I was motivated by a comment on Jim Pivarski’s recent post to speak about the Heisenberg Uncertainty Principle. Someone asked,
If uncertainty in quantum mechanics comes from (or is inseparable from) quantization, then where does it come from in its mathematical formulation i.e in terms of a space and its Fourier transform?
The Heisenberg Uncertainty Principle is a curious fact: it requires no physical intuition whatsoever and yet has profound physical ramifications. It is also interesting because it is among a small group of facts which are both physically and mathematically interesting. It is an important (to harmonic analysis) and commonly known fact that a function and its Fourier transform cannot both be compactly supported. There are stronger statements than that, though, of the following flavor: if a function is a narrow spike near a point, then its Fourier transform will be more spread out. The Heisenberg Uncertainty Principle is a quantitative statement about this kind of fact.
The Principle follows from several simple but fundamentally powerful aspects of the Fourier transform. First, polynomials in derivatives acting on a function can be pulled outside the Fourier transform into corresponding polynomials in frequency variables. More specifically,
where is a polynomial and means derivative with respect to Second, translation in space variables leads to modulation in frequency variables:
Third, modulation in space variables leads to translation in frequency variables:
Now, let’s derive the Uncertainty Principle. Let denote the functions which are square integrable — that is, We will write
On , define the operators and by
Here is Planck’s constant which is on the order of in the macroscopic units of Joules-seconds. The presence of is unimportant mathematically, so let’s assume we’re using units so that is 1. That corresponds to the position operator of a particle comes from the interpretation of the wave function: if is a solution to the Schrödinger equation, then it is a probability distribution — or rather, is — so that gives the expected value for position. That corresponds to the momentum operator basically comes from the fact that it’s like the velocity of the wave function, and momentum is mass times velocity. More precisely, the Schrödinger equation,
is nothing more than a statement about conservation of energy. The kinetic energy is which corresponds to the first term, and so our choice for the momentum operator reflects that.
One should note here that, as indicated above, taking the Fourier transform of momentum gives the position operator in the frequency variable:
The Fourier transform of position is basically the same thing as momentum, but it’s off by a constant multiple.
Observe that and do not commute; specifically,
We know that has as an inner product . Also, it’s pretty easy to see that both and are self-adjoint, meaning that and likewise for . Using these two facts, we find that for arbitrary real constants and ,
The first quantity is non-negative which means the last quantity is also, and so:
Picking and gives
For any fixed pair of numbers and , we can apply the above relation to and deduce that
The second term in the product on the left side of above comes from a fact called Plancherel’s formula:
For any wave function , we must have that since must be a probability distribution and thus have a total mass of 1. By choosing
the expected position, and
the expected momentum, we get that
where denotes variance — the variance is the square of the standard deviation and so for an arbitrary distribution the square root of variance is the standard way to measure deviation from the mean.
More commonly, the above relation is written as
By carefully retracing how Planck’s constant fits in, one can see that the physically relevant equation is
From a physical point of view, this means that our confidence in a measurement of position is at best inversely proportional to our confidence in a measurement of momentum. This gives us the usual qualitative interpretation of the Uncertainty Principle: one cannot simultaneously know with perfect certainty both position and momentum, or, as Heisenberg himself said, “The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.”