6.3: Example - Oscillation in a fully mixing two state system
( \newcommand{\kernel}{\mathrm{null}\,}\)
Consider the expectation value of a quantity S in a system which has two non-degenerate energy eigenstates |1⟩ and |2⟩, and where the Hermitian operator ˆS is defined by ˆS|1⟩=|2⟩, ˆS|2⟩=|1⟩.
The general state can be written:
|ϕ⟩=c1 exp(−iE1t/ℏ)|1⟩+c2 exp(−iE2t/ℏ)|2⟩
if we assume real c1, c2 it follows that the expectation value ⟨ˆS⟩ will be:
⟨ˆS⟩=⟨ϕ|ˆS|ϕ⟩=[c1eiE1t/ℏ⟨1|+c2eiE2t/ℏ⟨2|][c∗1e−iE1t/ℏ|2⟩+c∗2e−iE2t/ℏ|1⟩]=c1c2[eiω21t+e−iω21t]=2c1c2cos(ω21t)
Thus the expectation value of ˆS oscillates in time at frequency ω21=(E2−E1)/ℏ. This arises because ˆS is not compatible with the hamiltonian, and hence does not define a constant of the motion.