What is Wave Function?

Introduction

At a particular time, The wave function contains all the information about the particle.  It is not an observable quantity. But it does not have any physical significance.  However, the square of the absolute value of the wave function has a physical significance. We will find |ψ(x,t)|² as a probability density. Probability density is a probability per unit length or we can say  probability of finding the particle at a time t at position x.

The wave function ψ associated with a moving particle is a complex quantity. We know the representation of complex wave function.

ψ(x, y, z, t) = a + ib

and its complex conjugate is

ψ*(x, y, z, t) = a – ib.

The product of wave function and its complex conjugate is

ψ(x, y, z, t)ψ*(x, y, z, t) = (a + ib) (a – ib) = a² + b²

a² + b² is a real quantity. 

However, this can represent the probability density of locating the particle at a point in a given instant of time. 

The positive square root of ψ(x, y, z, t) ψ*(x, y, z, t) is represented as |ψ(x, y, z, t)|, called the modulus of ψ. The quantity |ψ(x, y, z, t)|² is called the probability. However, the product of a complex number with its complex conjugate is a real and non-negative number. Hence, this interpretation is possible.

Physical significance of Wave Function

• ‘Ѱ’ is a complex quantity representing the variation of a matter wave ‘Ѱ’ at a place at any instant of time. Although it does not have any  physical significance.
• The Ѱ(r,t) describes the position of particle at a given time.
• We  can consider it as ‘probability amplitude’ since it is used to find the location of the particle.
• Probability density of the particle is represented by the Ѱ(r,t) state function itself. Thus, the square of the wave function gives the probability density of the particle.
• If the value of probability density is high that simply means the probability to find the particle in that region is high.

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