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,