Chapter #7 Solutions - Modern Physics for Scientists and Engineers - Andrew Rex, Stephen T. Thornton - 4th Edition

1. Assume that the electron in the hydrogen atom is constrained to move only in a circle of radius a in the xy plane. Show that the separated Schrodinger equation for Ø becomes ... where Ø is the angle describing the position on the circle. Explain why this is similar to the Bohr assumption. Get solution

1q. Do the radial wave functions depend on m/? Explain your answers. Get solution

2. Solve the equation in the previous problem for ψ. Find the allowed energies and angular momenta. Compare your results with the Bohr theory. Get solution

2q. Would the radial wave functions be different for a potential V(r) other than the Coulomb potential? Explain. Get solution

3. After separating the Schrodinger equation using ψ = R(r)f(θ)g(Ø), the equation for Ø is ... where k = constant. Solve for g(Ø) in this equation and apply the appropriate boundary conditions. Show that k must be 0 or a positive or negative integer (k = m/, the magnetic quantum number). Get solution

3q. For what energy levels in the hydrogen atom will we not find / = 2 states? Get solution

4. Using the transformation equations between Cartesian coordinates and spherical polar coordinates given in Figure 7.1, transform the Schrodinger Equation (7.2) from Cartesian to spherical coordinates as given in Equation (7.3). ... ... ... Get solution

4q. What are the differences and similarities between atomic levels, atomic states, and atomic spectral lines? When do spectral lines occur? Get solution

5. Show that the radial wave function R20 for n = 2 and / = 0 satisfies Equation (7.13). What energy E results? Is this consistent with the Bohr model? ... Get solution

5q. What are the differences and similarities between the quantization of angular momentum in the Bohr model and the Schrodinger theory? Get solution

6. Show that the radial wave function R21 for n = 2 and / = 1 satisfies Equation (7.10). What energy results? Is this consistent with the Bohr model? ... Get solution

6q. Can the magnetic moment of an atom line up exactly with an external magnetic field? Explain. Get solution

7. Show that the radial wave function R21 for n = 2 and / = 1 is normalized. Get solution

7q. What are the possible magnetic quantum numbers for an f state? Get solution

8q. List all the reasons why a fourth quantum number (intrinsic spin) might have helped explain the complex optical spectra in the early 1920s. Get solution

9. List all the possible quantum numbers (n,/,m/ ) for the n = 5 level in atomic hydrogen. Get solution

9q. Is it possible for the z component of the orbital magnetic moment to be zero, but not the orbital angular momentum? Explain. Get solution

10. For a 3p state give the possible values of n,/,m/, L, Lz, Lx, and Ly. Get solution

10q. A close examination of the spectral lines coming from starlight can be used to determine the star’s magnetic fi eld. Explain how this is possible. Get solution

11. List all the wave functions for the 3p level of hydrogen. Identify the wave functions by their quantum numbers. Use the solutions in Tables 7.1 and 7.2. ... Get solution

11q. If a hydrogen atom in the 2p excited state decays to the 1s ground state, explain how the following properties are conserved: energy, linear momentum, and angular momentum. Get solution

12. Prove that ...by performing the summation for Equation (7.24). ... Get solution

13. What is the degeneracy of the n = 6 shell of atomic hydrogen considering (n,/,m/ ) and no magnetic field? Get solution

14. Draw for a 3d state all the possible orientations of the angular momentum vector ... . What is Lx 2 + Ly 2 for the m/ = -1 component? Get solution

15. What is the smallest value that ... may have if ... is within 10° of the z axis? Get solution

16. Prove that the degeneracy of an atomic hydrogen state having principal quantum number n is n2. (Ignore the spin quantum number.) Get solution

17. Write out the hydrogen wave functions ...for ...values of (2, 1, -1), (2, 1, 0), and (3, 2, -1). Get solution

18. Show that the hydrogen wave functions ψ200 and ψ21-1 are normalized. If the integrals required are not in Appendix 3, consult a table of integrals or use computer integration. Get solution

19. Calculate the possible z components of the orbital angular momentum for an electron in a 3p state. Get solution

20. For hydrogen atoms in a 4d state, what is the maximum difference in potential energy between atoms when placed in a magnetic field of 3.5 T? Ignore intrinsic spin. Get solution

21. Show that the wavelength difference between adjacent transitions in the normal Zeeman effect is given approximately by ... Get solution

22. For hydrogen atoms in a d state, sketch the orbital angular momentum with respect to the z axis. Use units of ...along the z axis and calculate the allowed angles of L with respect to the z axis. Get solution

23. For a hydrogen atom in the 6f state, what is the minimum angle between the orbital angular momentum vector and the z axis? Get solution

24. The red line of the Balmer series in hydrogen (λ = 656.5 nm) is observed to split into three spectral lines with Δ λ = 0.040 nm between two adjacent lines when placed in a magnetic field B. What is the value of B if Δ λ is due to the energy splitting between two adjacent m/ states? Get solution

25. A hydrogen atom in an excited 5f state is in a magnetic field of 3.00 T. How many energy states can the electron have in the 5f subshell? (Ignore the magnetic spin effects.) What is the energy of the 5f state in the absence of a magnetic field? What will be the energy of each state in the magnetic field? Get solution

26. The magnetic field in a Stern-Gerlach experiment varies along the vertical direction as dBz/dz = 20.0 T/cm. The horizontal length of the magnet is 7.10 cm, and the speed of the silver atoms averages 925 m/s. The average mass of the silver atoms is 1.81 × 10-25 kg. Show that the z component of its magnetic moment is 1 Bohr magneton. What is the separation of the two silver atom beams as they leave the magnet? Get solution

