Quantum Mechanics: Concepts and Applications · Problem 3.23

One-Dimensional Problems — Nouredine Zettili

Physics — Quantum Mechanics · two-level-system-energy-time-evolution

Problem

Consider a system whose Hamiltonian is

H = \mathcal{E} \begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix}

, where

\mathcal{E}

is a real constant with the dimensions of energy. (a) Find the eigenenergies,

E_1

and

E_2

, of

H

. (b) If the system is initially (i.e.,

t = 0

) in the state

|\psi_0\rangle = \begin{pmatrix} 1 \\ 0 \end{pmatrix}

, find the probability so that a measurement of energy at

t = 0

yields: (i)

E_1

, and (ii)

E_2

. (c) Find the average value of the energy

\langle \hat{H} \rangle

and the energy uncertainty

\sqrt{\langle \hat{H}^2 \rangle - \langle \hat{H} \rangle^2}

. (d) Find the state

|\psi(t)\rangle