Chapter #8 Solutions - Introduction to Quantum Mechanics - David J. Griffiths - 2nd Edition

1. Use the WKB approximation to find the allowed energies (En) of an infinite square well with a “shelf,” of height V0 extending half-way across (Figure 6.3):...Express your answer in terms of V0 and ... (the nthallowedenergy for the infinite square well with no shelf). Assume that ..., but do not assume that En >> V0. Compare your result with what we got in Example 6.1 using first-order perturbation theory. Note that they are in agreement if either V0 is very small (the perturbation theory regime) or n is very large (the WKB—semi-classical—regime). Get solution

2. An illuminating alternative derivation of the WKB formula (Equation 1) is based on an expansion in powers of ћ. Motivated by the free-particle wave function, ψ= A exp(± ipx/ћ), we write...where f(x) is some complex function. (Note that there is no loss of generality here—any nonzero function can be written in this way.)(a) Put this into Schrödinger's equation (in the form of Equation 8.1), and shov that...(b) Write f(x) as a power series in ћ:...and, collecting like powers of ћ, show that... etc.(c) Solve for f0(x) and f1(x), and show that—to first order in ћ—you recover Equation 1.Note: The logarithm of a negative number is defined by ln(‒z) = ln(z) + inπ, where n is an odd integer. If this formula is new to you, try exponentiating both sides, and you'll see where it comes from.Equation 1... Get solution

3. . Potential well with one vertical wall. Imagine a potential well that has one vertical side (at x = 0) and one sloping side (Figure 8.10). In this case ..., so Equation 8.46 says ... ... Get solution

4. Calculate the lifetimes of U238 and Po212, using Equations 1 and 8.25. Hint: The density of nuclear matter is relatively constant (i.e., the same for all nuclei), so (r1)3 is proportional to A (the number of neutrons plus protons). Empirically,...[8.29]...FIGURE 8.6: Graph of the logarithm of the lifetime versus 1/√E (where E is the energy of the emitted alpha particle), for uranium and thorium.The energy of the emitted alpha particle can be deduced by using Einstein's formula (E = mc2):... [8.30]where mp is the mass of the parent nucleus, md is the mass of the daughter nucleus, and mα is the mass of the alpha particle (which is to say, the He4 nucleus). To figure out what the daughter nucleus is, note that the alpha particle carries off two protons and two neutrons, so Z decreases by 2 and A by 4. Look up the relevant nuclear masses. To estimate v, use E = (1/2)mav2; this ignores the (negative) potential energy inside the nucleus, and surely underestimates v, but it's about the best we can do at this stage. Incidentally, the experimental lifetimes are 6 x 109 yrs and 0.5 μs, respectively.Equations 1... Get solution

5. Consider the quantum mechanical analog to the classical problem of a ball (mass m) bouncing elastically on the floor.13(a) What is the potential energy, as a function of height x above the floor? (For negative x, the potential is infinite—the ball can't get there at all.)(b) Solve the Schrödinger equation for this potential, expressing your answer in terms of the appropriate Airy function (note that Bi(z) blows up for large z, and must therefore be rejected). Don’t bother to normalize ψ(x).(c) Using g = 9.80 m/s2 and m = 0.100 kg, find the first four allowed energies, in joules, correct to three significant digits. Hint: See Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover, New York (1970), page 478; the notation is defined on page 450.(d) What is the ground state energy, in eV, of an electron in this gravitational field? How high off the ground is this electron, on the average? Hint: Use the virial theorem to determine ...x.... Get solution

6. Analyze the bouncing ball (Problem 1) using the WKB approximation.(a) Find the allowed energies, En, in terms of m, g, and h.(b) Now put in the particular values given in Problem 8.5(c), and compare the WKB approximation to the first four energies with the “exact” results.(c) About how large would the quantum number n have to be to give the ball an average height of, say, 1 meter above the ground?Problem 1Consider the quantum mechanical analog to the classical problem of a ball (mass m) bouncing elastically on the floor.13(a) What is the potential energy, as a function of height x above the floor? (For negative x, the potential is infinite—the ball can't get there at all.)(b) Solve the Schrödinger equation for this potential, expressing your answer in terms of the appropriate Airy function (note that Bi(z) blows up for large z, and must therefore be rejected). Don’t bother to normalize ψ(x).(c) Using g = 9.80 m/s2 and m = 0.100 kg, find the first four allowed energies, in joules, correct to three significant digits. Hint: See Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover, New York (1970), page 478; the notation is defined on page 450.(d) What is the ground state energy, in eV, of an electron in this gravitational field? How high off the ground is this electron, on the average? Hint: Use the virial theorem to determine ...x.... Get solution

