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Chapter 41 Problems

1, 2, 3 = straightforward, intermediate, challenging

Section 41.1 An Interpretation of Quantum Mechanics

1. A free electron has a wave function

[pic]

where x is in meters. Find (a) its de Broglie wavelength, (b) its momentum, and (c) its kinetic energy in electron volts.

2. The wave function for a particle is

[pic]

for a > 0 and –∞ < x < +∞. Determine the probability that the particle is located somewhere between x = –a and x = +a.

Section 41.2 A Particle in a Box

3. An electron is confined to a one-dimensional region in which its ground-state (n = 1) energy is 2.00 eV. (a) What is the length of the region? (b) How much energy is required to promote the electron to its first excited state?

4. An electron that has an energy of approximately 6 eV moves between rigid walls 1.00 nm apart. Find (a) the quantum number n for the energy state that the electron occupies and (b) the precise energy of the electron.

5. An electron is contained in a one-dimensional box of length 0.100 nm. (a) Draw an energy-level diagram for the electron for levels up to n = 4. (b) Find the wavelengths of all photons that can be emitted by the electron in making downward transitions that could eventually carry it from the n = 4 state to the n = 1 state.

6. An alpha particle in a nucleus can be modeled as a particle moving in a box of length 1.00 × 10–14 m (the approximate diameter of a nucleus). Using this model, estimate the energy and momentum of an alpha particle in its lowest energy state. (mα = 4 × 1.66 × 10–27 kg)

7. A ruby laser emits 694.3-nm light. Assume light of this wavelength is due to a transition of an electron in a box from its n = 2 state to its n = 1 state. Find the length of the box.

8. A laser emits light of wavelength λ. Assume this light is due to a transition of an electron in a box from its n = 2 state to its n = 1 state. Find the length of the box.

9. The nuclear potential energy that binds protons and neutrons in a nucleus is often approximated by a square well. Imagine a proton confined in an infinitely high square well of length 10.0 fm, a typical nuclear diameter. Calculate the wavelength and energy associated with the photon emitted when the proton moves from the n = 2 state to the ground state. In what region of the electromagnetic spectrum does this wavelength belong?

10. A proton is confined to move in a one-dimensional box of length 0.200 nm. (a) Find the lowest possible energy of the proton. (b) What If? What is the lowest possible energy of an electron confined to the same box? (c) How do you account for the great difference in your results for (a) and (b)?

11. Use the particle-in-a-box model to calculate the first three energy levels of a neutron trapped in a nucleus of diameter 20.0 fm. Do the energy-level differences have a realistic order of magnitude?

12. A photon with wavelength λ is absorbed by an electron confined to a box. As a result, the electron moves from state n = 1 to n = 4. (a) Find the length of the box. (b) What is the wavelength of the photon emitted in the transition of that electron from the state n = 4 to the state n = 2?

Section 41.3 The Particle Under Boundary Conditions

Section 41.4 The Schrödinger Equation

13. Show that the wave function ψ = Aei(kx – ωt) is a solution to the Schrödinger equation (Eq. 41.13) where k = 2π/λ and U = 0.

14. The wave function of a particle is given by

[pic]

where A, B, and k are constants. Show that ψ is a solution of the Schrödinger equation (Eq. 41.13), assuming the particle is free (U = 0), and find the corresponding energy E of the particle.

15. Prove that the first term in the Schrödinger equation, –(ћ2/2m)(d2ψ/dx2), reduces to the kinetic energy of the particle multiplied by the wave function (a) for a freely moving particle, with the wave function given by Equation 41.3 and (b) for a particle in a box, with the wave function given by Equation 41.17.

16. A particle in an infinitely deep square well has a wave function given by

[pic]

for 0 ≤ x ≤ L and zero otherwise. (a) Determine the expectation value of x. (b) Determine the probability of finding the particle near L/2, by calculating the probability that the particle lies in the range 0.490L ≤ x ≤ 0.510L. (c) What If? Determine the probability of finding the particle near L/4, by calculating the probability that the particle lies in the range 0.240L ≤ x ≤ 0.260L. (d) Argue that the result of part (a) does not contradict the results of parts (b) and (c).

17. The wave function for a particle confined to moving in a one-dimensional box is

[pic]

Use the normalization condition on ψ to show that

[pic]

Suggestion: Because the box length is L, the wave function is zero for x < 0 and for x > L, so the normalization condition (Eq. 41.6) reduces to

[pic]

18. The wave function of an electron is

[pic]

Calculate the probability of finding the electron between x = 0 and x = L/4.

