Demystifying h and h-bar

h was the beginning of quantum physics.  It represents the indivisible quanta of light.  To get a photon's energy, you multiply its frequency by h.  The frequency is, mathematically, the inverse of the wavelength.  In other words, h represents the circumference of a circle, the amount of ink you would use drawing one wavelength.

All images are as found on the internet.  Credit to whomever made the things in the first place.  I can't draw.

That's easy enough.  But what is h-bar (ħ)?  It is h divided by 2π.  That makes ħ the radius of the circle.  (The circumference of a circle is 2πr.)  In other words, ħ is the null-to-peak amplitude of a photon.  Which photon?  Every photon.  That was the insight that eventually led to quantum physics.

While we're here, the Heisenberg uncertainty theorem states that the uncertainty in the momentum of a particle, times the uncertainty in its position, cannot be less than ħ/2.  In other words, their product cannot be less than half the amplitude of a photon.  Which certainly makes sense, because you couldn't possibly measure anything less than that size by any means.  Notice that this distance is minuscule!  So when you hear people say that you can't know the position and momentum at the same time, they're talking nonsense.  Of course you can.  But there is an absolute limit to the ability to measure something of that ridiculously small scale of precision.  (Notice we're not talking about absolute size, we're talking about the precision of a measurement.)  Not just because of the tooling involved, but because it really doesn't make any sense to talk about distances smaller than ħ/2, because a circle smaller than that is less than the amplitude of a photon, and there is no such measurement.  Not with any tools.  Not even conceptually.  In addition, the minimum arises because when you work out the math, at the smallest scales, the uncertainties (precision) in position and momentum sort of end up being reciprocals, just like frequency and wavelength.  The less you have of one, the more you get of the other.

Contemplate and compare this relationship with the minimum sampling rate of digital audio, which must be at least twice the maximum frequency of the audio signal being sampled.  If you tried to sample at a lower rate, the result would have no relationship to the original signal.