Heisenberg uncertainty principle

location or momentum

The Heisenbery uncertainty principle (HUP) is one of those things from Physics that, to me at least, is really odd.

In this post I'll try to explain what it is, and also why we might not really need it!

This theorem was an attempt to try and figure out what quantum mechanics actually means and has become part of what is called the  Copenhagen interpretation of quantum mechanics.

This is how odd HUP is, take the title of the paper itself

 "Ueber den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik".

which roughly translates as


"On the anschaulich content of quantum theoretical kinematics and mechanics"

See the word anschaulich, as no direct translation into English and by choosing different translations actually introduces a certain uncertainty into the title of the paper!

The HUP was discovered by Heisenberg in 1927, the idea being that on an atomic scale we cannot know the momentum of an object and its location simultaneously. To understand HUP it is necessary to understand what was just said in the previous sentence, well the bolded bit.

So, thought experiment, imagine an electron and its momentum is known, in other words, we know its mass, we have applied a known force to it for a certain amount of time, so we know its momentum. Let's say that it starts of stationary. We apply a force, we know

F = m a,  which is the same as F/m = a

so if we know F and m we can calculate a. We also know that

v = a t ,

where v is velocity, a is acceleration and t is time. But hold on one moment, we can replace "a"

v = (F t)/m , which becomes

mv = F t , but mv is just the definition of momentum. So if the particle starts out stationary we apply a force for t seconds then its momentum is just F multiplied by t (ignoring relativistic effects, this is being done a low velocities).

A little while later someone says, were has that electron got to? So we decide to take a look, using something like light or by firing other particles at it. This is were we run into trouble according to Heisenberg. By trying to measure were the particle is, using the technique described, the interaction will alter the momentum by an unknown and indeterminable amount. So while we may now have some idea of where it is, we no longer know its momentum accurately.

What Heisenberg is saying here is that this inability to know position and momentum is a fundamental property of quantum mechanics. It is not the case that we are poor experimenters.

That is it, that is the Heisenberg Uncertainty Principle.

A year later a guy called Kennard proved the theorem to give us

Δ_ψp Δ_ψq ≥ ℏ/2

the equation that is now associated with the HUP. Over the years it seems to me that this idea has come to be accepted without people actually stopping to think what it means.

By taking a second measurement it is possible to work out the change in momentum from the first measurement. What this implies is that the HUP is not true of the past. Heisenberg also says that before taking the second measurement we don't know the momentum. Furthermore, after the measurement we no longer know the momentum.

As with many things in physics, HUP, while being one of the most famous aspects of quantum mechanics, is not the only player. In 1932 Dirac, while pondering the positron was also dreaming up  an alternative idea to Schrödingers and Heisenberg's work. This lay mostly forgotten until a visiting physicist called Jehle told Richard Feynman about it in 1941.

The Dirac-Feynman idea is based on path integrals and will be covered in another post. It turns out to be equivalent to Schrödingers and Heisenbergs ideas. So it will contain the HUP.

Is it the best of the three? No one really knows. Each has its place and each can be used to solve problems that are difficult using the other theories.

Interestingly Dirac-Feynmans theory was mostly ignored until the 1970s because it was generally hard to work with. Schrödingers and Heisenberg's work offered more productive alternatives especially to undergrads. Then a Russian paper used the idea to advance Quantum Field Theory and since then it has made a serious come back.

This raises the question of whether we will still need HUP if Dirac-Feynman's approach gains more acceptance over time? Will we still be discussing it in 30 years? Who knows?

I can't help thinking that there is probably another solution out there that is even more elegant than those proposed so far. Once this is discovered maybe all three will be retired!