Heisenberg's Uncertainty Principle

Section 3.3 Heisenberg's Uncertainty Principle

Okay, back to Figure 3.1. We have already commented that a particle moving with a well-defined momentum \(p\) has a wavefunction \(\psi(x)\) that is a pure sine wave, as in Figure 3.1(a). But where is the particle when it is in this state? The answer: it doesn't have a well-defined position. There is a non-zero probability density for finding the particle anywhere where the wave is non-zero. But a pure sine wave with a definite wavelength extends all the way from \(x = -\infty\) to \(x = +\infty\text{.}\) So a particle with a well-defined momentum could be found anywhere. Phrased another way, the spread in the particle's location is infinite. So, if the momentum of a particle is defined precisely, the particle's position is completely undetermined — it could be found anywhere. (And remember: it isn't just that we don't know where the particle is located, but rather that the particle simply doesn't have a location.)

But what if the particle's wavefunction looks like the one in Figure 3.1(c)? In this case, \(\psi(x) = 0\) everywhere except in a narrow region where the sharp blip is. So, its position is well-defined and predictable. It is theoretically possible to make an arbitrarily narrow pulse-like wavefunction, i.e., with a spread in position which is arbitrarily small. But what is the wavelength — and therefore the momentum — of this particle?

Formally, we use the symbol \(\sigma_x\) to denote the spread in a particle's position and \(\sigma_p\) to denote the spread in a particle's momentum.¹ For waves, there is an important theorem called Fourier's theorem that says that any function \(f(x)\) — which might be describing the height of a water wave or the pressure of a sound wave as a function of position or a quantum wavefunction \(\psi\) — can be decomposed into a sum of sines and cosines.² We won't prove it here, but Fourier's theorem shows that a pulse-like wave with \(\sigma_x = 0\) can be produced by adding sine waves with wavelengths ranging from 0 up to \(\infty\text{.}\) But since each wavelength corresponds to a different momentum, that means that a particle with a well-defined position (i.e., \(\sigma_x = 0\)) could have any momentum from infinity down to zero (i.e., \(\sigma_p = \infty\)).

So, a particle can have a well-defined momentum (but completely uncertain position) or a well-defined position (but completely uncertain momentum). There is a third possibility: Figure 3.1(b) shows a wavepacket that is reasonably (but not perfectly) localized and one that can be decomposed into the addition of sine waves with a limited (but non-zero) spread of wavelengths. In this case, the particle is neither perfectly localized nor does it have a perfectly determined momentum, but neither \(\sigma_x\) nor \(\sigma_p\) are infinite either.

A strict application of Fourier's theorem to matter waves shows that there is a minimum total spread in the particle's position and momentum:

\begin{equation} \sigma_x \sigma_p \geq \frac{\hbar}{2}\text{,}\label{eq_heisenberg_uncertainty}\tag{3.14} \end{equation}

where \(\hbar \equiv h/2\pi\text{.}\) This is known as the Heisenberg uncertainty relation. It is a profound result that caused Newton to roll over in his grave³, because it says that we cannot simultaneously know a particle's position and momentum. A precise position and a precise momentum would mean both \(\sigma_x\) and \(\sigma_p\) are zero, but that is not allowed.

Recall when we solved for the motion of an object in classical mechanics, say a blow dart shot straight upward, we needed to know the initial position of the dart and the initial velocity. The Heisenberg uncertainty relation says that is not possible. The more precisely you are able to determine the position of some object, the less you will be able to know about its momentum.

Now, \(\hbar\) is a really small constant. For distances and momenta on our macroscopic scale, Heisenberg's uncertainty places no practical limitations. But down at the atomic scale the Heisenberg uncertainty relation has a very big impact. And, of course, experiments have confirmed the uncertainty relation, so it is not just a proposal but it is part of reality.

Many people, when encountering the uncertainty relation for the first time, assume that this is a statement about the limits of our ability to do experiments. But that is not the case. It's a property of nature obeyed by all matter, even when we're not looking. As we shall see, the Heisenberg uncertainty relation is essentially the reason that atoms are stable.

Mathematically, \(\sigma_x\) is the same as the standard deviation

\begin{equation} \sigma_x = \sqrt{\langle x^2 \rangle - \langle x\rangle^2 }\text{.}\tag{3.12} \end{equation}

where the averages \(\langle x\rangle\) and \(\langle x^2\rangle\) are defined as

\begin{equation} \langle x\rangle = \int x\, P(x)\, dx \qquad \langle x^2\rangle = \int x^2\, P(x)\, dx\tag{3.13} \end{equation}

with \(P(x)\) being the probability density defined in the previous section.

Fourier's theorem is really cool and forms the basis for the theory of music among other things. The basilar membrane in your ear performs an operation very close to a Fourier transform, with different portions of the membrane responding to different sound frequencies. This process helps you to identify, say, the individual sounds produced by flutes, cellos and bass guitars that are all blended together during a concert. And Fourier analysis is used to determine the characteristic “fingerprint” of different musical instruments and, in fact, is used by designers of electronic music synthesizers to mimic the sounds of real instruments.

Well, okay, maybe Newton didn't literally roll over in his grave, but this result is very different than anything envisioned by classical physics