Question

please do questions g and h... ONLY G AND H

The three spin operators for an electron (which is a spin-1/2 particle) are $. - 1 (1 :). $=(: ;), $- (-). Suppose the electron is pinned in space but is subject to a magnetic field B = (0,0,B), so that its Hamiltonian H = -1B-S = - BS. Suppose an initial state of the electron is prepared so that (0)) = (?) a. Show that (0)) is a unit vector, i.e., that ((O) (O) = 1. b. Find (t)), and show that (()) = 1. c. What is the probability that a measurement of S, at time t yields a value 1/2? d. What is the probability that a measurement of S, at time t yields a value -ħ/2? e. What is the probability that a measurement of S, at time t yields a value ħ/2. f. Is it possible to perform a simultaneous measurement of any two of these spin operators at any point in time? 8. Calculate (S.), ($), and (S.). h. If a measurement of S, at time t is made and yields the value -ħ/2, what is the state vector immediately after the measurement is made? i. Find the state vector at a time t' later than the time t at which the measurement in part f was made, assuming that no measurements are made in the interim.

Answer 1

we hime three spin operatox fin= t 108) 2 h 20 have three ] and entral State function 2 LA 1410) = 1 in How Sn>= <40 15h/%) (12)56,06172 173) * $ 1912 19 (63) = 5 x2122 $12 This il is required Expectation value For <sys <p>* * (2) * L 14 (13) 2593 = 5 [A21 i) = 5 (- 421 +in2 ] = 6 <ses =oThij e' required Expectation halua 2525 = ta (2) 6,772 103) = (9217[ 23) - [7-1)= <S2y = - 5 Tore in is required Expectation value Scanned with CamScanner

2 2 If < = -th what is state nector ?_ Immediately Z5y467fylt > Syly+ = + + luz Fox -t § 14.3 = - 14- Now 197 = 1/1 277 / 12 -> thostattu) Now requined State Vector SAPNA Teacher's Signature : Scanned with CamScanner