 | || | | | � � �� ��MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II�Course Instructors: Professor Robert Field and Professor Andrei Tokmakoff MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.74 Quantum Mechanics II Spring, 2004 Professor Robert W. Field Problem Set # 5 DUE: At the start of Lecture on Monday, April 5. Reading: HLB–RWF 3.2.1 (optional), 9.1.1, 9.4.1 - 9.4.3. Problems: 1. Photon Plucks The first excited state of Ba is the metastable 6s5d 3D state. Consider the J = MJ = 3 component of 3D |3D3 3 = s0αd2α . A photon pluck promotes one e− via a ∆ = ±1, ∆m =0, ±1, ∆s =∆ms = 0 transition. If the photon is linearly polarized along the z–direction, the selection rule is ∆m =0. A. What are all of the 1m 1 s1ms1 2m 2 s2ms2 basis states accessible via a z–polarized photon pluck? B. Consider first only the 5d → np excitation to 6snp Rydberg complexes. What L−S−J−MJ eigenstates are coherently populated at t = 0? What are their relative amplitudes? C. Now consider the 6s np (n = n) excitation to doubly excited 5dnp configurations. What → L−S−J−MJ eigenstates are coherently populated at t = 0? What are their relative amplitudes? D. The only things not specified are the relative amplitudes within the 6snp Rydberg series, within the 5dnp Rydberg series, and for the 6snp series relative to the 5dnp series. These are given by the radial integrals µn� p5d = 6p z 5d (n/6)−3/2 µnp6s = 6p z 6s (n/6)−3/2 What is the form of the Ψ(t) that results from this single-photon pluck of Ba �6s 5d 3D3 3 ? I have not provided sufficient information about eigen–energies, photon center-frequency, and photon pulse duration. You should make reasonable choices for these quantities. E. The ionization threshold for the 6sn Rydberg series (n →∞) is 42032·4cm−1, which is consid-erably lower than that for the 5dn Rydberg series (n →∞) at 46906·3cm−1 (for J =3/2). Suppose you probe the coherent superposition state from part D with a detection pulse at an energy just above the 6s∞ limit but below the 5d∞ limit. You detect photo-ions as a function of delay between the excitation and detection pulses. What will you see? Which coherences will be detected by this pump/probe scheme and which coherences will be destroyed? F. The 3P states of the np5d and np6s Rydberg series interact with each other via the 1/r12 in-terelectronic repulsion operator. The Kepler period of a Rydberg wavepacket is proportional to n−3 −1. Owing to the difference in energy of the series limits, the isoenergetic members of the np5d and np6s series have n <n, thus the Kepler period of the np5d wavepacket is shorter than that of the np6s wavepacket. The interaction between the two wavepackets is largest when both are inside the ion-core. If you could monitor the amplitude in the np6s wavepacket as a function of time, what would you expect to see?� � � � |  2. Atomic Hyperfine Structure The Heff for 137Ba (I =3/2) is = Hel + HSO + HmhfsH . For Rydberg series (δ is the quantum defect,  = 109, 737 cm−1) Hel = [ /(n − δ )2] nnnn.− | |For two-electron atoms HSO = ξ(r1)1 · s1 + ξ(r2)2 · s2. For magnetic hyperfine structure of a two-electron atom Hmhfs =[a(r1)1 + a(r2)2 + b(r1)s1 + b(r2)s2] I.·Under special conditions (to be specified by you), HSO and Hmhfs simplify to “HSO”= ζ(N, L, S)L S·“Hmhfs”= C(N, L, S, J)I J·and Hmhfswhere N refers to the electronic configuration. These simplified forms of HSO are useful for depicting the pattern of splittings within an “isolated” state. Electronic transitions are controlled by � j + zˆ� n k  k m k � xˆi + yˆk� nkmk k , which is a matrix element of a one-electron operator that operates exclusively on the spatial (not spin) part of a single spin-orbital. and HmhfsA. Use the simplified forms of HSO to construct the spin-hyperfine structure of 137Ba in the 6s5d 3D and 6snp 3P states. HINT: (L + S)2 = J2 (J + I)2 = F2 B. Starting from the 137Ba 6s5d 3D3 level (with hyperfine F –components F =3/2, 5/2, 7/2, and 9/2 thermally populated), draw a level diagram on which you illustrate all of the allowed fine-hyperfine transitions from 3D3 to 3PJ,F . The rigorous selection rule for electric dipole transitions (a vector operator) is ∆F =0, ±1. The nearly rigorous selection rules ∆J =0, ±1, ∆L =0, ±1, and ∆S = 0 may also be taken seriously here. C. In order to derive the fine-hyperfine quantum beat signal obtained by pulsed excitation of 3PJ� ,F � ←3DJ=3,F transitions, you need to compute all of the relative transition amplitudes for the short-pulse excitation “pump” transition and for the delay–scanned detection “probe” transition. For simplicity you can use the 6snp 3PJ� ,F � → 6s6d 3D3,F probe transition. In order to calculate the relative transition amplitudes in the s1s212LSJIF MF  basis set, you must perform a series of |coupled→uncoupled transitions: s1s212LSJIF MF|  |  |  |  | | | |⇓ s1s212LSJMJ IMI ⇓ s1s212LMLSMS IMI ⇓ 1m 1 2m 2 s1s2SMS IMI ⇓ 1m 1 s1ms1 2m 2 s2ms2 IMI . All of the beat notes in your quantum beating signal are explicitly known half-integer multiples of a common factor with relative amplitudes controlled by 6d z npnp z 5d times factors computed by you. Would the relative intensities and phases of the beat notes be affected if the pump and probe lasers were polarized perpendicularly (i.e., x, z) rather than parallel (i.e., z, z)? Optional: compute and compare the beat patterns for probe (z), pump (z) to probe (x), pump (z). D. As the principal quantum number of the 6snp 3P state increases, the spin-orbit coupling constant decreases as n−3 but the contribution of the 6sms spin-orbital to the b(r1)s1·I hyperfine term remains constant. At some point the hyperfine splittings become larger than the spin-orbit split-tings. What happens to the level structure and quantum beat amplitudes? A qualitative answer is acceptable. HINT: The hyperfine structure of Ba+ 6s2S is highly relevant