Note: All of the following questions refer to the system for which the Hamiltonian operator is Ĥ = -yBoŜ₂. In all of the

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Note All Of The Following Questions Refer To The System For Which The Hamiltonian Operator Is H Ybos In All Of The 1 (60.09 KiB) Viewed 74 times

Note: All of the following questions refer to the system for which the Hamiltonian operator is Ĥ = -yBoŜ₂. In all of the questions below, x(t)) is the state you calculated above and assume that the coefficients a and b in the state are non-zero unless mentioned specifically otherwise. 9. Which one of the following gives the correct outcomes of measuring S, in the state x(t))? (a) ħ/2 with a probability aeBot/2 and -ħ/2 with a probability be-Bot/2 (b) ħ/2 with a probability a²eyBot and -ħ/2 with a probability b²e-Bot (c) ħ/2 with a probability |a2 and -ħ/2 with a probability b2. (d) ħ/2 and -ħ/2 with equal probability. 10. Consider the following conversation between Andy and Caroline about measuring Ŝ, in the state Ix(t)): • Andy: Since the probability of measuring ħ/2 is |a|², -ħ/2 is |b|² and |a|² + |b|² = 1, we can choose our a and b as a = e cos(a) and b = ei sin(a) where a, 1 and 2 are real numbers. • Caroline: I agree. Since cos² (a) + sin²(a) = 1 it gives the same relation as a2 + |b|² = 1 and there is no loss of generality. Do you agree with Andy and Caroline? (a) Yes. (b) No. 11. Calculate the expectation value of Ŝz in the state x(t)). Express your answer in terms of a defined earlier as a = e¹¹ cos(a) and b = ei sin(a).