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p137bp9 - ˆx is incident on the atom If the system is...

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Physics 137B, Fall 2007 Problem set 9 : radiative transitions and selection rules Assigned Saturday, 3 November. Due in box Friday, 9 November. 1. Calculate the rate of spontaneous emission for hydrogen 2p to 1s transitions. (Does this rate depend on the initial value of the m quantum number?) You can check that you’re on the right path by looking at the same calculation done in Bransden, p. 530-531. For N atoms initially in the 2p state, what is the initial radiated power? 2. Suppose now that a hydrogen atom is initially in the 2p state with quantum number m = 0, i.e., the eigenstate of L z is 0. Along what direction(s) should incident radiation be polarized to generate transitions to the 1s state? Why? For this polarization, what intensity per unit frequency interval I ( ω ba ) will generate one stimulated emission transition from 2p to 1s per nanosecond? 3. Suppose that instead of moving in the Coulomb field of a nucleus, an electron is confined in a 3D harmonic oscillator potential: H = p 2 2 m + k | r | 2 2 . (1) Now assume that an electromagnetic wave with polarization along
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Unformatted text preview: ˆx is incident on the atom. If the system is initially in a state with harmonic oscillator quantum numbers n x ,n y ,n z , and energy E = ¯ hω (3 / 2 + n x + n y + n z ) (2) where ω is the classical oscillation frequency p k/m , what transitions are allowed within the dipole approximation? Hint: your goal is to find which states n x ,n y ,n z have | D x ba | 2 6 = 0. It may help to think about how x is expressed in terms of the raising and lowering operators for the harmonic oscillator. 4. Bransden 11.8. (Hint: Proving this “Thomas-Reiche-Kuhn sum rule” is a little tricky. It may help you to think about how the action of the expression ∑ k | ψ k i E k h ψ k | on a state is exactly the same as that of an operator you know well, and to remember that ∑ k | ψ k ih ψ k | is the identity operator.) 5. Bransden 13.2. The “laboratory coordinate system” means the one in which B particles are initially stationary. 1...
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