19. An electron in an infinitely deep square well has a wave function that is given by

[pic]

for 0 ≤ x ≤ L and is zero otherwise. What are the most probable positions of the electron?

20. A particle is in the n = 1 state of an infinite square well with walls at x = 0 and x = L. Let ℓ be an arbitrary value of x between x = 0 and x = L. (a) Find an expression for the probability, as a function of ℓ, that the particle will be found between x = 0 and x = ℓ. (b) Sketch the probability as a function of ℓ/L. Choose values of ℓ/L ranging from 0 to 1.00 in steps of 0.100. (c) Find the value of ℓ for which the probability of finding the particle between x = 0 and x = ℓ is twice the probability of finding the particle between x = ℓ and x = L. You can solve the transcendental equation for ℓ/L numerically.

21. A particle in an infinite square well has a wave function that is given by

[pic]

for 0 ≤ x ≤ L and is zero otherwise. (a) Determine the probability of finding the particle between x = 0 and x = L/3. (b) Use the result of this calculation and symmetry arguments to find the probability of finding the particle between x = L/3 and x = 2L/3. Do not re-evaluate the integral. (c) What If? Compare the result of part (a) with the classical probability.

22. Consider a particle moving in a one-dimensional box for which the walls are at x = –L/2 and x = L/2. (a) Write the wave functions and probability densities for n = 1, n = 2, and n = 3. (b) Sketch the wave functions and probability densities. (Suggestion: Make an analogy to the case of a particle in a box for which the walls are at x = 0 and x = L.)

23. A particle of mass m moves in a potential well of length 2L. Its potential energy is infinite for x < –L and for x > +L. Inside the region –L < x < L, its potential energy is given by

[pic]

In addition, the particle is in a stationary state that is described by the wave function ψ(x) = A(1 – x2/L2) for –L < x < +L, and by ψ(x) = 0 elsewhere. (a) Determine the energy of the particle in terms of ħ, m, and L. (Suggestion: Use the Schrödinger equation, Eq. 41.13.) (b) Show that A = (15/16L)1/2. (c) Determine the probability that the particle is located between x = –L/3 and x = +L/3.

24. In a region of space, a particle with zero total energy has a wave function (a) Find the potential energy U as a function of x. (b) Make a sketch of U(x) versus x.

[pic]

Section 41.5 A Particle in a Well of Finite Height

25. Suppose a particle is trapped in its ground state in a box that has infinitely high walls (Fig. 41.4). Now suppose the left-hand wall is suddenly lowered to a finite height and width. (a) Qualitatively sketch the wave function for the particle a short time later. (b) If the box has a length L, what is the wavelength of the wave that penetrates the left-hand wall?

26. Sketch the wave function ψ(x) and the probability density [pic] for the n = 4 state of a particle in a finite potential well. (See Fig. 41.8.)

Section 41.6 Tunneling Through a Potential Energy Barrier

27. An electron with kinetic energy E = 5.00 eV is incident on a barrier with thickness L = 0.200 nm and height U = 10.0 eV (Fig. P41.27). What is the probability that the electron (a) will tunnel through the barrier? (b) will be reflected?

[pic]

Figure P41.27

28. An electron having total energy E = 4.50 eV approaches a rectangular energy barrier with U = 5.00 eV and L = 950 pm as shown in Figure P41.27. Classically, the electron cannot pass through the barrier because E < U. However, quantum-mechanically the probability of tunneling is not zero. Calculate this probability, which is the transmission coefficient.

29. What If? In Problem 28, by how much would the width L of the potential barrier have to be increased for the chance of an incident 4.50-eV electron tunneling through the barrier to be one in a million?

30. An electron has a kinetic energy of 12.0 eV. The electron is incident upon a rectangular barrier of height 20.0 eV and thickness 1.00 nm. By what factor would the electron’s probability of tunneling through the barrier increase assuming that the electron absorbs all the energy of a photon with wavelength 546 nm (green light)?

Section 41.7 The Scanning Tunneling Microscope

31. A scanning tunneling microscope (STM) can precisely determine the depths of surface features because the current through its tip is very sensitive to differences in the width of the gap between the tip and the sample surface. Assume that in this direction the electron wave function falls off exponentially with a decay length of 0.100 nm; that is, with C = 10.0/nm. Determine the ratio of the current when the STM tip is 0.500 nm above a surface feature to the current when the tip is 0.515 nm above the surface.

