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-.A physical system is described by a Hamiltonian operator A, 0 Â = (¦ ¯ia) (ia where a is a constant. The energy eigenvalues of Ĥ are E₁ = α and E₂ = -α. (a) Find the normalised eigenvectors of Â, [1) and [2), corresponding to eigenvalues E₁ = α and E₂ = -a, respectively. (b) A spin polarised along the positive z direction is injected into the system in the following initial state |(t = 0)) = () = c₁|1) + c₂|2) (i) Determine the values of the expansion coefficients c₁ and C₂. (ii) Calculate the probability that an energy measurement at time t = 0 gives th result E = E₁. (c) Use your result from (b)(i) to determine an expression for the spin state [½(t)) a time t > 0.

-.A physical system is described by a Hamiltonian operator A, 0 Â = (¦ ¯ia) (ia where a is a constant. The energy eigenvalues of Ĥ are E₁ = α and E₂ = -α. (a) Find the normalised eigenvectors of Â, [1) and [2), corresponding to eigenvalues E₁ = α and E₂ = -a, respectively. (b) A spin polarised along the positive z direction is injected into the system in the following initial state |(t = 0)) = () = c₁|1) + c₂|2) (i) Determine the values of the expansion coefficients c₁ and C₂. (ii) Calculate the probability that an energy measurement at time t = 0 gives th result E = E₁. (c) Use your result from (b)(i) to determine an expression for the spin state [½(t)) a time t > 0.

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1. A physical system is described by a Hamiltonian operator A, Â = (₁ ¯ia) ia 0 where a is a constant. The energy eigenvalues of Ĥ are E₁ = α and E₂ =-α. (a) Find the normalised eigenvectors of Â, 11) and [2), corresponding to eigenvalues E₁ = α and E₂ = -a, respectively. (b) A spin polarised along the positive z direction is injected into the system in the following initial state |4(t = 0)) = (1) = c₁|1) + c₂|2) (i) Determine the values of the expansion coefficients C₁ and C₂. (ii) Calculate the probability that an energy measurement at time t = 0 gives th result E = E₁. (c) Use your result from (b)(i) to determine an expression for the spin state [(t)) a time t > 0.
Transcribed Image Text:1. A physical system is described by a Hamiltonian operator A, Â = (₁ ¯ia) ia 0 where a is a constant. The energy eigenvalues of Ĥ are E₁ = α and E₂ =-α. (a) Find the normalised eigenvectors of Â, 11) and [2), corresponding to eigenvalues E₁ = α and E₂ = -a, respectively. (b) A spin polarised along the positive z direction is injected into the system in the following initial state |4(t = 0)) = (1) = c₁|1) + c₂|2) (i) Determine the values of the expansion coefficients C₁ and C₂. (ii) Calculate the probability that an energy measurement at time t = 0 gives th result E = E₁. (c) Use your result from (b)(i) to determine an expression for the spin state [(t)) a time t > 0.
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