If there is a wave associated with a particle, then there must be a function to represent it. This function is called wave function.

Wave function is defined as that quantity whose variations make up matter waves. It is represented by Greek symbol ψ(psi), ψ consists of real and imaginary parts.

Ψ=A+iB

PHYSICAL SIGNIFICANCE OF WAVE FUNCTIONS (BORN’S INTERPRETATION):

Born’s interpretation

The wave function ψ itself has no physical significance but the square of its absolute magnitude |ψ²| has significance when evaluated at a particular point and at a particular time |ψ²| gives the probability of finding the particle there at that time.

The wave function ψ(x,t) is a quantity such that the product

P(x,t)=ψ^*(x,t)ψ(x,t)

Is the probability per unit length of finding the particle at the position x at time t.

P(x,t) is the probability density and ψ^*(x,t) is complex conjugate of ψ(x,t)

Hence the probability of finding the particle is large wherever ψ is large and vice-versa.

NORMALIZATION CONDITION

The probability per unit length of finding the particle at position x at time t is

P=ψ^*(x,t)ψ(x,t)

So, probability of finding the particle in the length dx is

Pdx=ψ^*(x,t)ψ(x,t)dx

Total probability of finding the particle somewhere along x-axis is

∫pdx =∫ ψ^*(x,t)ψ(x,t)dx

If the particle exists , it must be somewhere on the x-axis . so the total probability of finding the particle must be unity i.e.

∫ψ^*(x,t)ψ(x,t)dx=1                               (1)

This is called the normalization condition . So a wave function ψ(x,t) is said to be normalized if it satisfies the condition(1)

ORTHOGONAL WAVE FUNCTIONS

Consider two different wave functions ψ_m and ψ_n such that both satisfies Schrodinger equation.These two wave functions are said to be orthogonal if they satisfy the conditions.

Or                        ∫ ψ_n^* (x,t) ψ_m(x,t) dV=0 for n≠m]                          ( 1)

∫ ψ_n^* (x,t) ψ_m(x,t) dV=0 for m≠n ]

If both the wave functions are simultaneously normal then

∫ ψ_m ψ_m^* d V=1=∫ψ_nψ_n^* dV                                   (2)

Orthonormal wave functions:

The sets of wave functions, which are both normalized as well as orthogonal are called orthonormal wave functions.

Equations (16) and (17) are collectively written as

∫ψ^*_mψ_ndV={ o if   m≠n

=[1 if m=n

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