are three-dimensional entities with a typical diameter of 1.00 Å = 10–10 m. It would seem then that the better approximation would be an electron trapped in a three-dimensional infinite potential well (a potential cube with sides of 1.00 Å).a) Derive an expression for the wave function and the corresponding energies of an electron in a three-dimensional rectangular infinite potential well.b) Calculate the lowest energy allowed for the electron in this case.Example 37.1 Electron in a Box...... Get solution
31. An electron is confined to a potential well shaped as shown in Figure 37.10. The width of the well is 1.0·10–9 m, and U1 = 2.0 eV. Is the n = 3 state bound in this well?Figure 37.10 Finite potential energy well... Get solution
32. If a proton of kinetic energy 18.0 MeV encounters a rectangular potential energy barrier of height 29.8 MeV and width 1.00·10–15 m, what is the probability that the proton will tunnel through the barrier? Get solution
33. Suppose the kinetic energy of the neutron described in Example 37.3 is increased by 15%. By what factor does the neutron’s probability of tunneling through the barrier increase?Example 37.3 Neutron Tunneling...... Get solution
34. A beam of electrons moving in the positive x-direction encounters a potential barrier that is 2.51 eV high and 1.00 nm wide. Each electron has a kinetic energy of 2.50 eV, and the electrons arrive at the barrier at a rate of 1000. electrons/s. What is the rate IT (in electrons/s) at which electrons pass through the barrier, on average? What is the rate IR (in electrons/s) at which electrons bounce back from the barrier, on average? Determine and compare the wavelengths of the electrons before and after they pass through the barrier.... Get solution
35. Consider an electron approaching a potential barrier 2.00 nm wide and 7.00 eV high. What is the energy of the electron if it has a 10.0% probability of tunneling through this barrier? Get solution
36. Consider an electron in a three-dimensional box—with infinite potential walls—of dimensions 1.00 nm × 2.00 nm × 3.00 nm. Find the quantum numbers nx, ny, and nz and the energies (in eV) of the six lowest energy levels. Are any of these levels degenerate, that is, do any distinct quantum states have identical energies? Get solution
37. In a scanning tunneling microscope, the probability that an electron from the probe will tunnel through a 0.100-nm gap is 0.100%. Calculate the work function of the probe. Get solution
38. Consider a square potential, U(x) = 0 for x α, U(x) = –U0 for –α ≤ x ≤ α, where U0 is a positive constant, and U(x) = 0 for x > α. For E > 0, the solutions of the time-independent Schrodinger equation in the three regions will be the following:For ..., where ... and R is the amplitude of a reflected wave.For..., and ....For ... where T is the amplitude of the transmitted wave.Match ψ(x) and dψ(x)/dx at –α and α and find an expression for R. What is the condition for which R = 0 (that is, there is no reflected wave)? Get solution
39. a) Determine the wave function and the energy levels for the bound states of an electron in the symmetrical one-dimensional potential well of finite depth shown in the figure.b) If the penetration distance η into the classically forbidden region is defined as the distance at which the wave function decreases to 1/e of its value at the edge of the well, determine an expression for this penetration distance....c) The electrons in a typical GaAs-GaAlAs quantum-well laser diode are confined within a one-dimensional potential well like the one shown in the figure, of width 1 nm and depth 0.300 eV. Numerical solutions to the Schrodinger equation show that there is only one possible bound state for the electrons in this case, with energy of 0.125 eV. Calculate the penetration distance for these electrons. Get solution
40. An oxygen molecule has a vibrational mode that behaves approximately like a simple harmonic oscillator with frequency 2.99·1014 rad/s. Calculate the energy of the ground state and the first two excited states. Get solution
42. An experimental measurement of the energy levels of a hydrogen molecule, H2, shows that they are evenly spaced and separated by about 9·10–20 J. A reasonable model of one of the hydrogen atoms would then seem to be that of a particle in a simple harmonic oscillator potential. Assuming that the hydrogen atom is attached by a spring with a spring constant κ to the other atom in the molecule, what is the spring constant k? Get solution
43. Calculate the ground-state energy for an electron confined to a cubic potential well with sides equal to twice the Bohr radius (R = 0.0529 nm). Determine the spring constant that would give this same ground-state energy for a harmonic oscillator. Get solution
44. A particle in a harmonic oscillator potential has the initial wave function Ψ(x, 0) = A [ψ0(x) + ψ1(x)]. Normalize Ψ(x, 0). Get solution
