p137bp5sol - Physics 137B, Fall 2007, Moore Problem Set 5...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Physics 137B, Fall 2007, Moore Problem Set 5 Solutions 1. Our unperturbed Hamiltonian is H = g B B h S z , with eigenstates |i , E =- g B B 2 - , |i , E = + g B B 2 = + . At time t = 0 we introduce the perturbation H = g B B h S x , whose eigenstates we will label as | 1 i = 1 2 ( |i - |i ) , E 1 =- g B B 2 , | 2 i = 1 2 ( |i + |i ) , E 2 = + g B B 2 . (a) We start in the state | (0) i = |i at time t = 0. According to the formalism of time-dependent perturbation theory, we can expand our state | ( t ) i for t > 0 as | ( t ) i = c ( t ) e- iE t/ h |i + c ( t ) e- iE t/ h |i . To first order, the time-dependent coefficients c ( t ) and c ( t ) are given by c ( t ) = 1 + ( i h )- 1 Z t h | H | i dt = 1 , 1 c ( t ) = ( i h )- 1 Z t h | H | i e it dt = 1 i h g B B 2 Z t e it dt = 1 i h g B B 2 1 i ( e it- 1 ) =- i B B e it/ 2 sin t 2 , where = ( E - E ) / h = 2 / h is the Bohr frequency. Thus our time-evolved state can be written to first order as | ( t ) i = e + i t/ h |i +- i B B e it/ 2 sin t 2 e- i t/ h |i = e + it/ 2 |i - i B B sin t 2 |i . The transition probability is given by P ( t ) = |h | ( t ) i| 2 = B B 2 sin 2 t 2 , exactly as you calculated in the previous homework assignment. To calculate the expectation value of H , we need h ( t ) | H | ( t ) i = e- it/ 2 h| + i B B sin t 2 h| e- it/ 2 (- ) |i + i B B sin t 2 (+ ) |i =- " 1- B B 2 sin 2 t 2 # . This is of order ( B /B ) 2 , whereas | ( t ) i is only normalized to order B /B . So we should divide by h ( t ) | ( t ) i = 1 + B B 2 sin 2 t 2 . Expanding the denominator to leading order we obtain the expectation value h H i = h ( t ) | H | ( t ) i h ( t ) | ( t ) i =- " 1- 2 B B 2 sin 2 t 2 # ....
View Full Document

This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at University of California, Berkeley.

Page1 / 8

p137bp5sol - Physics 137B, Fall 2007, Moore Problem Set 5...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online