**The hydrogen atom: **

**Schrödinger equation for hydrogen atom: **

Schrödinger equation can be solved completely for hydrogen atoms as well as hydrogen type atoms, like, He^{1+ }, Li^{2+} ( *Z * = 1). For the other atoms only approximate solution can be achieved.

For most calculations, it is simpler to solve the wave equation if the Cartesian coordinates * x *, *y *, and *z * are converted to polar coordinates, *r *, *θ *, and *Φ *.

**Figure 1.5. **Cartesian and polar coordinates.

It can be seen from Figure 1E that two sets of coordinates are related to each other by the following relation,

z = *r *cos θ

*y * = *r * sin θ sin Φ

*x = r *sin θ cos Φ

The Schrödinger equation is written as,

(xix) |

Where,

Changing to polar coordinates, ∇^{2}Ψ becomes,

Now we can write equation (xix) as,

(xx) |

(xxi) (Potential energy ( |