32. The design criterion for a typical scanning tunneling microscope specifies that it must be able to detect, on the sample below its tip, surface features that differ in height by only 0.002 00 nm. What percentage change in electron transmission must the electronics of the STM be able to detect, to achieve this resolution? Assume that the electron transmission coefficient is e–2CL with C = 10.0/nm.

Section 41.8 The Simple Harmonic Oscillator

33. Show that Equation 41.24 is a solution of Equation 41.22 with energy E = ½ ħω.

34. A one-dimensional harmonic oscillator wave function is

[pic]

(a) Show that ψ satisfies Equation 41.22. (b) Find b and the total energy E. (c) Is this a ground state or a first excited state?

35. A quantum simple harmonic oscillator consists of an electron bound by a restoring force proportional to its position relative to a certain equilibrium point. The proportionality constant is 8.99 N/m. What is the longest wavelength of light that can excite the oscillator?

36. (a) Normalize the wave function for the ground state of a simple harmonic oscillator. That is, apply Equation 41.6 to Equation 41.24 and find the required value for the constant B, in terms of m, ω, and fundamental constants. (b) Determine the probability of finding the oscillator in a narrow interval –δ/2 < x < δ/2 around its equilibrium position.

37. Two particles with masses m1 and m2 are joined by a light spring of force constant k. They vibrate along a straight line with their center of mass fixed. (a) Show that the total energy

[pic]

can be written as ½ μv2 + ½ kx2 where [pic] is the relative speed of the particles and μ = m1m2/(m1 + m2) is the reduced mass of the system. This result demonstrates that the pair of freely vibrating particles can be precisely modeled as a single particle vibrating on one end of a spring that has its other end fixed. (b) Differentiate the equation

½ μv2 + ½ kx2 = constant

with respect to x. Proceed to show that the system executes simple harmonic motion. Find its frequency.

38. The total energy of a particle–spring system in which the particle moves with simple harmonic motion along the x axis is

[pic]

where px is the momentum of the particle and k is the spring constant. (a) Using the uncertainty principle, show that this expression can also be written

[pic]

(b) Show that the minimum energy of the harmonic oscillator is

[pic]

39. Keeping a constant speed of 0.8 m/s, a marble rolls back and forth across a shoebox. Make an order-of-magnitude estimate of the probability of its escaping through the wall of the box by quantum tunneling. State the quantities you take as data and the values you measure or estimate for them.

40. A particle of mass 2.00 × 10–28 kg is confined to a onedimensional box of length 1.00 × 10–10 m. For n = 1, what are (a) the particle’s wavelength, (b) its momentum, and (c) its zero-point energy?

41. An electron is represented by the time-independent wave function

[pic]

(a) Sketch the wave function as a function of x. (b) Sketch the probability density representing the likelihood that the electron is found between x and x + dx. (c) Argue that this can be a physically reasonable wave function. (d) Normalize the wave function. (e) Determine the probability of finding the electron somewhere in the range

[pic]

42. Particles incident from the left are confronted with a step in potential energy shown in Figure P41.42. Located at x = 0, the step has a height U. The particles have energy E > U. Classically, we would expect all of the particles to continue on, although with reduced speed. According to quantum mechanics, a fraction of the particles are reflected at the barrier. (a) Prove that the reflection coefficient R for this case is

[pic]

where k1 = 2π/λ1 and k2 = 2π/λ2 are the wave numbers for the incident and transmitted particles. Proceed as follows. Show that the wave function ψ1 = Ae i k1 x + Be– i k1 x satisfies the Schrödinger equation in region 1, for x < 0. Here Ae i k1 x represents the incident beam and Be– i k1 x represents the reflected particles. Show that ψ2 = Ce i k2 x satisfies the Schrödinger equation in region 2, for x > 0. Impose the boundary conditions ψ1 = ψ2 and dψ1 / dx = dψ2 / dx at x = 0, to find the relationship between B and A. Then evaluate R = B2/A2. (b) A particle that has kinetic energy E = 7.00 eV is incident from a region where the potential energy is zero onto one in which U = 5.00 eV. Find its probability of being reflected and its probability of being transmitted.

[pic]

Figure P41.42

43. Particles incident from the left are confronted with a step in potential energy shown in Figure P41.42. The step has a height U, and the particles have energy E = 2U. Classically, all the particles would pass into the region of higher potential energy at the right. However, according to quantum mechanics, a fraction of the particles are reflected at the barrier. Use the result of Problem 42 to determine the fraction of the incident particles that are reflected. (This situation is analogous to the partial reflection and transmission of light striking an interface between two different media.)

