1. The Hamiltonian operator H for a certain physical system is represented by the matrix (1 0 0) H = hw 0 20 002 while two other observables A and B are represented by the matrices A = 1 0 0 0 0 21 (2u 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 cy are complex constants and (i) find the relationship between c, c2 and c3 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 vectorlu For each possible result find the wave function in the matrix representation immediately after the measurement.
1. The Hamiltonian operator H for a certain physical system is represented by the matrix (1 0 0) H = hw 0 20 002 while two other observables A and B are represented by the matrices A = 1 0 0 0 0 21 (2u 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 cy are complex constants and (i) find the relationship between c, c2 and c3 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 vectorlu For each possible result find the wave function in the matrix representation immediately after the measurement.
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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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