Note on Consistency of Complimentary Variables
This note is to explore the consistency of two forms of the Heisenberg Undertainty Principle. Our approach will be a cross between mathematics and hand-waving. What can go wrong?
The first form of the Uncertainty Principle is the best-known, with complimentary variables momentum p and position x:
ΔpΔx ≥ h/4π.
Can we show this to be consistent with the form with complimentary variables energy E and time interval t?
Let's take the original form and multiply the left side by Δt/Δt = 1:
ΔpΔt(Δx/Δt) ≥ h/4π.
Since Δx/Δt can be interpreted as speed v,
vΔpΔt ≥ h/4π. *
Here we take a detour to consider energy E. If we consider particles of kinetic energy T in free space or with a relatively constant potential energy V, we can write the change in energy as
ΔE = Δ(T + V) = ΔT + ΔV = ΔT
since we are treating V as constant. The classic expression for kinetic energy in terms of momentum p and mass m is T = p2/2m, so
ΔE = ΔT = Δ(p2/2m) = 2pΔp/2m = pΔp/m.
But if p = mv, then p/m = v, so
ΔE = vΔp.
So what? So, remember that before our detour, we were at this expression derived from the Unvertainty Principle for the complementary variables p and x:
vΔpΔt ≥ h/4π. *
We can replace that vΔp with ΔE, giving us
ΔEΔt ≥ h/4π.
Well, how about that?
There's more, thanks to QuarkNet fellow Rick Dower:
- Interested in the relativistic case? Try this Heisenberg and Energy.
- Learn about how single-slit diffrection of particles yields transverse Δp: Heisenberg and Diffraction,