Quantum Mechanics: Concepts and Applications · Problem 3.7

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · quantum-measurement-matrix-operators

Problem

Consider a system whose Hamiltonian HH and an operator AA are given by the matrices H=E0(1−10−11000−1),A=a(040401010),H = \mathcal{E}_0 \begin{pmatrix} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \quad A = a \begin{pmatrix} 0 & 4 & 0 \\ 4 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}, where E0\mathcal{E}_0 has the dimensions of energy. (a) If we measure the energy, what values will we obtain? (b) Suppose that when we measure the energy, we obtain a value of −E0-\mathcal{E}_0. Immediately afterwards, we measure AA. What values will we obtain for AA and what are the probabilities corresponding to each value? (c) Calculate the uncertainty ΔA\Delta A.

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