Quantum Physics

Wave function and its physical significance


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.



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


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.


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


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


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)


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

Or                        ∫ ψn* (x,t) ψm(x,t) dV=0 for n≠m]                          ( 1)

∫ ψn* (x,t) ψm(x,t) dV=0 for m≠n ]

If both the wave functions are simultaneously normal then

∫ ψm ψm* d V=1=∫ψnψn* 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*mψndV={ o if   m≠n

=[1 if m=n

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