27. An experimenter wants to separate silver atoms in a Stern-Gerlach experiment by at least 1 cm (a large separation) as they exit the magnetic field. To heat the silver she has an oven that can reach 1000°C and needs to order a suitable magnet. What should be the magnet specifications (magnet length and magnetic field gradient)? Get solution

28. In an external magnetic field, can the electron spin vector ... point in the direction of ...? Draw a diagram with ...showing ...and Sz. Get solution

29. Use all four quantum numbers (n,/,m/,ms ) to write down all possible sets of quantum numbers for the 4f state of atomic hydrogen. What is the total degeneracy? Get solution

30. Use all four quantum numbers (n,/,m/,ms ) to write down all possible sets of quantum numbers for the 5d state of atomic hydrogen. What is the total degeneracy? Get solution

31. The 21-cm line transition for atomic hydrogen results from a spin-flip transition for the electron in the parallel state of the n = 1 state. What temperature in interstellar space gives a hydrogen atom enough energy (5.9 × 10-6 eV) to excite another hydrogen atom in a collision? Get solution

32. Prove that the total degeneracy for an atomic hydrogen state having principal quantum number n is 2n2. Get solution

33. Show that for transitions between any two n states of atomic hydrogen, no more than three different spectral lines can be obtained for the normal Zeeman effect. Get solution

34. Find whether the following transitions are allowed, and if they are, find the energy involved and whether the photon is absorbed or emitted for the hydrogen atom: ... Get solution

36. Find the most probable radial position for the electron of the hydrogen atom in the 2s state. Compare this value with that found for the 2p state in Example 7.11. ... Get solution

37. Sketch the probability function as a function of r for the 2s state of hydrogen. At what radius is the position probability equal to zero? Get solution

38. Calculate the probability of an electron in the ground state of the hydrogen atom being inside the region of the proton (radius = 1.2 × 10-15 m). (Hint: Note that r a0.) Get solution

39. Calculate the probability that an electron in the ground state of the hydrogen atom can be found between 0.95a0 and 1.05a0. Get solution

40. Find the expectation value of the radial position for the electron of the hydrogen atom in the 2s and 2p states. Get solution

41. Calculate the probability of an electron in the 2s state of the hydrogen atom being inside the region of the proton (radius ... 1.2 × 10-15 m). Repeat for a 2p electron. (Hint: Note that r a0.) Get solution

42. Find the most probable radial position of an electron in the 3d state of the hydrogen atom. Get solution

43. What is the probability that an electron in the 3d state is located at a radius greater than a0? Get solution

44g. Assume the following (incorrect!) classical picture of the electron intrinsic spin. Take the electrical energy of the electron to be equal to its mass energy concentrated into a spherical shell of radius R: ... Calculate R (called the classical electron radius). Now let this spherical shell rotate and calculate the tangential speed v along the sphere’s equator in order to obtain the electron intrinsic spin. Use the equation ... where I = moment of inertia of a spherical shell = 2mR2/3. Is the value of v obtained in this manner consistent with the theory of relativity? Explain. Get solution

45g. As in the previous problem, we want to calculate the speed of the rotating electron. Now let’s assume that the diameter of the electron is equal to the Compton wavelength of an electron. Calculate v and comment on the result. Get solution

46g. Consider a hydrogen-like atom such as He+ or Li++that has a single electron outside a nucleus of charge +Ze. (a) Rewrite the Schrodinger equation with the new Coulomb potential. (b) What change does this new potential have on the separation of variables? (c) Will the radial wave functions be affected? Explain. (d) Will the spherical harmonics be affected? Explain. Get solution

47g. For the preceding problem find the wave function ψ100. Get solution

48g. Consider a hydrogen atom in the 3p state. (a) At what radius is the electron probability equal to zero? (b) At what radius will the electron probability be a maximum? (c) For m/ = 1, at what angles θ will the electron probability be equal to zero? What about for m/ = -1? Get solution

49g. Consider a “muonic atom,” which consists of a proton and a negative muon, symbol μ-. Compute the ground-state energy following the methods used for the hydrogen atom. Get solution

50g. The lifetime of the excited component of the n = 1 state (parallel spins) that produces the 21-cm line transition in stellar hydrogen is approximately 107 years. What is the energy line width of this state? Get solution

51g. One way to establish which transitions are forbidden is to compute the expectation value of the electron’s position vector r using wave functions for both the initial and final states in the transition. That is, compute ... where ... represents an integral over all space, and ψf and ψi are the final and initial states. If the value of the integral is zero, then the transition is forbidden. Use this procedure to show that a transition from one / = 0 state to another / = 0 state is forbidden. (Transitions considered this way are sometimes called electric dipole transitions, because the electric dipole moment ... is proportional to ....) (Hint: It is helpful to break the vector r into its Cartesian components x, y, and z.) Get solution

52g. Use the same method as in the preceding problem to show that a transition from a / = 2,m/ = 0 state to a / = 0 state is forbidden. Get solution

53g. Use the same method as in the two preceding problems to argue that a transition from a / = 1,m/ = 0 to a / = 0 state should be allowed. Get solution

54g. For the 3d state of hydrogen, at what radius is the electron probability a maximum? Compare your answer with the radius of the Bohr orbit for n = 3. Get solution

55g. (a) Find the average orbital radius for the electron in the 3p state of hydrogen. Compare your answer with the radius of the Bohr orbit for n = 3. (b) What is the probability that this electron is outside the radius given by the Bohr model? Get solution