Solutions Manual – Modern Physics with Modern Computational Methods (3rd Ed) Morrison

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SOLUTIONS MANUAL

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The Wave-Particle Duality - Solutions




1. The energy of photons in terms of the wavelength of light is given by Eq.
(1.5). Following Example 1.1 and substituting λ = 200 eV gives:
hc 1240 eV · nm
Ephoton = = = 6.2 eV
λ 200 nm
2. The energy of the beam each second is:
power 100 W
Etotal = = = 100 J
time 1s
The number of photons comes from the total energy divided by the energy
of each photon (see Problem 1). The photon’s energy must be converted to
Joules using the constant 1.602 × 10−19 J/eV , see Example 1.5. The result
is:
Etotal 100 J
Nphotons = = = 1.01 × 1020
Ephoton 9.93 × 10−19
for the number of photons striking the surface each second.

3. We are given the power of the laser in milliwatts, where 1 mW = 10−3 W .
The power may be expressed as: 1 W = 1 J/s. Following Example 1.1, the
energy of a single photon is:
hc 1240 eV · nm
Ephoton = = = 1.960 eV
λ 632.8 nm
We now convert to SI units (see Example 1.5):
1.960 eV × 1.602 × 10−19 J/eV = 3.14 × 10−19 J
Following the same procedure as Problem 2:
1 × 10−3 J/s photons
Rate of emission = = 3.19 × 1015
3.14 × 10−19 J/photon s

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4. The maximum kinetic energy of photoelectrons is found using Eq. (1.6)
and the work functions, W, of the metals are given in Table 1.1. Following
Problem 1, Ephoton = hc/λ = 6.20 eV . For part (a), Na has W = 2.28 eV :

(KE)max = 6.20 eV − 2.28 eV = 3.92 eV

Similarly, for Al metal in part (b), W = 4.08 eV giving (KE)max = 2.12 eV
and for Ag metal in part (c), W = 4.73 eV , giving (KE)max = 1.47 eV .

5. This problem again concerns the photoelectric effect. As in Problem 4, we
use Eq. (1.6):
hc
(KE)max = −W
λ
where W is the work function of the material and the term hc/λ describes
the energy of the incoming photons. Solving for the latter:
hc
= (KE)max + W = 2.3 eV + 0.9 eV = 3.2 eV
λ
Solving Eq. (1.5) for the wavelength:
1240 eV · nm
λ= = 387.5 nm
3.2 eV
6. A potential energy of 0.72 eV is needed to stop the flow of electrons. Hence,
(KE)max of the photoelectrons can be no more than 0.72 eV. Solving Eq. (1.6)
for the work function:
hc 1240 eV · nm
W = − (KE)max = − 0.72 eV = 1.98 eV
λ 460 nm
7. Reversing the procedure from Problem 6, we start with Eq. (1.6):
hc 1240 eV · nm
(KE)max = −W = − 1.98 eV = 3.19 eV
λ 240 nm
Hence, a stopping potential of 3.19 eV prohibits the electrons from reaching
the anode.

8. Just at threshold, the kinetic energy of the electron is zero. Setting
(KE)max = 0 in Eq. (1.6),
hc 1240 eV · nm
W = = = 3.44 eV
λ0 360 nm

9. A frequency of 1200 THz is equal to 1200 × 1012 Hz. Using Eq. (1.10),

Ephoton = hf = 4.136 × 10−15 eV · s × 1.2 × 1015 Hz = 4.96 eV

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Next, using the work function for sodium (Na) metal and Eq. (1.6),

(KE)max = Ephoton − W = 4.96 ev − 2.28 eV = 2.68 eV

10. We start from Eq. (1.8) for the case of m = 2:
 
1 1 1
=R − 2
λ 22 n

Now invert the equation and plug in for the Rydberg constant, R:
 −1
1 1 1
λ= − 2
1.0971 × 105 cm−1 4 n

Subtract the fractions by getting a common denominator:

n2 − 4
 −1
1 cm
λ=
1.0971 × 105 4n2

Invert the term in the parenthesis and factor out the common factor of 4

n2
 
4 cm
λ=
1.0971 × 105 n2 − 4

Doing the division, we get Eq. (1.7) for the Balmer formula:

n2
 
λ = (3645.6 × 10−8 cm)
n2 − 4

11. Following Example 1.2,
 
13.6 eV 13.6 eV
∆E = − − − = 2.86 eV
52 22

Using Eq. (1.12):

hc 1240 eV · nm
λ= = = 434 nm
∆E 2.86 eV
12. Since the initial state has m = 2, we can use Eq. (1.7) with n = 4:
 2 
4
λ = (364.56 nm) 2
= 486.1 nm
4 −4

To get the energy of the photon, use Eq. (1.5):

hc 1240 eV · nm
Ephoton = = = 2.551 eV
λ 486.1 nm