Chapter 13 Answers to Selected Problems
Additional Problems
A 0.6 (a) \(-5560\, \hat \jmath \Xunits{N/C}\text{;}\) (b) \(0\text{;}\) (c) \(5560\, \hat \jmath \Xunits{N/C}\text{.}\)
A 0.7 (a) \(k\lambda / a\text{;}\) (b) \(-k\lambda / a\text{.}\)
A 0.8 \(\frac{-2qk}{\pi R^2}(\hat\imath + \hat\jmath)\text{.}\)
A 0.9 (a) \(50,000\Xunits{eV}\text{;}\) (b) \(8.0 \times 10^{-15}\Xunits{J}\text{;}\) (c) \(1.33 \times 10^8\Xunits{m/s}\text{.}\)
A 0.11 left side: \(1600\Xunits{V/m}\) to left; right side: \(300\Xunits{V/m}\) to right.
A 0.18 \(v_f = \sqrt{2IHBD/m}\text{.}\)
A 0.20.b \(4 \times 10^{-5}\Xunits{T}\text{.}\)
A 0.22 \(5.8 \times 10^{-3}\Xunits{T}\text{.}\)
A 0.23 \(0.013\Xunits{s}\text{.}\)
A 0.24 \(\pi r^2 B_0(6t-1)/R\text{.}\)
A 0.31 (a) \(\pm z\text{,}\) (b) \(\pm x\text{,}\) (c) \(\pm z\text{,}\) (d) \(\pm z\text{,}\) (e) \(\pm x\text{,}\) \(\pm y\text{.}\)
A 0.36 (a) \(20\Xunits{m}\text{,}\) \(10\Xunits{m}\text{,}\) \(6.67\Xunits{m}\text{;}\) (b) \(\pi /10,\ \pi /5,\ 3\pi /10\text{;}\) (c) \(17\Xunits{Hz}\text{,}\) \(34\Xunits{Hz}\text{,}\) \(51\Xunits{Hz}\text{.}\)
A 0.45 (a) \(717\Xunits{nm}\text{;}\) (b) \(55,800\text{.}\)
A 0.48 (a) \(\lambda = 1257\Xunits{nm}\text{,}\) \(T = 4.19\times 10^{-15}\Xunits{s}\text{,}\) \(v = 3.0\times 10^8\Xunits{m/s}\text{,}\) infrared, (b) \(B = 0.2\Xunits{mT}\text{;}\)
A 0.51 (b) \(2 \times 10^{-14}\Xunits{m}\text{;}\) (c) \(p = 3.3 \times 10^{-20}\Xunits{kg \cdot m/s}\text{,}\) \(K = .3.3 \times 10^{-13}\Xunits{J}\text{.}\)
A 0.55 \(1.3 \times 10^{-21} \Xunits{m/s}\text{.}\)
A 0.59 (a) \(\Delta t = 6.6 \times 10^{-18}\Xunits{s}\text{;}\) (b) \(v = 4.2 \times 10^6\Xunits{m/s}\text{;}\) (c) \(L = 2.8 \times 10^{-11}\Xunits{m}\text{;}\) (d) \(0.13\text{.}\)
A 0.72 \(5.4\Xunits{cm}\text{.}\)
A 0.73 \(19.4\Xunits{mT}\text{.}\)
A 0.80 \(1.41A\text{.}\)
A 0.82 (a) minimum; (b) \(9I_0\text{.}\)
A 0.84 \(y(x,t) = 2\cos{(\pi x - 2\pi t)}\text{.}\)
A 0.86 (b) \(2.95\text{;}\) (c) \(0.5\Xunits{rad}\text{.}\)
A 0.88 (a) \(\frac{1}{1600\pi}\Xunits{m\)^{-2}\(}\text{;}\) (b) \(1/4\text{;}\) (c) \(P_A = \frac{1}{1000\pi}\Xunits{m\)^{-2}\(}\text{,}\) \(P_B = \frac{1}{2000\pi}\Xunits{m\)^{-2}\(}\text{;}\) (d) \(13/20\text{.}\)
A 0.90 (a) \(1 + i\theta - \theta^2/2! -i\theta^3/3! + \theta^4/4! + i\theta^5/5! - \theta^6/6! + \dots\text{;}\) (c) \(1 - i\theta - \theta^2/2! + i\theta^3/3! + \theta^4/4! - i\theta^5/5! - \theta^6/6! + \dots\text{;}\) (g) \(0.88 + 0.48i\text{;}\) (h) \(0.88 -0.48i\text{;}\) (i) \(1.76\text{;}\) (j) \(0.96i\text{;}\) (k) \(\left(e^{0.3i} + e^{-0.3i}\right)/2\text{;}\) (l) \(\left(e^{0.3i} - e^{-0.3i}\right)/2i\text{;}\) (m) \(1\text{;}\) (n) \(e^{0.6i}\text{.}\)
