Units being used in the graphs for ψ and radius (in nm) and ψ^2 and ra

[tennispro1213]

We've just started learning about ψ in the quantum mechanics section of our atomic structure chapter. So while reading, I found these graphs where they used certain units along the y-axis for the graphs for ψ and radius (in nm), and ψ^2 and radius.

What units are these exactly?

My chemistry teacher wasn't able to give a reasonable explanation, and trying to learn about it online has left me overwhelmed with all sorts of complicated formulae.
So please help, and don't forget: this is the first time I'm reading about all this.

 

[mfb]

The units of ψ are probably $$nm^{-\frac{3}{2}}$$
If you square it and integrate it over the 3D volume, the result has to be dimensionless (1, if properly normalized). So the unit of ψ² is just $\frac{1}{nm^3}$.

 

[Yanick]

My limited understanding of wavefunctions is that you cannot physically interpret a wavefunction in and of itself. The square of a wavefunction is interpreted in a probabilistic manner such that the square of the absolute value of psi can be interpreted as the probability of finding the particle in a region (x + dx). You can read a bit more about it here. You can see in your attachments that the psi² gives some peaks which are areas where you are most likely to find the particle. Additionally you have many areas where the probability of finding the particle is non-zero, so you can find the particle in those regions some of the time but less often. This is the reason why you may have been hearing about "electron clouds" and such and the spooky nature of the quantum world. I'm sure others may have better answers for you.

 

[mfb]

My limited understanding of wavefunctions is that you cannot physically interpret a wavefunction in and of itself.

You can.
Interpretations of quantum mechanics
In some of them, the wavefunction is a real thing.

 

[LudusRex]

The units being used in the graphs for ψ and radius (in nm) and ψ^2 and radius are likely representing the wave function (ψ) and its square (ψ^2) in units of nm^-1 (inverse nanometers) and nm^-2 (inverse nanometers squared), respectively. These units are commonly used in quantum mechanics to represent the spatial distribution of an electron's probability density in an atom. The radius, in this case, is also likely measured in nanometers (nm).

It is normal to feel overwhelmed when first learning about quantum mechanics, as it is a complex and abstract topic. I would suggest seeking out additional resources, such as textbooks or online lectures, to gain a better understanding of these concepts. It is also important to practice using the equations and units in various problems to become more comfortable with them. With time and practice, you will become more familiar with these units and their meanings in the context of quantum mechanics.