Quantum Mechanics: Concepts and Applications · Problem 3.17

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · energy-measurement-expectation-value

Problem

Consider a physical system whose Hamiltonian and initial state are given by H=E0(1−10−11000−1),∣ψ0⟩=16(112),H = \mathcal{E}_0 \begin{pmatrix} 1 & -1 & 0 \\ -1 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}, \quad |\psi_0\rangle = \frac{1}{\sqrt{6}} \begin{pmatrix} 1 \\ 1 \\ 2 \end{pmatrix}, where E0\mathcal{E}_0 has the dimensions of energy. (a) What values will we obtain when measuring the energy and with what probabilities? (b) Calculate the expectation value of the Hamiltonian ⟨H^⟩\langle \hat{H} \rangle.

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