A 0.91 (a) \(11E_1/5\text{;}\) (b) \(\sqrt{3/5}\ e^{-iE_1t/\hbar}|1\rangle + \sqrt{2/5}\ e^{-i4E_1t/\hbar}|2\rangle\text{;}\) (c) \(11E_1/5\text{.}\)
A 0.92 (a) \(|\Psi(0)\rangle = | + x\rangle = \sqrt{1/2}\left(| + z\rangle + | - z\rangle \right)\text{;}\) (b) \(E_+ = -\mu B_0\text{,}\)\(E_- = +\mu B_0\text{;}\) (c) \(|\Psi(t)\rangle = \sqrt{1/2}\left(e^{i\omega t/2}| + z\rangle + e^{-i\omega t/2}| - z\rangle\right)\text{;}\) (d) \(\left(1 -\sin{\left(\omega t\right)}\right)/2\text{;}\) (e) \(\left(1 +\sin{\left(\omega t\right)}\right)/2\text{.}\)
A 0.96 (b) \(2\text{.}\)
A 0.99 (a) \(e^+\text{,}\) \(W^+\text{;}\) (b) \(e^+\text{,}\) \(e^+\text{;}\) (c) \(Ru\text{,}\) \(Gd\text{.}\)
A 0.103 \(4.3\times 10^{-6}\Xunits{T}\text{.}\)
A 0.104 (a) \(0.1\Xunits{mT}\text{;}\) (b) \(0.2\Xunits{mT}\text{;}\) (c) \(0.1\Xunits{mT}\text{.}\)
Exercise 1.9.2. (b) \(x(t) = 30 e^{i(\frac{\pi}{3}t + \pi/2)}\Xunits{cm}\text{.}\)
Exercise 1.9.5. \(3\sqrt{5}\text{.}\)
Exercise 1.9.7. \(1.00 A\text{.}\)
Exercise 1.9.9. (b) \(\theta = 0.45^\circ\text{,}\) \(0.90^\circ\text{,}\) \(1.35^\circ\text{.}\)
Exercise 2.7.1. \(E_{ph} = 2.48\Xunits{eV}\text{,}\) \(K = 6.0 \times 10^{-6}\Xunits{eV}\text{.}\)
Exercise 2.7.3. (a) \(f_c = 5.4 \times 10^{14}\Xunits{Hz}\text{.}\)
Exercise 2.7.4. (a) \(10^{-9} \Xunits{m}\) or smaller, \(E_{ph} = 1240 \Xunits{eV}\text{;}\) (b) \(10^{-9}\Xunits{m}\) or smaller, \(K_{electron} = 1.5\Xunits{eV}\text{.}\)
Exercise 2.7.5. \(p = 4140\Xunits{eV/c}\) or \(2.21 \times 10^{-24}\Xunits{kg m/s}\text{.}\)
Exercise 2.7.6. \(1.1\Xunits{eV}\text{.}\)
Exercise 2.7.7. (a) \(E_{ph} = 2.48 \times 10^{-13}\Xunits{eV}\text{;}\) (b) \(E_{ph} \ll U_{bind}\text{,}\) so no chemical bonds affected, including those in DNA.
Exercise 2.7.10. (a) \(\frac{hc}{2L}j\text{;}\) (b) \(\frac{Lk_BT}{hc}\frac{1}{j}\text{;}\) (c) \(j_{max}=\frac{Lk_BT}{hc}\text{.}\)
Exercise 2.7.13. \(K_{max} = 7.1\Xunits{eV}\text{.}\)
Exercise 3.6.4. (a) 0.42; (b) 0.27.
Exercise 3.6.5. (a) \(1.06 \times 10^{-24}\Xunits{kg m/s}\text{;}\) (b) \(1.15 \times 10^6\Xunits{m/s}\text{.}\)
Exercise 3.6.6. \(2.1\times 10^{-26}\Xunits{m/s}\text{.}\)
Exercise 3.6.7. (a) \(6.6\times 10^{-13}\Xunits{m}\text{;}\) (b) \(2.3\times 10^{-12}\Xunits{m}\text{;}\) (c) (Hint: do you think the binding energy for a proton in a nucleus is 1 eV?)
