The spin of an electron is described by a vector [psi] = mat([psi_up],[psi_down]) and the spin operator S = S x i +S y j

The spin of an electron is described by a vector [psi] = mat([psi_up],[psi_down]) and the spin operator S = S x i +S y j +S z k with components S x = (h/2)*mat([0,1],[1,0]), S y = (h/2)*mat([0,-i],[i,0]), S z = (h/2)*mat([1,0],[0,-1]). a)i) State the normalisation condition for [psi]. ii) Give the general expressions for the probabilities to find S z =+-(h/2) in a measurement of S z . iii) Give the general expression of the expectation value . b)i) Calculate the commutator [S y ,S z ]. State whether S y and S z are simultaneous observables. ii) Calculate the commutator [S x , S 2 ], where S 2 = S x 2 + S y 2 + S z 2 . State whether S x and S 2 are simultaneous observables. c)i) Show that state [phi] = (1/sqrt(2))*mat([1],[1]) is a normalised eigenstate of S x and determine the associated eigenvalue. ii) Calculate the probability to find this eigenvalue in a measurement of S x , provided the system is in the state [phi] = (1/5)*mat([4],[3]). iii) Calculate the expectation values , , in the state [psi]. Image with better formatted question is attached! Transcribed Image Text: The spin of an electron is described by a vector p:
and the spin operator S = S,i+
(? ).
0 -i
0 1
1 0
1
0.
Sj+ S,k with components S,
Sy
i
1
(a) (i) State the normalisation condition for v.
(ii) Give the general expressions for the probabilities to find Sz = +h/2 in a measure-
ment of S.
(iii) Give the general expression of the expectation value (S.).
(b) (i) Calculate the commutator [Šy, Š]. State whether S, and S, are simultaneous ob-
%3D
servables.
(ii) Calculate the commutator [S„, Š³], where S = S? + S? + S?. State whether S, and
S° are simultaneous observables.
(c) (i) Show that the state o
V2
is a normalised eigenstate of S, and determine the
associated eigenvalue.
(ii) Calculate the probability to find this eigenvalue in a measurement of S, provided
1(4)
53
the system is in the state
(iii) Calculate the expectation values (S), (Š,) and (S;) in the state y.

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