Question

Consider the state of a spin-1/2 particle 14) = v1o (31+z) + i] – z)) where | z) are the eigenstates of the operator of the spin z-component $z. 1. Show that [V) is properly normalized, i.e. (W14) = 1. 2. Calculate the probability that a measurement of $x = 6x yields 3. Calculate the expectation value (Šx) for the state 14) and its dispersion ASx = V(@z) – ($()2. 4. Assume that the spin is placed in the magnetic field oriented along z axis such that the spin is described by the Hamiltonian H = woŜz. Assume that the system is prepared in the state ) at t = 0. Derive the probability to find the system in [) at later times: P(t) = [(#(0)¥(t)}|2 . Write down this probability for t = 5. Calculate (Šx(t)) = (y(t)|$x[¥(t)). Make sure you check that (Sx(0)) is consistent with the previous calculations. TT 211 - and WO wo

Answer 1

for spin & particle x for s = 12 sz= 1 / 2 ( for ta direction) 104 - direction) 14 yosto ( 31+27+ 11-27] (a) showe lu> is normalised T4147= 1 (to check) <41= £ [341 -1 <-21] Tylpyt [3<z-3<-zl] [31z7+al-zY] + [942/ 2Y + <-21-27 (Iter and 1-27 are ortho normal {as (2/27= 1 and anz|czy = 1] lates) X4147 = 1 [9+1 10 K4147 = 1

sol- (s) Calculate the meaurement of & h or yeilds the 147= L [31+ xy + il-z7] Je [this is the eigenstate corresponding 1+27= 1: to S4= 1-27 = (i {eigenstate consesponding to sx = -1/2 = L Sx. te [ (6) + 4(0)] to 1:1 Now, we will try to site it in the eigenstater of Eigennector of S = [1] for de the Eigenrector of Sx = √ [1] for t= - the 147 con le tol 1) - 911) b conresponding to tulz Cossesponding =G. to the

3 JO J2 and. A Add and 2 3ti Vio IGI 3ti 120 Probability that measurement of St yields this 191² 3ti 3-i 20 Ans 10 20 100 = =

They 20 234z1(833) 1z7 +1 <-z1 ss3)-z>] to [9 <z/s4z7 – 9<z15,2\z+ (-2/541-27 - <-z1 s -z] 9x 3 h² - 9X 2 * ² + 371 2 - the 9x2 + ² + 2 t2 2+² = I 20 ㅗ 20 4 Ixlox 2 t² 20 4. LS27 = 1th 2 So, AS x = 5 SXY - TSxY AS x = It 2 - (0) 4 Aspe = 2

190 P(A) = 144 (0)14(A) 71² He cose -twoszt /h - 14(o)re 14ct) 4 (o) e 'Etlh = 1<4(0) 14 (0)7 e i wosztlh 12 At t= I ധാ and 21 Probability will remains same (that is 1). This is because y is an eigenfunction of Sh. 80, its probability miell be time independent.

e-letto vy ly(t) >= 14loy H = wosz. (ginen) 1963) - ono (0) + Hol) ty (t)) = -iwost iwoszt + t t viö () (i) e Vio 3 e i e iwost for as here it is pointing -z direction -iwoszt ly(t)) = wowo По sagard expectatin value Sx at time it Ty Lt) I år 14 (t)) SNISAA ¿co o szt /h iw Sztlth 30 rie og TO STO of Asal 1xd [ 3e voletto iesselhalie i voseth 5(212&e coming sering Be toaleta . Liwoszthh WAT AA di wo fetth 3ie il - Bie 20

to 20 ziwo z t/th sile Szt/h enih é -20 szt /h you x 2i maofined culo 2i Togel transand =-ħ x 3 sin lliuret 10 lo I SA a Sul 10 ħ imes.) = -3h sin 12iwosat). It is be creator melap juntion of represents rotation about t-aris and this is knowen as precession As calculated before a Sx> at t= 0 = 0. from h also putting t=0 t=0 gives 24x7=0 This shoues Se is a time. This