todo

Given one system in isolation and a corresponding Hamiltonian with a gap (positive non-zero eigenstate), the (ratios of the) differences between eigenvalues is more relevant than the value of the eigenvalues themselves. This is because we may be able to rescale our units so that $w$, e.g. the lowest eigen-frequency, equals $2\pi$.

Now given eigenstates

$W\,|w\rangle = w\,|w\rangle$

$W\,|u\rangle = u\,|u\rangle$

Write the eigenvalue $u>w$ as

$u = w + \Delta_{wu}$

the evolution is described by individually non-measurable oscillations ${\mathrm e}^{-iw}$ resp. ${\mathrm e}^{-iu}={\mathrm e}^{-iw}{\mathrm e}^{-i\Delta_{wu}}$ and what's interesting is the second factor, capturing how much faster the second state oscillates.

For

$|\psi\rangle = c_w\,|w\rangle + c_u\,|u\rangle$

the action is

$W\, |\psi\rangle = w\,|\psi\rangle + \Delta_{wu}\,c_u\,|u\rangle$

The rate difference $\Delta_{wu}$ (the value at which the second rate/frequency $u$ exceeds the first one, $w$) determines how quickly a state $|\psi\rangle$ developes away from itself.

And then

$e^{itW}\, |\psi\rangle = e^{itw}(|\psi\rangle + (e^{it\Delta_{wu}}-1)\,c_u\,|u\rangle)$

Up to a phase, $(e^{it\Delta_{wu}}-1)$ generates oscillations about $|\psi\rangle$. This is somewhat related to Rabi-oscillations.

Vector space homomorphism

Normed vector space

Wikipedia: Quantum harmonic oscillator

Vector space homomorphism

Normed vector space