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Question

# Consider a two-state quantum system with a Hamiltonian$H \doteq\left(\begin{array}{cc} E_1 & 0 \\ 0 & E_2 \end{array}\right)$Another physical observable $A$ is described by the operator$A=\left(\begin{array}{ll} 0 & a \\ a & 0 \end{array}\right)$where $a$ is real and positive. Let the initial state of the system be $|\psi(0)\rangle=\left|a_1\right\rangle$, where $\left|a_1\right\rangle$ is the eigenstate corresponding to the larger of the two possible eigenvalues of $A$. What is the frequency of oscillation (i.e., the Bohr frequency) of the expectation value of $A$ ?

Solution

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In this problem, we are given a two-state system with a Hamiltonian

$H=\begin{bmatrix}E_1&0\\0&E_2\end{bmatrix}$

Let's consider an observable given as

$A=\begin{bmatrix}0&a\\a&0\end{bmatrix}$

If the system is in the state $\ket {a_1}$ we want to know the Bohr frequency of the expectation value of A.

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