# Chapter 3: Wave Function

Welcome to Chapter 3 of our Quantum Explainers Series! If you are inspired dive deeper into this topic, check out the Quantum Shorts Contest. You can take inspiration from this article (we will even hint at potential topics that you can make your video about) and provide your own take on wave functions for the contest!

By Albert Zhu

April 14, 2024

Take a look at the objects around you. Everything has a definite position: your laptop is a certain distance away from your eyes; your hand is in a certain position from your keyboard; and even if you wave your hands around in a chaotic motion, they will always have a definite position.

The position of objects in the macroscopic world around us is certain. However, that is not the case for objects at the quantum scale (e.g. electrons). While we often visualize electrons as tiny particles orbiting the nucleus of an atom, in reality this is not the case!

Visualization of electron orbiting nucleus (not physically correct).

How, then, should we think about the motion of electrons? We use a mathematical tool called a “wave function”, which describes the probability of finding an electron in a certain position in space. [Editor’s note: this is the definition of the wave function in the “position basis.” While we often think about wave functions in this way, there are other fun bases for us to define a wave function!]

Think of it this way: imagine you are playing hide and seek with your friends at a playground and you are the seeker. When you get ready to look for your friends, you don’t know where any of them are (unless they are really bad and standing in plain sight, but let’s assume they aren’t for now). However, you probably have an idea of where they might be hiding and thus a plan for which spots to check. The wavefunction is similar to your plan as a seeker: it contains the probabilities of finding an electron in a certain place. Once you find your friend though, you know with 100% certainty where he/she was hiding–this is akin to measuring the location of the electron, also known as “wave function collapse.”

Above: Probability distributions of finding your friend at different locations at the park. Below: Wave functions are probability distributions of finding an electron at different positions, here we drew it in 1D along a direction which we call "x."

To be more precise, a wave function actually contains complex-valued probability amplitudes of finding an electron in a certain position, which are different from probabilities. Unlike a probability, which is always a real number, a probability amplitude is a complex number (i.e. it may contain an imaginary part). We can solve for the probability by squaring the probability amplitudes given by the wavefunction.

The reason why we work with these complex probability amplitudes is because they allow us to describe interference, which is when the amplitudes of waves either add up or cancel each other out. You may recall that very small particles like electrons have wave-like properties and can thus interfere with each other, leading to a very high probability of finding the electron in a certain position and a very low or zero probability of finding it in other positions. The act of squaring the wavefunction is the mathematical way of mimicking interference.

Returning to the example of an electron orbiting an atomic nucleus: what is the wave function picture of its motion? Electron orbitals!

Sketch of atomic orbitals for an atom with one electron, with the probability of finding the electron at a location indicated by the darkness of the shading. The P orbitals are the same, just rotated from each-other.

Orbitals are the atomic wave function of electrons with a particular energy. For example, if an electron orbiting Hydrogen is in the lowest energy state (i.e. the ground state), it is in the S orbital. The electron prefers to be close to the nucleus, but does not prefer a particular direction. Promoting the electron to a higher energy (the first excited state), it will occupy a P orbital. In the Pz orbital, the electron prefers to be along the z direction along two lobes.

We will conclude on an interesting note. The uncertainty of an electron’s location (say in the S orbital) isn’t due to our lack of knowledge. Rather, the electron is in a superposition state of position (see Chapter 2) — the electron actually isn’t actually anywhere in particular! This is where the analogy to hide-and-seek falls apart; while your friends are definitely hiding somewhere, the electron is not hiding anywhere in particular. This last concept is one reason why quantum mechanics is so puzzling yet so interesting to many people, and we hope that it intrigues you to learn more as well!