1. 10. The probability of a 1 is $P_1=\frac{1}{6}$. Similarly, $P_6=\frac{1}{6}$. The probability of a 1 or a 6 is $P_{1\text{or}6}=\frac{1}{6}+\frac{1}{6}=\frac{1}{3}$. Thus the expected number is $30\left(\frac{1}{3}\right)=10$.

2. $\mathbf{A}\mathbf{>}\mathbf{B}\mathbf{=}\mathbf{D}\mathbf{>}\mathbf{C}$. $|A\left(x\right)|{}^2$ is proportional to the density of dots.

3. $x{}_C$. The probability is largest at the point where the square of $\psi \left(x\right)$ is largest.

4. b. The area $\frac{1}{2}a(2\text{mm})$ must equal 1.

5. b. $\text{Δ}t=1.0\times {10}^{-7}\text{s}$. The bandwidth is $\text{Δ}{f}_{\text{B}}=1/\text{Δ}t=1.0\times {10}^7\text{Hz}=\text{ MHz}$

6. A. Wave packet A has a smaller spatial extent $\text{Δ}x$. The wavelength isn’t relevant.

1. $P_{\text{C}}=20\%\text{,}{P}_{\text{D}}=10\%$

3.

1. 7.7%

2. 25%

5.

1. 1/6

2. 1/6

3. 5/18

7. 100 V/m

9. $4.0\text{m}{}^{-1}$

11.

• a.

  

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• b.

  

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• c.

  

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13.

1. $5.0\times {10}^{-3}$

2. $2.5\times {10}^{-3}$

3. $2.5\times {10}^{-3}$

15.

• a. $0.25{\text{fm}}^{-1}$

• b.

  

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• c. 0.75

17.

• a. $\sqrt{{\displaystyle \frac{3}{8}}}{\text{mm}}^{-3/2}$

• b.

  

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• c. 6.3%

19. $1.0\times {10}^5$

21. 10.0 kHz

23. $-0.65\times {10}^{-36}\text{m/s}\le v_x\le 0.65\times {10}^{-36}\text{m}\text{/}\text{s}$

25. 4 nm

27.

1. 8 cycles

2. 0.938 MHz to 1.063 MHz

29.

• a.

  

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• b.

  

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• c.

  

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31.

• a. $\sqrt{3}{\text{nm}}^{-{\displaystyle \frac{1}{2}}}$

• b.

  

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• c.

  

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33.

• a. 4 nm

• b. 0.0 cm

• c. $-2.0\text{cm}\le x\le 2.0\text{cm}$

• d.

  

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35.

1. 0.27%

2. 32%

37.

• a. $\sqrt{{\displaystyle \frac{2}{L}}}$

• b.

  

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• d. 40%

39.

• a. $a=b$

• b. $a=b=0.84$

• c.

  

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• d. 58.1%

41. $18\mu \text{m}$

43.

1. $1.7\times {10}^6\text{m}\text{/}\text{s}$

2. $1.1\times {10}^{20}\text{reflections}\text{/}\text{s}$

45. No, $1.4\times {10}^{-27}\text{m}$

47. 19%

49.

• a. $b=c$

• b.

  

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• c. 91%

51. 200 m