A Lesson in Collapsing

In this lesson, I'd like to address a common misconception about quantum mechanics.

To explain this misconception, I need to describe two of the most basic ideas in quantum mechanics - the way the wavefunction encodes probabilities and uncertainties, and the collapse of the wavefunction upon measurement.

Let's take as an example an electron in an atom. There are many energy levels (or more precisely, orbitals) that the electron can occupy, but lets just imagine that it is in the lowest possible energy state, called the ground state. The electron's wavefunction is often called an electron cloud, to help us imagine the fact that the electron behaves as a wave. The electron is not just at one point in space, but, like sound waves or water waves, it is at many places at once. Its position is uncertain. (If you think that the electron's position is uncertain, just because we don't know exactly where it is, please read here before continuing).

If we measure precisely where the electron is, we will certainly find it somewhere, and the wavefunction tells us the probability of finding the electron at various positions. Where the wave has a large amplitude (dark in the images below), we are relatively likely to find the electron, but don't expect to find it where the wave amplitude is small (light in the images below).


If we make a measurement and find the electron somewhere, it is no longer in many places all at once like it used to be. This is called the collapse of the wavefunction, when a measurement causes the wavefunction to change suddenly - in this case, changing from an electron cloud in many places all around the atom to a very localized electron that we found in a very definite place.

The common misconception is that, now that the wavefunction has collapsed, the uncertainty is gone. Now we know where the electron in the ground state is, ta-da! But, no, that is not true. We have gotten rid of (or made very small) the uncertainty in position, but now the electron is not in the ground state energy level. In fact the energy level of the electron is uncertain.

Why is that? Well, each energy level possible for the electron in the atom has a corresponding electron-cloud wavefunction, as shown in the images above. An electron with a precise position does not have a wavefunction that looks like any of the electron clouds in the images (or any not in these images, for that matter), therefore it is not in any of the possible energy levels. Instead it is in a superposition of many energy levels, and its energy is uncertain.

Measurements which collapse a wavefunction make the measured quantity quite certain, but at the expense of making other quantities less certain. This is one application of the Heisenberg Uncertainty Principle (HUP), which, in one form, states that the product of the uncertainty in position and the uncertainty in momentum must be at least as large as Planck's constant divided by 4pi. A wavefunction collapse that took away all uncertainty - in both position and momentum, for example - would violate the HUP. If the position is very, very certain, its uncertainty is very, very small. That means the uncertainty in momentum must not be very, very small, too, or the product of the two won't be larger than Planck's constant divided by 4pi.

So that's the lesson of the day - no wavefunction collapse ever makes all uncertainty go away. Quantum systems always have uncertainty, in at least some quantities.