# Probability distribution of an electron

What are the different methods to describe the probability distribution of an electron within an atom? I don't at all understand the difference between the radial wave function and the radial probability function.

P.S: Please keep in mind that I have only just graduated high school so I'm not aware of very much advanced topics.

• I don't know what you mean by 'method to describe...'. And, you basically square the wave-function to get the probability density, see the Born rule. – lemon Aug 19 '16 at 10:47
• I mean the graphs! – user106570 Aug 19 '16 at 10:49
• The graphs of the orbitals are something you should never look at if you want to do physics. – gented Aug 19 '16 at 10:51
• @Gennaro Tedesco: Oh, really? Why not? And bad luck but I've got to learn the basics of those things for an upcoming exam. Never mind the graphs. If u or somebody else could just explain exactly what ψ(r) and ψ(θ, ϕ) are, that would be great! – user106570 Aug 19 '16 at 10:56
• @GennaroTedesco I disagree. It's important to understand the symmetry, and the easiest way for many people is to look at a picture. – garyp Aug 19 '16 at 11:17

If u or somebody else could just explain exactly what ψ(r) and ψ(θ, ϕ), that would be great.

The functions that are used to describe the orbitals of the electron are, because we are using a spherical co ordinate system, dependent on the variables $r, \phi, \theta$. These can be expressed as the product of two functions , one based on $r$ and the other Spherical Harmonic Functions based on ($\phi, \theta$).

So the radial wave function is the part of the overall function that describes exactly what it says, the radial distance from the nucleus.

Functions that satisfy Laplace's Equation are often said to be harmonic, hence the name spherical harmonics.

Just bear in mind that, despite their name, they don't necessarily give you a spherical orbit, as you can see below. In this picture, $r$ is taken to be a given value.

And the first few are:

Image Source: en.citizendium.org

• No no, I didn't know this! You see, it's taught to us that ψ² gives the location of the electron and that we can solve the Schrödinger's equation to obtain values for the first 3 quantum numbers. Nothing more is taught but the questions are so much more trickier, and mostly out of our syllabus. I wasn't even aware that we use a spherical coordinate system. What are these three variables, can u say? – user106570 Aug 19 '16 at 12:17
• Its up to you, but I think you should probably make sure you get your course material learned off first. The S.E. is calculated in the cartesian coordinate system, for 1 D problems and then, when you need to study the H atom, you switch to spherical co ordinates. You can use any co ordinate system you want, but you obviously use the coordinate system best suited to the problem. Stand on the ground, straight up/down is $r$. If you fall left or right, thats $\phi$, if you fall forwards or backwards, that's $\theta$. They are orthogonal, (at right angles, like x, y, z), to each other. – user108787 Aug 19 '16 at 12:30
• Oh, okay, so those are the three directions(so to speak) in the spherical coordinate system? And what's the definition of the "radial wave function"? What does it give? – user106570 Aug 19 '16 at 12:32
• The clue is in the name :). It's an indication of "how far away" the electron is, but this is just a picture you learn for the moment, it's more complicated than that. Google H atom, and read the answers you've got so far. – user108787 Aug 19 '16 at 12:36
• Oh, I see, OK. If the radial wave function gives how far away the electron is, what do the other functions give? – user106570 Aug 19 '16 at 12:38

The wave function is an abstract quantity that is related to the probability, but not equal to it. You obtain the probability by finding the square modulus of the wave function. $$P(r) = \Psi(r)^*\Psi(r)$$

More correctly, what we have is the probability density. That is, $P(r)\,r^2\mathrm{d}r$ is the probability that the particle will be found between $r$ and $\mathrm{d}r$.

• $r^2 P(r)dr$ gives back the actual probabilty (keep in mind to integrate against the Jacobian determinant $r^2$). – gented Aug 19 '16 at 11:29