Answers to Selected Problems

Chapter 13 Answers to Selected Problems

Additional Problems

A 0.6 (a)

- 5560 \hat{ȷ} N / C;

(b)

0;

(c)

5560 \hat{ȷ} N / C .

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A 0.7 (a)

k λ / a;

(b)

- k λ / a .

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A 0.8

\frac{- 2 q k}{π R^{2}} (\hat{ı} + \hat{ȷ}) .

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A 0.9 (a)

50, 000 eV;

(b)

8.0 \times 10^{- 15} J;

(c)

1.33 \times 10^{8} m / s .

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A 0.11 left side:

1600 V / m

to left; right side:

300 V / m

to right.

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A 0.18

v_{f} = \sqrt{2 I H B D / m} .

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A 0.20.b

4 \times 10^{- 5} T .

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A 0.22

5.8 \times 10^{- 3} T .

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A 0.23

0.013 s .

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A 0.24

π r^{2} B_{0} (6 t - 1) / R .

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A 0.31 (a)

\pm z,

(b)

\pm x,

(c)

\pm z,

(d)

\pm z,

(e)

\pm x,

\pm y .

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A 0.36 (a)

20 m,

10 m,

6.67 m;

(b)

π / 10, π / 5, 3 π / 10;

(c)

17 Hz,

34 Hz,

51 Hz .

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A 0.45 (a)

717 nm;

(b)

55, 800 .

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A 0.48 (a)

λ = 1257 nm,

T = 4.19 \times 10^{- 15} s,

v = 3.0 \times 10^{8} m / s,

infrared, (b)

B = 0.2 mT;

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A 0.51 (b)

2 \times 10^{- 14} m;

(c)

p = 3.3 \times 10^{- 20} kg \cdot m / s,

K = .3 .3 \times 10^{- 13} J .

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A 0.55

1.3 \times 10^{- 21} m / s .

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A 0.59 (a)

Δ t = 6.6 \times 10^{- 18} s;

(b)

v = 4.2 \times 10^{6} m / s;

(c)

L = 2.8 \times 10^{- 11} m;

(d)

0.13 .

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A 0.72

5.4 cm .

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A 0.73

19.4 mT .

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A 0.80

1.41 A .

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A 0.82 (a) minimum; (b)

9 I_{0} .

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A 0.84

y (x, t) = 2 \cos (π x - 2 π t) .

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A 0.86 (b)

2.95;

(c)

0.5 rad .

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A 0.88 (a) \(\frac{1}{1600\pi}\Xunits{m\)^{-2}

Extra close brace or missing open brace

(b)

1 / 4;

Extra close brace or missing open brace

\(P_B = \frac{1}{2000\pi}\Xunits{m\)^{-2}

Extra close brace or missing open brace

(d)

13 / 20 .

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A 0.90 (a)

1 + i θ - θ^{2} / 2! - i θ^{3} / 3! + θ^{4} / 4! + i θ^{5} / 5! - θ^{6} / 6! + \dots;

(c)

1 - i θ - θ^{2} / 2! + i θ^{3} / 3! + θ^{4} / 4! - i θ^{5} / 5! - θ^{6} / 6! + \dots;

(g)

0.88 + 0.48 i;

(h)

0.88 - 0.48 i;

(i)

1.76;

(j)

0.96 i;

(k)

(e^{0.3 i} + e^{- 0.3 i}) / 2;

(l)

(e^{0.3 i} - e^{- 0.3 i}) / 2 i;

(m)

1;

(n)

e^{0.6 i} .

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A 0.91 (a)

11 E_{1} / 5;

(b)

\sqrt{3 / 5} e^{- i E_{1} t / ℏ} | 1 ⟩ + \sqrt{2 / 5} e^{- i 4 E_{1} t / ℏ} | 2 ⟩;

(c)

11 E_{1} / 5 .

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A 0.92 (a)

| Ψ (0) ⟩ = | + x ⟩ = \sqrt{1 / 2} (| + z ⟩ + | - z ⟩);

(b)

E_{+} = - μ B_{0},

E_{-} = + μ B_{0};

(c)

| Ψ (t) ⟩ = \sqrt{1 / 2} (e^{i ω t / 2} | + z ⟩ + e^{- i ω t / 2} | - z ⟩);

(d)

(1 - \sin (ω t)) / 2;

(e)

(1 + \sin (ω t)) / 2 .

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A 0.96 (b)

2 .

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A 0.99 (a)

e^{+},

W^{+};

(b)

e^{+},

e^{+};

(c)

R u,

G d .

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A 0.103

4.3 \times 10^{- 6} T .

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A 0.104 (a)

0.1 mT;

(b)

0.2 mT;

(c)

0.1 mT .

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Chapter 1

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Exercise 1.9.2. (b)

x (t) = 30 e^{i (\frac{π}{3} t + π / 2)} cm .

