In classical mechanics, electrons find themselves in classical states in which they have both definite positions and definite momenta. See, for example, Figure 2 in Chapter 1. The situation is different in quantum mechanics. An electron cannot be in a quantum state in which it has both a definite position and a definite momentum.

Consider an electron that has a definite momentum. It is represented by a wavefunction that is a definite-momentum eigenfunction such as the one shown in Figure 12, left. The function looks like this infinitely far in both directions. This is one end of the uncertainty principle spectrum: if an electron has an exact momentum, then it is in an equal superposition of all positions. We are entirely uncertain about its position.

In Chapter 5, we created a wavefunction for which the position of the electron was roughly contained within ±50. That wavefunction is reproduced in Figure 12, center. We did this by superposing eleven different definite-momentum eigenfunctions with momenta ranging from 1.9 to 6.1. The superposition resulted in destructive interference for positions less than −50 and greater than 50. Here we see that in order to reduce the uncertainty in position, we have to increase the uncertainty in momentum.

What if we want to know the position of the electron more precisely? For the wavefunction shown in Figure 12, right, the position of the electron is roughly contained within ±3.5. To make this happen, we have to superpose definite-momentum eigenfunctions with a range of momenta that is about 15 times the range of momenta used for the wavefunction in Figure 12, center. We see that in order to further reduce the uncertainty in position, we have to further increase the uncertainty in momentum.

To create a wavefunction representing an electron with a definite position (which would be a definite-position eigenfunction), we would have to superpose definite-momentum eigenfunctions including all values of momenta. This is the other end of the uncertainty principle spectrum.

This is the essence of the uncertainty principle, which was first proposed by the physicist Werner Heisenberg in 1927. As an equation, this principle states that

ΔxΔp ≥ ℏ ∕ 2

where
Δx = the uncertainty in position
Δp = the uncertainty in momentum
ℏ = h ∕ 2π

Rearranged, this equation shows that Δx can never be smaller than ℏ/(2Δp). Since the value of ℏ is very small, this uncertainty only becomes important when we probe the position of things on very small scales.

The uncertainty principle tells us how precisely we can know position and momentum at the same time. It is entirely possible to at one time know the precise position, then at a later time know the precise momentum, and vice versa. Suppose an electron is represented by any old wavefunction, then we measure the electron’s position. The wavefunction would collapse to a definite-position eigenfunction, and we would know the electron’s precise position. At the moment of position measurement, the uncertainty in the electron’s momentum would be enormous. But then, a short time later, we could perform a momentum measurement. Now the wavefunction would collapse to a definite-momentum eigenfunction, and we would know the electron’s precise momentum. At the moment of momentum measurement, the uncertainty in the electron’s position would be enormous.