Question

4. 10 points The Spin operators for a spin-1/2 particle can be described by the Pauli matrices: 0 1 0 0 ,02= 0 -1 1 ¿ a) Write the normalized eigenvectors of Oz, I+) and 1-) which are defined such that 0z|+) = 1+) and 0z1-) = -1-), as column vectors in the same basis as the Pauli matrices given above. (You can assume without loss of generality that these eigenvectors are real.) (3 pts) b) Consider an eigenvector (V) = alt) + b1-). Assuming that a is a real number, osa<1, show that b = ei® /1 - a?, where o is an arbitrary angle. (3 pts) c) Find the expectation values of Oz, Oy and o, in the state ) in terms of a and 0. (4 pts)

Answer 1

(a) Oz () 00) 0 1+) -> (6) 17 alt) + bl-> 14) is normalised 2 o's a att bl a & Given that a is real a² 2 al 2 2 at lbl 2 161² = 1-a² =) Ibl 1-a² π > b = e Nl-a² ted )

CLASSMate Date Page (0) Do It) (4) & H 603 - <+1 O oxlt) (i 071) 071-) = -17 (0) = (0) = 1) 0 Tx lt) (-1 Oxla) = (0) (0) = (4) = 1) Q O 0 = Ogla) It) a Oult) = ( 10 = (0) - i() = ily ... こ Sylt) oyl-) (ii)0) = (-1) = -f(1) =-ilt) Dyl-) - Filt) Tox * (a <+1+62-1) (217) +61- >) (+1 + 5* G1) (a 1-) +b 17) aat (+/-) + at b <+1+) + ba (+-) + b belt) & ? म M * abtab -a a is real & bidra afbtb)

classmate Date Page (0x) Leid - a2 + eida) a t e it te ant- leit (O) za 1-a² cos (6) (y) = LX 4 <4oyly) a Atb Hoy og (alt) +677) la (+1 + b 1) lail) iblt). in a (+1-3-ia b [+/+) tiba (-+-) - ibb (-17) ♡ * * ata -i ab tiba = ile b-ba) -i (ab-ba) =-ia (6-6) ele id Via) IL -ial e l-a e 70 =-2 a 1-a² sin (6)

classmate Date Page (2) = (dal TEL) (alt) + ->) (1) (alt) t JI>) 세 +b y tbt that a Atb ¥ lal 와비 ㅕ 0 0 e)