44. An electron is trapped in a quantum dot. The quantum dot may be modeled as a one-dimensional, rigid-walled box of length 1.00 nm. (a) Sketch the wave functions and probability densities for the n = 1 and n = 2 states. (b) For the n = 1 state, calculate the probability of finding the electron between x1 = 0.150 nm and x2 = 0.350 nm, where x = 0 is the left side of the box. (c) Repeat part (b) for the n = 2 state. (d) Calculate the energies in electron volts of the n = 1 and n = 2 states. Suggestion: For parts (b) and (c), use Equation 41.5 and note that

[pic]

45. An atom in an excited state 1.80 eV above the ground state remains in that excited state 2.00 μs before moving to the ground state. Find (a) the frequency and (b) the wavelength of the emitted photon. (c) Find the approximate uncertainty in energy of the photon.

46. An electron is confined to move in the xy plane in a rectangle whose dimensions are Lx and Ly. That is, the electron is trapped in a two-dimensional potential well having lengths of Lx and Ly. In this situation, the allowed energies of the electron depend on two quantum numbers nx and ny. The allowed energies are given by

[pic]

(a) Assuming Lx = Ly = L, find the energies of the lowest four energy levels for the electron. (b) Construct an energy-level diagram for the electron, and determine the energy difference between the second excited state and the ground state.

47. For a particle described by a wave function ψ(x), the expectation value of a physical quantity f(x) associated with the particle is defined by

[pic]

For a particle in a one-dimensional box extending from x = 0 to x = L, show that

[pic]

48. A particle is described by the wave function

[pic]

(a) Determine the normalization constant A. (b) What is the probability that the particle will be found between x = 0 and x = L/8 if its position is measured? (Suggestion: Use Eq. 41.5.)

49. A particle has a wave function

[pic]

(a) Find and sketch the probability density. (b) Find the probability that the particle will be at any point where x < 0. (c) Show that ψ is normalized, and then find the probability that the particle will be found between x = 0 and x = a.

50. A particle of mass m is placed in a one-dimensional box of length L. What If? Assume the box is so small that the particle’s motion is relativistic, so that K = p2/2m is not valid. (a) Derive an expression for the kinetic energy levels of the particle. (b) Assume the particle is an electron in a box of length L = 1.00 × 10–12 m. Find its lowest possible kinetic energy. By what percent is the nonrelativistic equation in error? (Suggestion: See Eq. 39.23.)

51. Consider a “crystal” consisting of two nuclei and two electrons as shown in Figure P41.51. (a) Taking into account all the pairs of interactions, find the potential energy of the system as a function of d. (b) Assuming the electrons to be restricted to a one-dimensional box of length 3d, find the minimum kinetic energy of the two electrons. (c) Find the value of d for which the total energy is a minimum. (d) Compare this value of d with the spacing of atoms in lithium, which has a density of 0.530 g/cm3 and an atomic mass of 7 u. (This type of calculation can be used to estimate the density of crystals and certain stars.)

[pic]

Figure P41.51

52. The simple harmonic oscillator excited. The wave function

[pic]

is a solution to the simple harmonic oscillator problem. (a) Find the energy of this state. (b) At what position are you least likely to find the particle? (c) At what positions are you most likely to find the particle? (d) Determine the value of B required to normalize the wave function. (e) What If? Determine the classical probability of finding the particle in an interval of small width δ centered at the position x = 2(ħ/mω)1/2. (f) What is the actual probability of finding the particle in this interval?

53. Normalization of wave functions. (a) Find the normalization constant A for a wave function made up of the two lowest states of a particle in a box:

[pic]

(b) A particle is described in the space –a ≤ x ≤ a by the wave function

[pic]

Determine the relationship between the values of A and B required for normalization. (Suggestion: Use the identity sin 2θ = 2 sin θ cos θ.)

54. The normalized wave functions for the ground state, ψ0(x), and the first excited state, ψ1(x), of a quantum harmonic oscillator are

[pic] [pic]

where a = mω/ħ. A mixed state, ψ01(x), is constructed from these states:

[pic]

The symbol s denotes the expectation value of the quantity q for the state ψs(x). Calculate the following expectation values: (a) 0 (b) 1 (c) 01.

55. A two-slit electron diffraction experiment is done with slits of unequal widths. When only slit 1 is open, the number of electrons reaching the screen per second is 25.0 times the number of electrons reaching the screen per second when only slit 2 is open. When both slits are open, an interference pattern results in which the destructive interference is not complete. Find the ratio of the probability of an electron arriving at an interference maximum to the probability of an electron arriving at an adjacent interference minimum. (Suggestion: Use the superposition principle.)