The dependence of ψ on *r *, *θ *, and *Φ * can not be shown directly with equation (xxi). Because, it would require a four dimensional graph. However, the equation in this form can be express as follows,

(xxii) |

*R*(*r*) is a function that depends on the distance from the nucleus. It depends on the quantum numbers*n*and*l*.- Θ (θ) is a function of θ and depends on the quantum numbers
*l*and*m*. - Φ (Φ) is a function of Φ and depends on the quantum numbers
*m*.

Therefore, equation (xxii) can be express as,

This splits wave function into two parts which can be solved separately,

*R*(*r*) is a radial function that depends on the quantum numbers*n*and*l*.*A*is the total angular wave function that depends on the quantum numbers_{ml}*m*and*l*.

**Radial part of wave functions, R : **

The radial function *R * has no meaning. *R *^{2} gives the probability of finding the electron in a small volume d *v * near the point at which *R * is measured.

**Figure 1.6.
Showing volume difference
**

For a given value of r the total volume will be,

We may consider that an atom is composed of thin layers of thickness d *r *. The volume d *v * for between *r *and *r *+d *r * will be then (Figure 1F),

The probability of finding the electron in that volume will be,

**Figure 1.7. ** Radial probability functions for n = 1, 2,3 for the hydrogen atom. The radial density is along *y * axis.