Its about time . note
Note
This entry is concerned with the notion of time and is a spinoff of On physical units . note.
Observation
A core assumption of physics that's easy to adopt is that we are able to enumerate some moments in an ordered fashion.
$t_1, t_2, t_3, ...$
Usually they are considered as point on some real axis, but I guess the only thing that really matters is that this enables us to count (other) sorts of events between such moments. If indeed the moments correspond to real numbers, $t_i, t_j$ say, then $\dfrac{1}{t_j-t_i}$ is what we want to call a frequency and as such a unit for all kinds of rates.
Energy as frequency
In On physical units . note we consider the Schrödinger equation
$h_\Psi\cdot\dfrac{{\mathrm d}}{{\mathrm d}t}\Psi = (-2\pi{\mathrm i})\,H\,\Psi$
and note that if $h_\Psi$ is chosen unitless, the conserved quantity of the system (what is also called energy) is a frequency. The cost associated with a process is basically inversely proportional to how quick it takes place, so that Reaching higher frequencies is more expensive. But for a more precise desciption in QM, one must look at the state spaces in the models. A Hamiltonian $H(\langle q,p\rangle)$ is like an $\langle q,p\rangle$-indexed list of prices. A Hamiltonian operator $H$ acting on state vectors in QM is like a stamp with prices.
To understand many formulas, units like e.g. mass ($kg=s/m^2$, after setting $\hbar$ unitless) can or should be multiplied by natural constants (e.g. $c^2$) so that they represent a frequency or time scale.
Canonical variables
If
${\mathcal H} \propto {\dot q}\frac{\partial {\mathcal L}}{\partial {\dot q}} \equiv {\dot q} \, p$
is supposed have the unit of a frequency, then
$[p]=\dfrac{1}{[q]}$.
That's also clear from
$[q,p] \propto \hbar$.
List of quantities that multiply to frequency/energy
(this is a temporary list for me to read formulas in terms of units)
Electrical field $[E]=\dfrac{1}{[t][x][q_x]}$ and dipole moments $[d]=[x][q_x]$.