7. Use the WKB approximation to find the allowed energies of the harmonic oscillator. Get solution

8. Consider a particle of mass m in the nth stationary state of the harmonic oscillator (angular frequency-ω) (a) Find the turning point, x2. (b) How far (d) could you go above the turning point before the error in the linearized potential (Equation 8.32, but with the turning point at x2) reaches 1%? That is, if ... (c) The. asymptotic form of Ai (z) is accurate to 1% as long as z > 5. For the d in part (b), determine the smallest it such that ad > 5. (For any it larger than this there exists an overlap region in which the linearized potential is good to 1% and the large-z form of the Airy function is good to 1%.) ... Get solution

9. Derive the connection formulas at a downward-sloping turning point, and confirm Equation 1.Equation 1... Get solution

10. Use appropriate connection formulas to analyze the problem of scattering from a barrier with sloping walls (Figure 8.12). Hint: Begin by writing the WKB wave function in the form ... Do not assume C = 0. Calculate the tunneling probability, ..., and show that your result reduces to Equation 8.22 in the case of a broad, high bather. ... ... Get solution

11. Use the WKB approximation to find the allowed energies of the general power-law potential:...where v is a positive number. Check your result for the case v = 2. Answer:14... [8.53] Get solution

12. Use the WKB approximation to find the bound state energy for the potential in Problem 2.51. Compare the exact answer. Answer: ‒[(9/8) ‒ (1/√2)] ћ2a2/m. Get solution

13. For spherically symmetrical potentials we can apply the WKB approximation to the radial part (Equation 1). In the case l = 0 it is reasonable15 to use Equation 8.47 in the form... [8.54]where r0 is the turning point (in effect, we treat r = 0 as an infinite wall) Exploit this formula to estimate the allowed energies of a particle in the logarithmic potential...(for constants V0 and a). Treat only the case l = 0. Show that the spacing between the levels is independent of mass. Partial answer:...Equation 1... Get solution

14. Use the WKB approximation in the form... [8.55]to estimate the bound state energies for hydrogen. Don’t forget the centrifugal term in the effective potential (Equation 1). The following integral may help:... [8.56]Note that you recover the Bohr levels when n >> l and n >> 1/2. Answer:... [8.57]Equation 1... Get solution

15. Consider the case of a symmetrical double well, such as the one pictured in Figure 8.13. We are interested in bound states with E (a) Write down the WKB wave functions in regions ..., and .... Impose the appropriate connection formulas at xi and x2 (this has already been done, in Equation 8.46, for x2; you will have to work out x1 for yourself), to show that ... ... Get solution

16. Tunneling in the Stark effect. When you turn on an external electric field, the electron in an atom can, in principle, tunnel out, ionizing the atom. Question: Is this likely to happen in a typical Stark effect experiment? We can estimate the probability using a crude one-dimensional model, as follows. Imagine a particle in a very deep finite square well (Section 2.6). (a) What is the energy of the ground state, measured up from the bottom of the well? Assume .... Hint: This is just the ground state energy of the infinite square well (of width 2a). (b) Now introduce a perturbation H' = -aα (for an electron in an electric field ...we would have .... Assume it is relatively weak .... Sketch the total potential, and note that the particle can now tunnel out, in the direction of poTitiecr (c) Calculate the tunnelisfactor y (Equation 8.22), and estimate the time it would take itirthe particle to escape (Equation 8.28). Answer: γ = .... ... Get solution

17. About how long would it take for a can of beer at room temperature to topple over spontaneously, as a result of quantum tunneling? Hint: Treat it as a uniform cylinder of mass in, radius R, and length h. As the can tips, let x be the height of the center above its equilibrium position (h/2). The potential enemy is mgx, and it topples when x reaches the critical value ... Calculate the tunneling probability (Equation 8.22), for E = 0. Use Equation 8.28, with the thermal energy ... to estimate the velocity. Put in reasonable numbers, and give your final answer in years.17 ... Get solution