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Exercise 1.9.5.

3 \sqrt{5} .

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Exercise 1.9.7.

1.00 A .

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Exercise 1.9.9. (b)

θ = {0.45}^{\circ},

{0.90}^{\circ},

{1.35}^{\circ} .

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Chapter 2

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Exercise 2.7.1.

E_{p h} = 2.48 eV,

K = 6.0 \times 10^{- 6} eV .

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Exercise 2.7.3. (a)

f_{c} = 5.4 \times 10^{14} Hz .

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Exercise 2.7.4. (a)

10^{- 9} m

or smaller,

E_{p h} = 1240 eV;

(b)

10^{- 9} m

or smaller,

K_{e l e c t r o n} = 1.5 eV .

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Exercise 2.7.5.

p = 4140 eV / c

2.21 \times 10^{- 24} kg m / s .

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Exercise 2.7.6.

1.1 eV .

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Exercise 2.7.7. (a)

E_{p h} = 2.48 \times 10^{- 13} eV;

(b)

E_{p h} ≪ U_{b i n d},

so no chemical bonds affected, including those in DNA.

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Exercise 2.7.10. (a)

\frac{h c}{2 L} j;

(b)

\frac{L k_{B} T}{h c} \frac{1}{j};

(c)

j_{m a x} = \frac{L k_{B} T}{h c} .

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Exercise 2.7.13.

K_{m a x} = 7.1 eV .

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Chapter 3

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Exercise 3.6.4. (a) 0.42; (b) 0.27.

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Exercise 3.6.5. (a)

1.06 \times 10^{- 24} kg m / s;

(b)

1.15 \times 10^{6} m / s .

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Exercise 3.6.6.

2.1 \times 10^{- 26} m / s .

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Exercise 3.6.7. (a)

6.6 \times 10^{- 13} m;

(b)

2.3 \times 10^{- 12} m;

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Exercise 3.6.10. (a)

6.1 \times 10^{- 19} J

3.8 eV;

(b)

1.5 \times 10^{- 33} J

9.6 \times 10^{- 15} eV;

(c)

6.9 \times 10^{- 61} J

4.3 \times 10^{- 42} eV .

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Exercise 3.6.13. (a) Works if

A = 5 / 6

and

B = 3;

(b) Doesn't work for all values of

x .

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Exercise 3.6.14. (b) Doesn't work; (c) Works if

k = \pm \sqrt{2 m (E - U_{0})} / ℏ,

which is fine since

E > U_{0};

(d) Would work if

κ = \pm \sqrt{2 m (U_{0} - E)} / ℏ,

but

E > U_{0},

so this would be an imaginary

κ .

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Exercise 3.6.16.

E = 2

for a solution.

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Chapter 4

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Exercise 4.7.1. (b)

2 L / 3;

(c)

E = \frac{9 h^{2}}{8 m L^{2}} .

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Exercise 4.7.2. (a)

0.38 eV;

(b)

1.51 eV .

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Exercise 4.7.4.

L / 4

and

3 L / 4 .

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Exercise 4.7.10. (a)

7.2 \times 10^{9}

^{- 1};

(b)

4.8 \times 10^{- 11} m;

(c)

5.2 \times 10^{- 36} m .

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Exercise 4.7.12.

47.2 μ m,

27.5 μ m,

22.0 μ m,

66.0 μ m,

41.2 μ m,

110 μ m .

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Exercise 4.7.13.

5.18 nm .

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Exercise 4.7.14. (a)

315 nm;

(b)

240 nm .

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Exercise 4.7.16.

15 μ m

and

7.5 μ m .

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Chapter 5

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Exercise 5.7.3. (a)

\frac{1}{2};

(b)

0;

(c)

\frac{1}{2} .

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Exercise 5.7.5. (a)

\frac{1}{5};

(b)

\frac{1}{10};

(c)

\frac{1}{2};

(d) 0.

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Exercise 5.7.6. (a)

\frac{1}{2};

(b)

\frac{1}{2} .

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Exercise 5.7.7. (b)

\cos^{2} θ,

| X ⟩;

(c)

\cos^{2} θ,

| θ ⟩;

(d)

\sin^{2} θ;

(e)

\cos^{2} θ \sin^{2} θ .

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Exercise 5.7.10. (a)

0.75;

(b)

0.067 .

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Exercise 5.7.11.

0.50 .

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Exercise 5.7.14.

21 cm .

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Exercise 5.7.17. (a)

\frac{13}{36};

(b) 0; (c) There are numerous answers to this question. The easiest are

\frac{\sqrt{2}}{2}

- \frac{\sqrt{2}}{2}

i \frac{\sqrt{2}}{2}

- i \frac{\sqrt{2}}{2} .