45. The ground-state wave function for a harmonic oscillator is given by ....a) Determine the normalization constant, A2.b) Determine the probability that a quantum harmonic oscillator in the n = 0 state will be found in the classically forbidden region. Get solution
46. A particle is in a square infinite potential well of width L and is in the n = 3 state. What is the probability that, when observed, the particle is found to be in the rightmost 10.0% of the well? Get solution
47. An electron is confined between x = 0 and x = L. The wave function of the electron is ψ(x) = Asin(2πx/L). The wave function is zero for the regions x x > L.a) Determine the normalization constant, A.b) What is the probability of finding the electron in the region 0 ≤ x ≤ L/3? Get solution
48. Find the probability of finding an electron trapped in a one-dimensional infinite well of width 2.00 nm in the n = 2 state between 0.800 and 0.900 nm (assume that the left edge of the well is at x = 0 and the right edge is at x = 2.00 nm). Get solution
49. An electron is trapped in a one-dimensional infinite potential well that is L = 300. pm wide. What is the probability that the electron in the first excited state will be detected in an interval between x = 0.500 L and x = 0.750 L? Get solution
50. Find the uncertainty of x for the wave function ... Get solution
51. Write a wave function ... for a nonrelativistic free particle of mass m moving in three dimensions with momentum ..., including the correct time dependence as required by the Schrodinger equation. What is the probability density associated with this wave? Get solution
52. Suppose a quantum particle is in a stationary state with a wave function ... (x,t). The calculation of x, the expectation value of the particle’s position, is shown in the text. Calculate ... Get solution
53. Although quantum systems are frequently characterized by their stationary states, a quantum particle is not required to be in such a state unless its energy has been measured. The actual state of the particle is determined by its initial conditions. Suppose a particle of mass m in a one-dimensional potential well with infinite walls (a box) of width a is actually in a state with the wave function....where ... denotes the stationary state with quantum number n = 1 and ... denotes the state with n = 2. Calculate the probability density distribution for the position x of the particle in this state. Get solution
54. In Chapter 40, you will see that a nuclear-fusion reaction between two protons (creating a deuteron, a positron, and a neutrino) releases 0.42 MeV of energy. Nuclear fusion is what causes the stars to shine, and if we can harness it, we can solve the world’s energy problems. Compare the energy released in this fusion reaction with what would be released by the annihilation of a proton and an antiproton. Get solution
55. Particle-antiparticle pairs are occasionally created out of empty space. Considering energy-time uncertainty, how long would such pairs be expected to exist at most if they consist ofa) an electron and a positron?b) a proton and an antiproton? Get solution
56. A positron and an electron annihilate, producing two 2.0-MeV gamma rays moving in opposite directions. Calculate the kinetic energy of the electron when the kinetic energy of the positron is twice that of the electron. Get solution
57. Calculate the energy of the first excited state of a proton in a one-dimensional infinite potential well of width α = 1.00 nm. Get solution
58. Electrons in a scanning tunneling microscope encounter a potential barrier that has a height of U = 4.0 eV above their total energy. By what factor does the tunneling current change if the tip moves a net distance of 0.10 nm farther from the surface? Get solution
59. An electron is confined in a three-dimensional cubic space of L3 with infinite potentials.a) Write down the normalized solution of the wave function for the ground state.b) How many energy states are available between the ground state and the second excited state? (Take the electron’s spin into account.) Get solution
60. A mass-and-spring harmonic oscillator used for classroom demonstrations has the angular frequency ω0 = 4.45 s–1. If this oscillator has a total (kinetic plus potential) energy E = 1.00 J, what is its corresponding quantum number, n? Get solution
61. The neutrons in a parallel beam, each having kinetic energy ... (which approximately corresponds to room temperature), are directed through two slits 0.50 mm apart. How far apart will the peaks of the interference pattern be on a screen 1.5 m away? Get solution
62. Find the ground-state energy (in eV) of an electron in a one-dimensional box, if the box is of length L = 0.100 nm. Get solution
