# What is meant by the "probability density" of finding an electron?

Could someone please explain to me in layman's terms what probability density of finding an electron means, just as probability means chances of finding an electron. With due respect, please don't answer the formula or the distribution curves as I understand and can plot those, I just need the definition.

• Probability density means probability per unit volume, ie. If you multiply probability density with volume then you will get the probability of finding ē in that volume.
– JM97
Commented Sep 16, 2016 at 15:29
• The formula IS the definition.
– Greg
Commented Aug 21, 2020 at 2:43

In quantum mechanics we often need to find the probability at some position $x$. However, $x$ is a continuous variable with an infinite number of values so it does not make sense to ask what is the probability of being at some exact position , say $x=1/4$, as this will be vanishingly small. Instead we imagine that the probability is that of being at position x to $x + \mathrm{d}x$ where $\mathrm{d}x$ has some very small value. If $f(x)\,\mathrm{d}x$ is that probability then $f(x)$ is the probability density which must be a real non-negative number. It turns out that quantum mechanics postulates that the probability density is given in terms of the wavefunction is $|\psi|^2$.

In quantum mechanics, we clearly know that electron isboth a wave and a particle.Due to this characteristic feature of the electron, being a wave it can exist at numerous places at an instant. So the probability distribution function in relevance with the Schrödinger wave equation are the chances of finding an electron in the region which needs to be studied.

So imagine you have a friend who's an alcoholic, and your phone is broken. You know he's not home, so he's probably at his favorite bar. If you had an aerial map of the area and had to guess where he was, you'd draw a dot on his favorite bar stool [most likely place he is], a circle around the bar building [still could be here], and a dotted line encompassing all of the shrubs, taco truck, and parking lot around the bar [he MIGHT be getting a taco, beat up in the parking lot, or passed out in the bushes].

So in this little parable, your friend is an electron and you have drawn a probability density.

Extra credit: the reason we have to use a probability density is that, for complex mathematical reasons [Heisenberg uncertainty principle] you can't know your friend/electron's velocity AND position, so you're making a mathematical guess about where your friend/electron is, and then describing that mathematical guess in terms that are confusing to us plebes. But really it's just a heat map. Also, depending on what element/number of electrons/orbital you're dealing with, you can get different shapes. Don't be confused - it's just like knowing more friends each with their own different bar, and they don't like each other so they don't overlap.

• Forgot to mention - the name for the mathematical description of the probability density is the schrodinger equation, in case anyone throws that at ya. Commented Aug 21, 2020 at 0:42
• This is certainly a colorful answer, but might not be the most appropriate for SE ;) Commented Aug 21, 2020 at 14:09