= Three-level observable and missing spectral line
Consider an original ideal <quantum system> with positive scales $\epsilon,d$ and
$$
H=\epsilon\begin{pmatrix}0&0&0\\0&2&0\\0&0&5\end{pmatrix},\qquad D=d\begin{pmatrix}0&1&0\\1&0&2\\0&2&0\end{pmatrix}.
$$
Interpret $D$ as one component of an <electric dipole moment>, so its coupling to the corresponding <electric field> is the <electric-dipole interaction> $-\mathcal E(t)D$. The gaps are $2\epsilon,3\epsilon,5\epsilon$, but $D_{02}=0$: this interaction has no direct first-order $2\leftrightarrow0$ coupling. The nonzero squared <matrix elements> are $|D_{01}|^2=d^2$ and $|D_{12}|^2=4d^2$. These are coupling weights; actual <spectral line> intensities also depend on populations and radiation factors. In particular, there is no promise that an experiment populates the upper state of every allowed transition.
\Image[/matrix-mechanics-transitions-and-elements.png]
{title=<Matrix mechanics>: three <energy levels>, an <observable> array and two nonzero coupling weights}
{description=Original ideal three-state model. The gap $5\epsilon$ exists even though its <electric dipole> <matrix element> vanishes. The plotted weights are squared couplings, not measured emission intensities.}
A measurement of $D$ itself has possible outcomes $0,\sqrt5d,-\sqrt5d$, since
$$
\det(\lambda I-D/d)=\lambda(\lambda^2-5).
$$
These are <eigenvalues> of $D$, not the <energy eigenvalues> $0,2\epsilon,5\epsilon$. Moreover, $(D^2)_{02}=D_{01}D_{12}=2d^2$ although $D_{02}=0$: two factors connect through $|1\rangle$. This algebraic statement does not turn $D_{02}$ into a nonzero direct coupling. Higher-order <probability amplitudes> for transitions require the appropriate dynamical calculation.
Back to article page