There are also numerous

a + i b

combinations that would work, as long as

| a + i b |^{2} = \frac{1}{2} .

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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} = + ℏ / 2

and the other with

S_{z} = - ℏ / 2 .

(b) A beam will emerge from just one of the two exits with

S_{x} = - ℏ / 2 .

This beam will be 1/2 the intensity of the initial electron beam.

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Chapter 6

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1. (a) electrons are indistinguishable, (b)

\frac{1}{\sqrt{2}} | E_{1} ↑ E_{1} ↓ ⟩ - \frac{1}{\sqrt{2}} | E_{1} ↓ E_{1} ↑ ⟩ .

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2. Yes for both (a) and (b). Electrons and muons are distinguishable.

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3. (b) 4 for classical, 3 for bosons, 1 for fermions; (c)

\frac{1}{2}

for classical,

\frac{2}{3}

for bosons.

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84 eV .

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^{3}

He is a fermion.

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10. 0 (that's the point!).

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Chapter 7

🔗

6.84 \times 10^{- 34} J \cdot s .

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- 1.51 eV .

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7. (a)

2.8 \times 10^{- 18},

0.017

and

0.40,

(b)

0.026 eV,

0.26 eV

and

2.6 eV;

(c)

0,

0,

and

7 .

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9. Would expect poor conductivity at

300 K,

moderate conductivity at

3, 000 K,

and excellent conductivity at

30, 000 K .

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11. (a)

867 nm

(near-IR); (b)

549 nm

(yellowish green); (c)

365 nm

(near-UV).

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15. (b)

E = - \frac{m k^{2} e^{4}}{2 ℏ^{2}} .

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Chapter 8

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1. (a) 3/10, (b) 7/10.

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3. (a)

c_{+} = \sqrt{3 / 10},

c_{-} = \sqrt{7 / 10},

| ϕ_{1} ⟩ = \sqrt{1 / 3} | ↑ ⟩ + \sqrt{2 / 3} | ↓ ⟩,

| ϕ_{2} ⟩ = \sqrt{2 / 7} | ↑ ⟩ + \sqrt{5 / 7} | ↓ ⟩ .

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4. (a) 0.55; (b)

| ψ_{new} ⟩ = 1.0 | ↑ ⟩ | ϕ_{1} ⟩;

(c)

\frac{1}{3} .

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7. 0.854.

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9. (a) 0.75, (b) 0.67.

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Chapter 9

🔗

Exercise 9.8.2. Mesons are bosons, baryons are fermions.

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Exercise 9.8.6. (a)

K^{+},

(b)

ν_{e},

(c)

K^{+} .

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Exercise 9.8.7. (a) yes, (b) no, (c) no, (d) yes.

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Exercise 9.8.10. Does not conserve charge.

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Exercise 9.8.11.

S = - 2 .

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Exercise 9.8.12. (a)

u d s,

(b)

d s s,

(c)

u \bar{d} .

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Exercise 9.8.13. This requires two

s

-quarks, with total charge

- 2 / 3 .

No single quark can add

q = 5 / 3 .

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Exercise 9.8.14. (a)

n

Δ^{0},

(b)

Σ^{+}

Σ^{* +},

(c)

K^{0} .

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Chapter 10

🔗

Exercise 10.8.2. (a) 1.022 \mathrm{MeV}, (b)

6.5 \times 10^{- 22} s .

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Exercise 10.8.3. (a)

91.2 GeV,

(b)

7 \times 10^{- 27} s,

(c)

2 \times 10^{- 18} m .

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Exercise 10.8.8. antigreen.

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Chapter 11

🔗

Exercise 11.6.3. About two protons should decay.

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Exercise 11.6.4. Only

Σ^{-} \to n + π^{-} .

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Exercise 11.6.5. (a) strong, (b) electromagnetic, (c) weak.

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Exercise 11.6.7. (a) There are no lighter baryons with

S = - 3,

(b) weak; strangeness is not conserved, (c) weak interaction is slower than strong or electromagnetic.

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Exercise 11.6.8.

Ξ^{-} (1535)

decays much faster; it can decay by the strong interaction;

Ξ^{-} (1535) \to Ξ^{-} (1322) + π^{0},

while the lighter

Ξ^{-}

must go by a weak interaction.

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Exercise 11.6.9. Photons or leptons. Because the weak decay to leptons is much slower than the electromagnetic decay to photons.

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Chapter 12

🔗

Exercise 12.6.2. about

10^{- 7}

10^{- 6} s

🔗

Exercise 12.6.6. about

100 MeV,

pions

🔗

Exercise 12.6.9. a) 5730

q,

5640

\bar{q},

330

l,

300

\bar{l},

b) 1910 baryons, 1880 antibaryons, c) 30 baryons, 30 leptons, 2180 photons, d) 72.7