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.

Say we start with a model $W$ and eigenstates $|w\rangle$, $|u\rangle$ and then decide to make it “more realistic” and add interaction terms, making for a new Hamiltonian $W'$. Usually $W'=W+qI$, where $q\in{\mathbb R}$ is a scalar called couple constant. One now said this describes an interacting system, but that's a relative notion: All Hamiltonians like $W$ or $W'$ that one considers are hermitean and thus diagonalizable, i.e. there are states $|w'\rangle$, $|u'\rangle$ for $W'$ which are not interacting. The nomenclature basically just comes from sticking to the old states and this is mostly done because those are the ones one can compare.

So given a system with $W'$ is e.g. in an eigenstate $|u'\rangle$ with oscillation frequency $u'$ (“total energy of the system”), then expressed as superposition of $|w\rangle$ and $|u\rangle$ (making for a state that undergoes an back and forth according to $\Delta_{wu}$) we have a notion of total energy flowing between those non-eigenstates of the “interacting system”.

Vector space homomorphism

Normed vector space

Wikipedia: Quantum harmonic oscillator

Vector space homomorphism

Normed vector space