Question

4.8. A spin- particle, initially in a state with S h/2 with n sin i+ cos k, is in a constant magnetic field Bo in the z direction. Determine the state of the particle at time and determine how (S,), (S), and (S.) vary with time.

Answer 1

A spin-1/2 particle, initially in a state with S_{​​​​​​n}​​​ = h/2 with sin θ i +cos θ k, is a constant magnetic field B_{​​​​​​0}​​​​​ in the z direction.

Given that The Hanuiltanion of the system is Ĥ 1.8 - - S . BE 9.00 .B se ne T sin @i + cos Of J.(BOR) - he cos O Bo - Wo cos o energy stales of Ŝ. Ñ +Z) - ņ Wo Cos | +Z) À l-Z) - Wo cos Ol=2) The system of the state after time t' is 1 40 (49) - [FEET 1+2)+e-uri -3] - [1'14 00603 831-2) + eil?oc* 03) - [e' (4-1080) i lov z) = 25 (+126*} --) 1. Sa is the expectation value of the state ( 52). K< +2 11 41+>? [?]+\2-21146+T* [?] y even 1" [$]+ 1 cent to P 1-?] - O

Therefore the expectation value of Sz dosen't change by time Next, we consider 1(*2 1/4 (+)))[1+2 ) 1-2)]. [eilico(00) 1+2)721702090441 = (1,) 1 é?l} Wocos og+7 et [+wocoso]t "I(+* 119(E)) 1² = cos [z Cocos Ot] Similarly 1-21 14C+2>p- [51+z)-)-z]. [eliocoo]} 1+z)et (4 10 credit )-a] Illint) I sest lż wo cos o] +7 Š (1-2) 5 Let I 10 COS Oj+] - 1<21/4(+)373-i [ sin [ { ao cos ou] Then the expectation value is, (52)-|{+ x 11416)>1[*]+K-211 412311-17 al cos[{ wocos 0 t]\? ]+ sin(t woros ol]} 1-1 :.(S2). cos (Wo los ot]

The expectation value of Sy contains yields, ( 93) -|C+y114(+»>/* 74]+\2-41/4 (4) >> ++] | ' * sin (10C6 OL))* [ * 3 *) *570 (0 603 00271-27 (Sy) - sin (we cos ot)