63. An approximately one-dimensional potential well can be formed by surrounding a layer of GaAs with layers of AlxGa1–xAs. The GaAs layers can be fabricated in thicknesses that are integral multiples of the single-layer thickness, 0.28 nm. Some electrons in the GaAs layer behave as if they were trapped in a box. For simplicity, treat the box as a one-dimensional infinite potential well and ignore the interactions between the electrons and the Ga and As atoms (such interactions are often accounted for by replacing the actual electron mass with an effective electron mass). Calculate the energy of the ground state in this well for these cases:a) 2 GaAs layersb) 5 GaAs layers Get solution
64. Consider a water molecule in the vapor state in a room 4.00 m × 10.0 m × 10.0 m.a) What is the ground-state energy of this molecule, treating it as a simple particle in a box?b) Compare this energy to the average thermal energy of such a molecule, taking the temperature to be 300. K.c) What can you conclude from the two numbers you just calculated? Get solution
65. A neutron moves between rigid walls 8.4 fm apart. What is the energy of its n = 1 state? Get solution
66. A surface is examined using a scanning tunneling microscope (STM). For the range of the gap, L, between the tip of the probe and the sample surface, assume that the electron wave function for the atoms under investigation falls off exponentially as .... The tunneling current through the tip of the probe is proportional to the tunneling probability. In this situation, what is the ratio of the current when the tip is 0.400 nm above a surface feature to the current when the tip is 0.420 nm above the surface? Get solution
67. An electron is trapped in a one-dimensional infinite well of width 2.00 nm. It starts in the n = 4 state, and then goes into the n = 2 state, emitting radiation with energy corresponding to the energy difference between the two states. What is the wavelength of the radiation? Get solution
68. Two long, straight wires that lie along the same line have a separation at their tips of 2.00 nm. The potential energy of an electron in this gap is about 1.00 eV higher than it is in the conduction band of the two wires. Conduction-band electrons have enough energy to contribute to the current flowing in the wire. What is the probability that a conduction electron from one wire will be found in the other wire after arriving at the gap? Get solution
69. Consider an electron that is confined to a one-dimensional infinite potential well of width a = 0.10 nm, and another electron that is confined to a three-dimensional (cubic) infinite potential well with sides of length a = 0.10 nm. Let the electron confined to the cube be in its ground state. Determine the difference in energy ground state of the two electrons and the excited state of the one-dimensional electron that minimizes the difference in energy with the three-dimensional electron. Get solution
70. An electron with an energy of 129 KeV is trapped in a potential well defined by an infinite potential at x U1 extending from x = 529.2 fm to x = 2116.8 fm (1 fm = 1 · 10–15 m), as shown in the figure. It is found that the electron can be detected beyond the barrier with a probability of 10%. Calculate the height of the potential barrier.... Get solution
71. Consider an electron that is confined to the xy-plane by a two-dimensional rectangular infinite potential well. The width of the well is w in the x-direction and 2w in the y-direction. What is the lowest energy that is shared by more than one distinct state, that is, two different states having the same energy? Get solution
73. A 6.31-MeV alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.7 MeV and whose width is 13.7 fm (1 fm = 1·10–15 m). What is the value of the decay constant, γ (in fm–1)? (Hint: A potentially useful value is ħc = 197.327 MeV fm.) Get solution
75. A 8.59-MeV alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.9 MeV. The tunneling probability is measured to be 1.042·10–18. What is the width of the barrier? (Hint: A potentially useful value is ħc = 197.327 MeV fm.) Get solution
74. An alpha particle (mass = 3.7274 GeV/c2) inside a heavy nucleus encounters a barrier whose average height is 15.7 MeV and whose width is 15.5 fm (1 fm = 1·10–15 m). The decay constant, γ, is measured to be 1.257 fm–1. What is the kinetic energy of the alpha particle? (Hint: A potentially useful value is ħc = 197.327 MeV fm.)... Get solution
76. An electron with a mass of 9.109·10–31 kg is trapped inside a one-dimensional infinite potential well of width 13.5 nm. What is the energy difference between the n = 5 and the n = 1 states? Get solution
77. A proton with a mass of 1.673·10–27 kg is trapped inside a one-dimensional infinite potential well of width 23.9 nm. What is the quantum number, n, of the state that has an energy difference of 1.08 ·10–3 meV with the n = 2 state? Get solution
78. A particle is trapped inside a one-dimensional infinite potential well of width 19.3 nm. The energy difference between the n = 2 and the n = 1 states is 2.639·10–25 J. What is the mass of the particle? Get solution
