[asparagusrisotto]

Hi! Can someone help me differentiate between what exactly the radial distribution function, the radial wave function and the radial probability function is? I'm getting confused, as I thought two of them represented the same characteristic. Thanks!!

[Corribus]

The wavefunction is an eigenfunction of the Schrodinger equation. It basically contains all the information about the state of a quantum particle, including its energy, momentum, position, etc. In a three dimensional space, the wavefunction contains three dimensions, and under certain approximations it can be separated into its independent spatial counterparts. For objects or systems that have radial symmetry - like atoms - expressing the wavefunction in spherical polar coordinates is convenient. This means your wavefunction in 3D space has a 1D radial counterpart and a 2D spherical harmonic counterpart that are nominally independent of each other. The radial wavefunction contains all the information about the system's behavior as a function of its linear distance from the origin (usually, the atomic center). The spherical harmonic wavefunction likewise contains all the information about the systems behavior independent of its linear distance.

The square of the normalized wavefunction (specifically, the product of the wavefunction and its complex conjugate) is frequently called the probability distribution function because its amplitude at a point in space gives the probability that the particle may be found at that point instantaneously. (Although note that due to the uncertainty principle, we cannot actually locate a quantum particle at a specific point with infinite position.) The radial distribution function expresses just the radial counterpart- i.e., the square of the radial wavefunction, which specifies the probability that the quantum particle is likely to be found a a certain distance from the origin. Because the radial wavefunction contains only information about distance from the nucleus, the radial probability function does not tell you which direction from the nucleus the particle is likely to be - only the magnitude of the vector between the likely point and space and the origin.

Likewise, there is also a spherical probability distribution function that is the square of the spherical harmonic wavefunction. This provides the probability that the particle is located in a certain direction from the nucleus irrespective of its distance. In practice, this is what gives the shape of the atomic orbitals.

Radial distribution and radial probability function might be used interchangeably in this context. I believe radial distribution function has other uses related to multiparticle systems in statistical mechanics, but this is different from its meaning in quantum mechanics, which is always tied to the wavenfunction.