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The spin of an electron is described by a vector = (3) and the spin operator S = S₂i+ Ŝj + Sk with components S₂ -1 (16). Sy 1 (17), $ - 21 ( 19 ) ħ 01 ħ 0 -i ħ 0 10 2 i 0 0-1 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find S₂ = ±ħ/2 in a measure- ment of $₂. (iii) Give the general expression of the expectation value (Ŝ₂). [ (b) (i) Calculate the commutator [S,, S₂]. State whether S, and S, are simultaneous ob- servables. (ii) Calculate the commutator [S, S²], where S² = S²+S+S2. State whether S, and $² are simultaneous observables. [ 1 (1) is a normalised eigenstate of Ŝ₂ and determine the √2 (c) (i) Show that the state = associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of Ŝ, provided (3) (iii) Calculate the expectation values (S), (Sy) and (S₂) in the state . 1 the system is in the state & = E

The spin of an electron is described by a vector = (3) and the spin operator S = S₂i+ Ŝj + Sk with components S₂ -1 (16). Sy 1 (17), $ - 21 ( 19 ) ħ 01 ħ 0 -i ħ 0 10 2 i 0 0-1 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find S₂ = ±ħ/2 in a measure- ment of $₂. (iii) Give the general expression of the expectation value (Ŝ₂). [ (b) (i) Calculate the commutator [S,, S₂]. State whether S, and S, are simultaneous ob- servables. (ii) Calculate the commutator [S, S²], where S² = S²+S+S2. State whether S, and $² are simultaneous observables. [ 1 (1) is a normalised eigenstate of Ŝ₂ and determine the √2 (c) (i) Show that the state = associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of Ŝ, provided (3) (iii) Calculate the expectation values (S), (Sy) and (S₂) in the state . 1 the system is in the state & = E

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A4. The spin of an electron is described by a vector : 01 Suj + S₂k with components ST (18), Sy 10 ħ 2 = 24 &₁ ħ 2 1 √2 1, and the spin operator Ŝ = Ŝ₂i+ 0 0 (17) d), §. = 1/(1-9). i 0 2 0 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find S₂ = ±ħ/2 in a measure- ment of $₂. (iii) Give the general expression of the expectation value (Ŝ₂). [6] (b) (i) Calculate the commutator [Sy, S₂]. State whether S, and Ŝ, are simultaneous ob- servables. (ii) Calculate the commutator [S, $²], where S² = S²+²+S2. State whether S, and $² are simultaneous observables. [6] is a normalised eigenstate of S, and determine the (c) (i) Show that the state associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of Ŝ, provided 1/4 the system is in the state = - 5 3 (iii) Calculate the expectation values (Ŝz), (Sy) and (S₂) in the state . [8]
Transcribed Image Text:A4. The spin of an electron is described by a vector : 01 Suj + S₂k with components ST (18), Sy 10 ħ 2 = 24 &₁ ħ 2 1 √2 1, and the spin operator Ŝ = Ŝ₂i+ 0 0 (17) d), §. = 1/(1-9). i 0 2 0 (a) (i) State the normalisation condition for . (ii) Give the general expressions for the probabilities to find S₂ = ±ħ/2 in a measure- ment of $₂. (iii) Give the general expression of the expectation value (Ŝ₂). [6] (b) (i) Calculate the commutator [Sy, S₂]. State whether S, and Ŝ, are simultaneous ob- servables. (ii) Calculate the commutator [S, $²], where S² = S²+²+S2. State whether S, and $² are simultaneous observables. [6] is a normalised eigenstate of S, and determine the (c) (i) Show that the state associated eigenvalue. (ii) Calculate the probability to find this eigenvalue in a measurement of Ŝ, provided 1/4 the system is in the state = - 5 3 (iii) Calculate the expectation values (Ŝz), (Sy) and (S₂) in the state . [8]
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