Exercise 3.6.10. (a) \(6.1\times 10^{-19}\Xunits{J}\) or \(3.8\Xunits{eV}\text{;}\) (b) \(1.5\times 10^{-33}\Xunits{J}\) or \(9.6\times 10^{-15}\Xunits{eV}\text{;}\) (c) \(6.9\times 10^{-61}\Xunits{J}\) or \(4.3\times 10^{-42}\Xunits{eV}\text{.}\)
Exercise 3.6.13. (a) Works if \(A = 5/6\) and \(B = 3\text{;}\) (b) Doesn't work for all values of \(x\text{.}\)
Exercise 3.6.14. (b) Doesn't work; (c) Works if \(k = \pm \sqrt{2m(E-U_0)}/\hbar\text{,}\) which is fine since \(E > U_0\text{;}\) (d) Would work if \(\kappa = \pm \sqrt{2m(U_0-E)}/\hbar\text{,}\) but \(E > U_0\text{,}\) so this would be an imaginary \(\kappa\text{.}\)
Exercise 3.6.16. \(E = 2\) for a solution.
Exercise 4.7.1. (b) \(2L/3\text{;}\) (c) \(E=\frac{9h^2}{8mL^2}\text{.}\)
Exercise 4.7.2. (a) \(0.38\Xunits{eV}\text{;}\) (b) \(1.51 \Xunits{eV}\text{.}\)
Exercise 4.7.4. \(L/4\) and \(3L/4\text{.}\)
Exercise 4.7.10. (a) \(7.2\times 10^9\)m\(^{-1}\text{;}\) (b) \(4.8\times 10^{-11}\Xunits{m}\text{;}\) (c) \(5.2\times 10^{-36}\Xunits{m}\text{.}\)
Exercise 4.7.12. \(47.2\Xunits{ \mu m}\text{,}\) \(27.5\Xunits{ \mu m}\text{,}\) \(22.0\Xunits{ \mu m}\text{,}\) \(66.0\Xunits{ \mu m}\text{,}\) \(41.2\Xunits{ \mu m}\text{,}\) \(110\Xunits{ \mu m}\text{.}\)
Exercise 4.7.13. \(5.18\Xunits{nm}\text{.}\)
Exercise 4.7.14. (a) \(315\Xunits{nm}\text{;}\) (b) \(240\Xunits{nm}\text{.}\)
Exercise 4.7.16. \(15\Xunits{ \mu m}\) and \(7.5\Xunits{ \mu m}\text{.}\)
Exercise 5.7.3. (a) \(\frac{1}{2}\text{;}\) (b) \(0\text{;}\) (c) \(\frac{1}{2}\text{.}\)
Exercise 5.7.5. (a) \(\frac{1}{5}\text{;}\) (b) \(\frac{1}{10}\text{;}\) (c) \(\frac{1}{2}\text{;}\) (d) 0.
Exercise 5.7.6. (a) \(\frac{1}{2}\text{;}\) (b) \(\frac{1}{2}\text{.}\)
Exercise 5.7.7. (b) \(\cos^2\theta\text{,}\) \(\vert\mbox{X} \rangle\text{;}\) (c) \(\cos^2\theta\text{,}\) \(\vert \theta \rangle\text{;}\) (d) \(\sin^2\theta\text{;}\) (e) \(\cos^2\theta\sin^2\theta\text{.}\)
Exercise 5.7.10. (a) \(0.75\text{;}\) (b) \(0.067\text{.}\)
Exercise 5.7.11. \(0.50\text{.}\)
Exercise 5.7.14. \(21\Xunits{cm}\text{.}\)
Exercise 5.7.17. (a) \(\frac{13}{36}\text{;}\) (b) 0; (c) There are numerous answers to this question. The easiest are \(\frac{\sqrt{2}}{2}\) or \(-\frac{\sqrt{2}}{2}\) or \(i \frac{\sqrt{2}}{2}\) or \(-i \frac{\sqrt{2}}{2}\text{.}\) There are also numerous \(a+i b\) combinations that would work, as long as \(|a+i b|^2 = \frac{1}{2}\text{.}\)
Exercise 5.7.19. (a) Two equal intensity beams (1/4 the intensity of the intensity of the initial electron beam) will emerge, one with \(S_z = +\hbar/2\) and the other with \(S_z = -\hbar/2\text{.}\) (b) A beam will emerge from just one of the two exits with \(S_x = -\hbar/2\text{.}\) This beam will be 1/2 the intensity of the initial electron beam.
1. (a) electrons are indistinguishable, (b) \(\frac{1}{\sqrt 2}\ket{E_1\uparrow\; E_1\downarrow} -\frac{1}{\sqrt 2}\ket{E_1\downarrow\; E_1\uparrow}\text{.}\)
2. Yes for both (a) and (b). Electrons and muons are distinguishable.
3. (b) 4 for classical, 3 for bosons, 1 for fermions; (c) \(\frac{1}{2}\) for classical, \(\frac{2}{3}\) for bosons.
