The Quantum Botanist

July 22, 2026

What is a Wavefunction?

In quantum mechanics, the wavefunction is thought to be the complete description of a quantum system. There are debates about whether the wavefunction is complete or not (see my post on the EPR Paradox for more details), but for this post, we’ll assume that it’s complete. This means that, with the wavefunction of a system and any external forces, we can predict the reality of the system at any future point. There are a few requirements for a wavefunction to be valid. We can view the wavefunction as a statistical distribution, kind of. When we square it, we get a probability distribution function. For the purposes of this post, we’ll be assuming a one-dimensional wavefunction, but everything can be applied to more.

Valid Wavefunctions

For a wavefunction to be valid, there are a few requirements. First, we need the wavefunction to go to 0 as $x$ goes to infinity. If we think about this requirement logically, we can see why. Let’s assume that the wavefunction does not go to 0. Now, when we square it, we get a probability density function. For a probability density function to be valid, its integral must be 1, otherwise, we would be saying that the probability of any event occuring is over 100%, which wouldn’t make any sense. If a function does not go to zero as $x$ goes to infinity, then the integral diverges, and we would be saying that the probability of, for example, an electron being anywhere in the universe, is over 100%, which clearly makes no sense. Therefore, we must conclude that the wavefunction must go to 0.
Next, we need the square of the wavefunction to be normalizable. This means that we need to have a number we can multiply it by that will make the integral 1. This goes back to the thought of it as a probability density function. We have already eliminated the possibility of it being infinite, and therefore we only need to be concerned about the integral being 0, as any other number can be multiplied by its inverse to result in 1. Allowing the squared wavefunction to be 0 would imply that the electron whose position we’re measuring does not exist anywhere in the universe, which would not be a useful equation to have.

Common Wavefunctions

Commonly, wavefunctions will be either sinusoidal or exponential functions. In general, they are functions that will satisfy the wave equation as well as the requirements above. Frequently, the wavefunction will combine sinusoids and an exponential decay, or it will have a sinusoid for one region, then exponential decay in other parts. For a more in depth look at a derivation of a wavefunction, you can also check out this post on the square potential well.