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Unformatted text preview: PH 216, Problem set 3, Due 4/20 1. Consider the hydrogen atom in an excited n = 2 state, which is subjected to an external uniform electric field E. Do not neglect the spin of the electron. Assume that the field E is sufficiently weak so that eEa0 is small compared to the fine structure, but such that the "Lamb shift" ( = 1057 MHz) cannot be neglected. This "Lamb shift" is an effect from QED that breaks the degeneracy between the 2S1/2 and 2P1/2 states: if the Lamb-shift is neglected, then the 2s1/2 and 2p1/2 states are degenerate with energy, 1 5 E (0) = - 2 me c2 2 - 128 me c2 4 , as computed in class. Including the Lamb shift, 2s1/2 and 2p1/2 are no longer degenerate. Instead, we have E (0) (2s1/2 ) - E (0) (2p1/2 ) > 0 , which defines . Assume here that the Lamb shift can be included as a term HLS in the unperturbed Hamiltonian. Thus, for present purposes you can treat the problem as a two-level system consisting of the the 2S1/2 and 2P1/2 states of hydrogen. In particular, you may ignore the 2P3/2 state of hydrogen and the hyperfine interactions. (a) Compute the Stark effect for the 2S1/2 and 2P1/2 states of hydrogen by solving the two-level system exactly (i.e. find the four matrix elements of H between the two states, and diagonalize.) HINT: When electron spin is included, the hydrogen atom energy eigenstates are two-component wave functions given (in the coordinate representation) by:
j () = Rn (r)Ym (, ) , r where Rn (r) is the radial wave function of the hydrogen atom, and the spin spherical harmonics are defined by:1 Ym where 1 j = 1 , m = 1 ( + 1 m)1/2 , 1 ; m - 1 , 1 + ( + 1 m)1/2 , 1 ; m + 2 , - 1 . 2 2 2 2 2 2 2 2 2 + 1 (b) Show that for eEa0 h, the energy shifts due to the external electric field are quadratic in E, whereas for eEa0 h, they are linear in E. Determine the (perturbed) energy eigenstates in both limiting cases.
1 j= 1 2 (, ) , |j = 1 , m , 2 Note that the usual spherical harmonics can be written as: Ym (, ) , | , m. 3 (c) The critical field is defined as: h Ec , 3ea0 where the factor of 3 is conventional. The linear or quadratic behavior of the energy shifts obtained in part (b) depend on the magnitude of E as compared to Ec . Determine Ec in volts/cm. 2. Consider a two-level system with E1 < E2 , and a Hamiltonian H = H 0 + V , where |1 and |2 are the energy eigenstates of H 0 . V is a time-dependent potential that connects the two levels as follows: V11 = V22 = 0, V12 = eit , V21 = e-it ( real) , where Vij = i|V |j. At time t = 0, it is known that only the lower level is populated-- that is, c1 (0) = 1 and c2 (0) = 0. We saw in class that if |(t)I = where n=1 cn (t) |n, |(t)S , then the cn (t) satisfy a differential equation i |(t)I eiH 0 t/ where Vnm (t) n|V (t)|m and kn Ek - En . By solving the above system of differential equations exactly, find |c1 (t)|2 and |c2 (t)|2 for t > 0. HINT: It is convenient to define new coefficients, c1 (t) ei(21 -)t/2 c1 (t) , c2 (t) e-i(21 -)t/2 c2 (t) . dcn = Vnm (t)einm t cm , dt m=1 (2) Then, show that eq. (17) reduces to a matrix differential equation of the form d c1 (t) c (t) i =A 1 , c2 (t) dt c2 (t) (3) where A is a time-independent 22 traceless hermitian matrix. Verify that the solution to eq. (18) is c1 (t) -iAt/ c1 (0) =e . c2 (t) c2 (0) By writing A = (where the vector is uniquely determined), it is straightforward a a to compute e-iAt/ and complete part (a) of the problem. 3. Do the same problem using time-dependent perturbation theory to lowest nonvanishing order. Compare the two approaches for small values of . Treat the following two cases separately: (i) very different from 21 , and (ii) close to 21 . 4 ...
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This document was uploaded on 10/31/2011.
- Spring '09