4. \(84\Xunits{eV}\text{.}\)
8. \(^3\)He is a fermion.
10. 0 (that's the point!).
3. \(6.84 \times 10^{-34}\Xunits{J \cdot s}\text{.}\)
4. \(-1.51\Xunits{eV}\text{.}\)
7. (a)\(~2.8\times 10^{-18}\text{,}\) \(0.017\) and \(0.40\text{,}\) (b) \(0.026\Xunits{eV}\text{,}\) \(0.26\Xunits{eV}\) and \(2.6\Xunits{eV}\text{;}\) (c) \(0\text{,}\) \(0\text{,}\) and \(7\text{.}\)
9. Would expect poor conductivity at \(300\Xunits{K}\text{,}\) moderate conductivity at \(3,000\Xunits{K}\text{,}\) and excellent conductivity at \(30,000\Xunits{K}\text{.}\)
11. (a) \(867\Xunits{nm}\) (near-IR); (b) \(549\Xunits{nm}\) (yellowish green); (c) \(365\Xunits{nm}\) (near-UV).
15. (b)\(E = -\frac{mk^2e^4}{2\hbar^2}\text{.}\)
1. (a) 3/10, (b) 7/10.
3. (a) \(c_+=\sqrt{3/10}\text{,}\) \(c_-=\sqrt{7/10}\text{,}\) \(\ket{\phi_1} = \sqrt{1/3}\ket{\uparrow} + \sqrt{2/3}\ket{\downarrow}\text{,}\) \(\ket{\phi_2} = \sqrt{2/7}\ket{\uparrow} + \sqrt{5/7}\ket{\downarrow}\text{.}\)
4. (a) 0.55; (b) \(\ket{\psi_\text{ new } } = 1.0\ket{\uparrow}\ket{\phi_1}\text{;}\) (c) \(\frac{1}{3}\text{.}\)
7. 0.854.
9. (a) 0.75, (b) 0.67.
Exercise 9.8.2. Mesons are bosons, baryons are fermions.
Exercise 9.8.6. (a) \(K^+\text{,}\) (b) \(\nu_e\text{,}\) (c) \(K^+\text{.}\)
Exercise 9.8.7. (a) yes, (b) no, (c) no, (d) yes.
Exercise 9.8.10. Does not conserve charge.
Exercise 9.8.11. \(S=-2\text{.}\)
Exercise 9.8.12. (a) \(uds\text{,}\) (b) \(dss\text{,}\) (c) \(u\bar d\text{.}\)
Exercise 9.8.13. This requires two \(s\)-quarks, with total charge \(-2/3\text{.}\) No single quark can add \(q=5/3\text{.}\)
Exercise 9.8.14. (a) \(n\) or \(\Delta ^0\text{,}\) (b) \(\Sigma^+\) or \(\Sigma^{*+}\text{,}\) (c) \(K^0\text{.}\)
Exercise 10.8.2. (a) 1.022 \mathrm{MeV}, (b) \(6.5\times 10^{-22}\Xunits{s}\text{.}\)
Exercise 10.8.3. (a) \(91.2\Xunits{GeV}\text{,}\) (b) \(7\times 10^{-27}\Xunits{s}\text{,}\) (c) \(2\times 10^{-18}\Xunits{m}\text{.}\)
Exercise 10.8.8. antigreen.
Exercise 11.6.3. About two protons should decay.
Exercise 11.6.4. Only \(\Sigma^-\to n + \pi^-\text{.}\)
Exercise 11.6.5. (a) strong, (b) electromagnetic, (c) weak.
Exercise 11.6.7. (a) There are no lighter baryons with \(S=-3\text{,}\) (b) weak; strangeness is not conserved, (c) weak interaction is slower than strong or electromagnetic.
Exercise 11.6.8. \(\Xi^-(1535)\) decays much faster; it can decay by the strong interaction; \(\Xi^-(1535)\to \Xi^-(1322) + \pi^0\text{,}\) while the lighter \(\Xi^-\) must go by a weak interaction.
Exercise 11.6.9. Photons or leptons. Because the weak decay to leptons is much slower than the electromagnetic decay to photons.
Exercise 12.6.2. about \(10^{-7}\) to \(10^{-6}\Xunits{s}\)
Exercise 12.6.6. about \(100\Xunits{MeV}\text{,}\) pions
Exercise 12.6.9. a) 5730 \(q\text{,}\) 5640 \(\bar q\text{,}\) 330 \(l\text{,}\) 300 \(\bar l\text{,}\) b) 1910 baryons, 1880 antibaryons, c) 30 baryons, 30 leptons, 2180 photons, d) 72.7