Question

Problem 3.46 The Hamiltonian for a certain three-level system is represented by the matrix $H = hbaromega egin{pmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2 end{pmatrix}$. Two other observables, A and B, are represented by the matrices $A = lambda egin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 2 end{pmatrix}$, $B = mu egin{pmatrix} 2 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 end{pmatrix}$, where $omega$, $lambda$, and $mu$ are positive real numbers. (a) Find the eigenvalues and (normalized) eigenvectors of H, A, and B. (b) Suppose the system starts out in the generic state $|S(0) angle = egin{pmatrix} c_1 \ c_2 \ c_3 end{pmatrix}$, with $|c_1|^2 + |c_2|^2 + |c_3|^2 = 1$. Find the expectation values (at $t = 0$) of H, A, and B. (c) What is $|S(t) angle$? If you measured the energy of this state (at time $t$), what values might you get, and what is the probability of each? Answer the same questions for observables A and for B.

          Problem 3.46 The Hamiltonian for a certain three-level system is represented by the matrix
$H = hbaromega egin{pmatrix} 1 & 0 & 0 \ 0 & 2 & 0 \ 0 & 0 & 2 end{pmatrix}$.
Two other observables, A and B, are represented by the matrices
$A = lambda egin{pmatrix} 0 & 1 & 0 \ 1 & 0 & 0 \ 0 & 0 & 2 end{pmatrix}$, $B = mu egin{pmatrix} 2 & 0 & 0 \ 0 & 0 & 1 \ 0 & 1 & 0 end{pmatrix}$,
where $omega$, $lambda$, and $mu$ are positive real numbers.
(a) Find the eigenvalues and (normalized) eigenvectors of H, A, and B.
(b) Suppose the system starts out in the generic state
$|S(0)
angle = egin{pmatrix} c_1 \ c_2 \ c_3 end{pmatrix}$,
with $|c_1|^2 + |c_2|^2 + |c_3|^2 = 1$. Find the expectation values (at $t = 0$) of H, A,
and B.
(c) What is $|S(t)
angle$? If you measured the energy of this state (at time $t$), what values
might you get, and what is the probability of each? Answer the same questions for
observables A and for B.

Problem 3.46 The Hamiltonian for a certain three-level system is represented by the matrix
H = hbaromega eginpmatrix 1 0 0 0 2 0 0 0 2 endpmatrix.
Two other observables, A and B, are represented by the matrices
A = lambda eginpmatrix 0 1 0 1 0 0 0 0 2 endpmatrix, B = mu eginpmatrix 2 0 0 0 0 1 0 1 0 endpmatrix,
where omega, lambda, and mu are positive real numbers.
(a) Find the eigenvalues and (normalized) eigenvectors of H, A, and B.
(b) Suppose the system starts out in the generic state
|S(0)
angle = eginpmatrix c1 c2 c3 endpmatrix,
with |c1|^2 + |c2|^2 + |c3|^2 = 1. Find the expectation values (at t = 0) of H, A,
and B.
(c) What is |S(t)
angle? If you measured the energy of this state (at time t), what values
might you get, and what is the probability of each? Answer the same questions for
observables A and for B.

Added by Maria L.

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