#### Chemist20

Okay, this is a really basic question. I'm just learning the basics of QM now.

I can't wrap my head around the idea that the radial distribution function goes to zero as r-->0 but that the probability density as at a maximum as r-->zero. How can this be? they are opposite to each other!

Thanks!

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#### jtbell

Mentor
The probability density $|\Psi|^2$ gives you the probability per unit volume, of finding the electron in a little box dxdydz (rectangular coordinates) or $r^2 \sin \theta dr d\theta d\phi$ in spherical coordinates. For e.g. the ground state, this is maximum at the origin.

The radial probability distribution $4\pi r^2 |R|^2$ gives you the probability per unit radius, of finding the electron in a thin spherical shell with radius r and thickness dr. If you keep dr constant and decrease r, the volume of the shell decreases, which tends to decrease the probability also. As r approaches zero, the volume of the shell approaches zero, and so does the probability.

#### Chemist20

The probability density $|\Psi|^2$ gives you the probability per unit volume, of finding the electron in a little box dxdydz (rectangular coordinates) or $r^2 \sin \theta dr d\theta d\phi$ in spherical coordinates. For e.g. the ground state, this is maximum at the origin.

The radial probability distribution $4\pi r^2 |R|^2$ gives you the probability per unit radius, of finding the electron in a thin spherical shell with radius r and thickness dr. If you keep dr constant and decrease r, the volume of the shell decreases, which tends to decrease the probability also. As r approaches zero, the volume of the shell approaches zero, and so does the probability.
I don't really understand what you mean by "per unit volume" "per unit radius"... sorry. can you explain?

#### jtbell

Mentor
Assuming the probability density is uniform (constant) inside a box, you get the probability of the particle being inside the box by multiplying the probability density by the volume of the box

$$P = |\Psi|^2 V$$

which we usually think of in terms of an infinitesimally small box:

$$dP = |\Psi|^2 dxdydz$$

or

$$dP = |\Psi|^2 r^2 \sin \theta dr d\theta d\phi$$

If the probability density isn't uniform, we have to integrate a lot of infinitesimally tiny boxes.

Assuming the radial probability distribution Pr is constant between radius R1 and R2, you get the probability that the particle is in a spherical shell with inner radius R1 and outer radius R2 by multiplying the radial probability distribution by the distance between R1 and R2:

$$P = P_r (R_2 - R_1)$$

For an infinitesimally thin shell this becomes

$$dP = P_r dr$$

The difference between the two kinds of probability distributions is basically the volume of a thin shell: 4πr2dr.

If you want the probability that the particle is inside that infinitesimally thin shell, but you have the probability density and not the radial probability distribution, then you have to multiply by the volume of the shell:

$$dP = |\Psi|^2 4\pi r^2 dr$$

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#### billschnieder

Okay, this is a really basic question. I'm just learning the basics of QM now.

I can't wrap my head around the idea that the radial distribution function goes to zero as r-->0 but that the probability density as at a maximum as r-->zero. How can this be? they are opposite to each other!

Thanks!
In simple terms, r is a distance between two things (electrons). The radial distribution function gives you the probability of finding two electrons with a distance of r between them. Since they can not occupy the same location, the probability becomes zero as r--> zero.

The probability density of finding an electron at the zero position will be maximum. In other words, they are answering two different questions

1. What is the probability of finding **another** electron a distance r separated from the first: (Radial distribution function)

2. What is the probability of finding **any** electron a distance r from an electron: (Probability density)

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