Schrödingers wave equation correlates the wave property of the electron with its energy. During the derivation of this equation, he took the following under consideration.

1. de-Broglies wave particle duality equation
2. Heisenbergs uncertainty principle
3. Bohrs concept of quantized energy levels.

The equation for the standing wave comparable with that of a stretched string is

Where,.  (sigh) = amplitude at displacement x and is wave function

A = constant

x = displacement direction

l = wavelength.

Differentiating eq (i) twice with respect to x we get,

As total energy (E) = KE + PE

or [P.E. = U]

Again

According to de-Broglie s equation

Putting the value of v² from eq (iii) on eq (iv) we get,

Putting the value of eq (v) on eq (ii) we get

or 

or 

Equation (vi) is a Schrödinger wave equation for the wave motion in one dimension only ie x-axis. For electron moving in a three dimension space it is modified as:

Where Ѳ (del-square)[Laplacian operator] = 

Equation (vii) and (viii) are Schrödinger wave equation expressions.

The valid values of  are called Eigen functions, and the values of E corresponding to this Eigen functions are called Eigen values. The Eigen values are found to be more or less the same energy values given by Bohr s in different orbit.

For  to be valid it should satisfy following conditions:

1. must be finite and continuous
2. It should be single valued

iii. must be continuous functions of x, y, and z coordinates respectively.

Significance of  and ² 

 represent the three dimensional amplitude of electron wave at various points surrounding the nucleus, and is called orbital. However, ², gives the probability of finding electron of certain energy at a space inside the atom.

The value of  may be real or imaginary. If is real, ² also is real. Thus ²gives the probability of finding electron.

But if  is imaginary * gives the probability of finding electron wave

let  = a + ib (imaginary quantity)

* = (a+ib) (a–ib) = a²+b² (real)