I’ve heard electrons referred to as probability density functions, and I’ve also heard the term as a standalone entity. What does it mean?
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Probability density functions are just the way you calculate probability density.
Probability density is a way to figure out the likely value for a discrete outcome. In the case of electrons, it’s a matter of “is the electron here or not”. To be clear, the probability density isn’t the electron itself, but rather describes the region of space you’re most likely to encounter an electron based on it’s energy. If you plot the probability density for an electron, you get a couple of distinctive types of shapes, which we refer to as “orbitals” and label s, p, d, and f based on their specific shapes.
Probability is how likely a thing is to be true. When you flip a coin, the chance of it being heads is 50%, so we say probability=0.5
For things that exist in finite discrete states, we can define the probability of each state (in this case, p(heads)=0.5, p(tails)=0.5)
But what about something that does not have to exist in a discrete state? What if the question was if I throw a dart, where will it hit? I am not particularly good at throwing darts, so it probably won’t be exactly where I aim for. If you want to know the odds of me hitting one particular point, the probability of that is practically zero. However, if you ask me how likely I am to hit within a certain range, then my probability starts to increase. If you think about it, the larger the area you give me, the more likely I am to hit within that area. Probability density is a way of describing how much my odds of hitting a region are if you increase that region by a certain amount.
If the probability density of a certain location is 0.02/cm^2, that means that adding one square cenitmeter will increase my odds of hitting by 2%, or adding 0.5cm^2 will increase my odds by 1%, or adding 2cm^2 will increase my odds by 4%, etc.
However, not all spots are equally likely. I am, for example, more likely to hit an area that is close to where I am aiming than one directly behind me. The probability density of the area in front of me is larger than the probability density of the area behind me. We therefore call it a Probability Density Function because the probability density depends on where you are looking. If we make it a mathematically nice function then there are a bunch of things we can do with it that will let us calculate probabilities pretty easily.
For the example of the electron, one aspect of quantum mechanics is that things are never in one exact location. But if you measure a region they will either be there or not be there. The probability density of the electron now describes how much the probability of getting a ‘yes’ answer increases when you make the volume a little bigger.
Also for the electron, the rules of quantum mechanics describe how the PDF* of that electron evolve over time. Some PDFs are stable, so the electron will happily sit there forever without changing. An example of this would be the electron orbitals of an atom.
*Technically we usually say that the rules of QM describe how the wave function evolves, but the PDF is a result of the wave function.
Simply put, it’s the likelihood of something being in a certain range of samples within a group.
For example, adult humans range in height from 4′ to 8′. You would use a probably density function to determine the likelihood of someone being over 7′ tall, for example. Or between 6′ and 6’3″.
In more mathematical terms, it’s the area under two points of the curve of a probability distribution.
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