Plotting a Radial Probability Function

[Jimmy25]

I'm trying to plot the radial probability function for a hydrogen atom.

I have the function itself (Psi²*4*pi*r²) my problem is that when I plot the function with angstroms on the x-axis, the y-values are larger than they should be (they look about right if I divide them by the bohr radius in angstroms).

Here's what it should look like when plotted:
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hydr.html

I can't figure out what I'm doing wrong here!

 

[kuruman]

Look at the abscissa. It is in dimensionless units of r/a₀.

 

[Jimmy25]

I still am not seeing why I would have to multiply the y-axis by a_o nor do I understand what the form of the numbers on the y-axis take (I have looked at several resources, they all give different descriptions of the y-axis).

 

[kuruman]

I assume you wish to reproduce the hyperphysics plot, not others. Look at the y-axis. It represents a probability distribution that is dimensionless. dP is dimensionless, but dP/dr is not. Starting from the expression for dP, you multiply by the Bohr radius and divide by dr to get a dimensionless expression. Whatever is left on the right hand side is what you plot as the probability distribution.

 

[Jimmy25]

Okay, now I can see how they got there.

However, I still am very confused about the units in all these functions. I suppose the source of my confusion is at the wave function itself. When I plot the wave function for a 1s hydrogen orbital does it have any associated units? What about psi squared?

 

[kuruman]

The wave function does not have the same dimensions as the probability distribution. The probability distribution is dimensionless; the wave function has dimensions such that (in 3-d) ψ^*ψdV is dimensionless. Maybe that's what is confusing you.

 

[Jimmy25]

I've been trying to make sense of these curves (attached).

Fig 3-4 is the wave function and probability density. Fig 3-5 is the probability distribution.

I don't understand what units they are using on the y-axis. The probability density must be in P/a_o³. I don't understand what they're using in the wave function or distribution curve.

I've been working on this for several hours and am pulling my hair out because it seems like it should be simple but I just don't get it.

Source: http://www.chemistry.mcmaster.ca/esam/Chapter_3/section_2.html

 

[kuruman]

Look at the first plot showing the ground state wave function. The caption says that the function is in atomic units. This means that in the equation \psi(r)=\frac{1}{\sqrt{\pi} a^{3/2}_{0}}e^{-r/a_0} you must set a₀=1. Do that and evaluate the function at r = 0. Does what you get match what you see plotted for the value of the wave function? Square that. Do you get the value that the probability plot shows for r = 0?

 

[Jimmy25]

So, the y-axis of the first curve should really be labelled psi/a_o^-3/2, and the y-axis of the second curve should really be labelled P/a_o^-3. Correct?

 

[kuruman]

So, the y-axis of the first curve should really be labelled psi/a_o^-3/2, and the y-axis of the second curve should really be labelled P/a_o^-3. Correct?

Either that or say what they say, that the wave function is in atomic units.

 

[Jimmy25]

Okay, I think I got that part.

But back to my original question. You said:

dP is dimensionless, but dP/dr is not. Starting from the expression for dP, you multiply by the Bohr radius and divide by dr to get a dimensionless expression.

I see how they got to their solution but I'm a bit confused as to why they multiplied by a_o. When I integrate the function that has not been multiplied by a_o from zero to infinity I get 1. However, when I integrate the function that has been multiplied by the bohr radius I get the bohr radius. Why don't they just leave it and plot dp/dr?

 

[kuruman]

Because they want to plot a probability distribution and probability distributions are dimensionless. dP/dr has dimensions of L^-1; multiplying it by a₀ makes it dimensionless.