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The wave function of the $2\mathrm{p}_z$ orbital is

$$Ψ = \frac{1}{4\sqrt{2π}}\left(\frac Z a\right)^{5/2} r \mathrm e^{-Zr/a}\cos θ.$$

I'm confused if this function will be a three-dimensional function or a four-dimensional function?

My professor told me that

$$z/r = \cos θ,$$

where $θ$ is the angle made by $r$ with the $z$ axis. So, I guess it will be four-dimensional since we would have to then plug in

$$r = \sqrt{x^2 + y^2 + z^2}$$

in the equation.

But when he plotted the function, he plotted a three-dimensional one! What is the correct answer and why?

cqfd
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coconutmercury
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1 Answers1

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Hydrogen-like orbitals are always 3-dimensional. But, since we can't directly visualize functions $\mathbb R^3\mapsto \mathbb R$, to plot them you need some trick.

One of such tricks it to take a cross section of the 3D space, e.g. xOz plane, and plot the function inside this plane. Then, knowing that the function is symmetric with respect to rotation around the $z$ axis, you can imagine what it could look like in other cross sections (or just plot these too). Here's an example:

xOz plot of the orbital from the OP

Another way is to draw the function in a density plot. This will look like a cloud of colored points, with their opacity depending on the function value. Example (here blue is negative, orange is positive):

density plot of the same orbital

Ruslan
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  • So while plotting wave functions, the first axis is x,the second is y and the third is $Ψ$ instead of z, right? – coconutmercury Jan 27 '21 at 04:51
  • @DylanRodrigues that all depends on what cross section you choose. One of the axes should indeed be $\Psi$, but the others are allocated by the one who is plotting. – Ruslan Jan 27 '21 at 06:20
  • I have a similar doubt in https://math.stackexchange.com/questions/4001487/doubt-in-graph-of-wave-function Please answer this too if possible – coconutmercury Jan 27 '21 at 06:24