The hydrogen atom:

Schrödinger equation for hydrogen atom:

Schrödinger equation can be solved completely for hydrogen atoms as well as hydrogen type atoms, like, He¹⁺ , Li²⁺ ( 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, ∇²Ψ becomes,

Now we can write equation (xix) as,

(xx)

(xxi) (Potential energy ( E k ) = V , see equation vii)