Question

Problem 8.3 - A New Two-State System Consider a new two-level system with a Hamiltonian given by i = Ti 1461 – 12) (2) (3) Also consider an observable represented by the operator Ŝ = * 11/21 - *12/11: It should (hopefully) be clear that 1) and 2) are eigenkets of the Hamiltonian. Let $1) be an eigenket of S corresponding to the smaller eigenvalue of S and let S2) be an eigenket of S corresponding to the larger eigenvalue. (a) What are the possible results of an energy measurement in this system? What are the possible results of a measurement of the observable S? (b) Find the eigenkets of S in the energy basis. [Note: That is, find expressions for S1) and S2) in terms of the energy eigenkets 1) and 2).]

Answer 1

© # the 1961- du 12/28) Hy = <1*11) = : 11) - 13:30130 Note <min) = 5mm = { if man = < 1/10);> <l! - hve 132378°. the <b- the H22 = 20) f/2) = (a to 11) - $22) (277/23 tuo <213112940 0)27 <2/2) i f the = H21 = 0 .. secular can H-EI) = 0 7 Jowel (hes - E) ( - 2 E) = 0 Energy / Ei= to E= - + 7 Possible melyement of energy

Eignstalle l energy elgneton) og (9) men To whom man azo Normalisation 2307/23 * (*) </=(2, 0) 1 =4) =(,9 () -se? goes!] 1000S Eignsseite for Es, (2) = (0) Sid = 0, 980 After normalisation, 12) = 10 S SE5412201 - -1031) =( Note: this matrix , you can build as Ĥ. seculo y eah Is Allro 1-1 152 1

) it to = (4) dtet a = - temel big Eignrector (Eign kelt) of to, 18)= (x) + lu+iyo > x= -3y + y = in 19,7 = 63) (hy) = (1, -ix) 1= 26, 1f2) = (x,- ) = ² + 2 = m² x = 1/2 Eignfect his = 14) = (3) t-x+14) = 0 x = iy y=-11 Affe, normalisation 16) = + (!) s 194, +(!) 1 (1) + 119] the 19 +1 12>]

- Cou/ có 4 Hi ( kg) - Cho | Similarly 12404), ty W - HY - // - H 72 . A hthithet Que=che 20W) = (help e tat/2 Con I = Lino (c)! + Fra? (1) 4621? Cell + 4/1 3 1 2 411 ] F = (68 / ® CHIE [GEN!+ - (0780

00) = Coupon) - jan*) (...) I couleallien (ww.) o To covet) + Iain (38) ÛL) lehed To etiwala) oth o l eroth E DOH) = - o ! Hence Octs in a unitery operate <$) = 28,18/0,9 1,-1) ® = $(1,-1)(*) <$) +(5+) = & please do yourself (s) (t) = (210) 9/46) & d <S) (H/dd to
please raise your thumb up, thank you