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1. The Hamiltonian operator H for a certain physical system is represented by the matrix (100) H= ha 020 (002 while two other observables A and B are represented by the matrices O A O A = 20 0) (o o 21) (2µ 0 0 B-0 04 where à and u are real (non-zero) numbers. If the system is in a state described by the state vector where c, cz and cz are complex constants and (i) find the relationship between cj, cz and cz such that uis normalised to unity; and (ii) find the expectation values of H, A and B.

1. The Hamiltonian operator H for a certain physical system is represented by the matrix (100) H= ha 020 (002 while two other observables A and B are represented by the matrices O A O A = 20 0) (o o 21) (2µ 0 0 B-0 04 where à and u are real (non-zero) numbers. If the system is in a state described by the state vector where c, cz and cz are complex constants and (i) find the relationship between cj, cz and cz such that uis normalised to unity; and (ii) find the expectation values of H, A and B.

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Quantum Mechanics 
Parts i, ii, iii

1. The Hamiltonian operator H for a certain physical system is represented by the matrix 100 H = hw 020 002, while two other observables A and B are represented by the matrices A = 200 0 0 21 -) -- 2u 0 0 B = 0 0u O0 u 0 where à and u are real (non-zero) numbers. If the system is in a state described by the state vector where c, c2 and c3 are complex constants and (i) find the relationship between c1, cz and cz such that uis normalised to unity; and (ii) find the expectation values of H, A and B. (iii) What are the possible values of the energy that can be obtained in a measurement when the system is described by the state vector u? For each possible result find the wave function in the matrix representation immediately after the measurement.
Transcribed Image Text:1. The Hamiltonian operator H for a certain physical system is represented by the matrix 100 H = hw 020 002, while two other observables A and B are represented by the matrices A = 200 0 0 21 -) -- 2u 0 0 B = 0 0u O0 u 0 where à and u are real (non-zero) numbers. If the system is in a state described by the state vector where c, c2 and c3 are complex constants and (i) find the relationship between c1, cz and cz such that uis normalised to unity; and (ii) find the expectation values of H, A and B. (iii) What are the possible values of the energy that can be obtained in a measurement when the system is described by the state vector u? For each possible result find the wave function in the matrix representation immediately after the measurement.
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