Exam III Review

1. Check whether the following transition (specified in terms of the initial and final quantum numbers n, ℓ, m_ℓ, and m_s) is allowed, and if it is, find the energy involved. (6,5, -1, 1/2) -> (5, 4, -2, 1/2) [Pay attention to the signage. Use E₁ = -13.6 eV]

Δn = 1

Δl = -1

Δm_l = -1

Δm_s = 0

(5,4,-2,1/2)

n = 5

l = 4 < n

m_l = -2 <= |l|

m_s = +1/2

Allowed states

n = 1 -> ∞

l = 0 -> (n-1)

m_l = - l … 0 … l

m_s = +1/2, -1/2

Selection rules (allowed transitions)

Δn = anything

Δl = -1,+1

Δm_l = -1,0,+1

Δm_s = 0, -+1/2

(6,5,-1,1/2)

n = 6

l = 5 < n

m_l = -1 <= |l|

m_s = +1/2

E = -0.1662 eV

What the quantum numbers represent.

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n

l

n: Primarily determines distance electron is from nucleus.

l: Primarily determines shape of electron’s orbit.

m_l: Primarily determines elevation (or angle) of electron path relative to the nucleus.

m_s: Primarily determines direction of electron’s magnetic field relative to central axis angle.

What the quantum numbers represent.

θ

n, l*

l

m_l

+

-

m_s

*Not to scale

What the quantum numbers represent.

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2. A radioactive sample of ⁶⁰Co (t_1/2 = 5.271y) has a β^- activity of 2 x 10⁷ Bq. How many grams of ⁶⁰Co are present?

N = 32

Z = 27

A = 59

= 8.768 MeV

4. A large Michelson interferometer consisting of orthogonal optical cavities, where each cavity has a length of 10 km is used to detect gravitational waves. The maximum amplitude of strain (or fractional change in length) measured by the detectors was 10⁻²¹. What was the maximum change in length (in m) of the optical cavity due to the gravitational wave? (For comparison, the radius of a proton is about 10⁻¹⁵ m.)

10km

10km

ΔL/L = 10^-21 _{(Not to scale)}

5. A gravitational wave signal was the result of two black holes, orbiting around a common center of mass, which spiraled inward and merged to form a single black hole. The mass of the first black hole was 40 solar masses, and the second was 30 solar masses. The final black hole mass after the merger was 60 solar masses. The entire detected signal lasted about 20 seconds. What was the average power (in W) radiated away as gravitational waves during this time? (Take one solar mass to be 2E30 kg.)

M₁ = 40

M₂ = 30

M_f = 60

T = 20s

ΔM = (M₁ + M₂) – M_f = 10

E = mc² = (ΔM)c²

= (10*2E30 kg)*(3E8 m/s²)²

E = 1.8E48 J

6. An airplane flies at 300 m/s with an altitude of 10⁴ m and circles the world. It flies for a total of 45 hours. Calculate the gravitational red shift of a clock on the airplane assuming the airplane speed remains 300 m/s. Assume the circumference of the earth is 4E7 m, the acceleration due to gravity is 9.8 m/s², and the speed of light c=3E8 m/s.

H = 10⁴ m

T = 45 hr = 1.62E5 s

General form: