Normalisation of Wave Function

NORMALISATION OF WAVE FUNCTION

The constant A is determined by normalisation of wave function as follows. 

It is certain that the particle is some where inside the box. Thus, the probability of finding the particle inside the box of length a is given by

The second term of the integral becomes zero at both limits

On substituting. eqn (12) in egn (9), we have

The eigen function (Ψn) belongs to eigen energy values En and it is expressed as

This expression (13) is known as normalised eigen function. The energy En and normalised wave functions Ψn are shown in .fig. 6.10.

Special cases

From eqns (8) and (13), the following cases can be taken and they explain the motion of electron in one dimensional box.

Hence, Ψ1 (x) is maximum at exactly middle of the box as shown in fig. 6.10.

Hence, Ψ2 (x) is maximum at quarter distance from either sides of the box as shown in 6.10.

Hence, Ψ3 (x) is maximum at exactly middle and one-sixth distance from either sides of the box as shown in fig. 6.10.