# Physical Significance of ψ and ψ^{2}

## Physical Significance of Wave Function ψ and ψ^{2}

In Schrodinger's wave equation, ψ represents the amplitude of spherical wave. According to the theory of propagation of light and sound waves, the square of the amplitude of the wave is proportional to the intensity of the sound or light. A similar concept, modified to meet the requirement of uncertainty principle has been developed for the physical interpretation of wave function. This may be stated as

*the probability of finding an electron in an extremely small volume (d𝜏) around a point is proportional to the square of the function at that point*.

Hence the square of modulus of ψ is the

**probability density**. If wave function ψ is imaginary, ψψ* becomes a real quantity where ψ* is a complex conjugate of ψ. This quantity represents the probability of ψ

^{2}as a function of x, y and z co-ordinates of the system, and it varies from one space region to another. Thus the probability of finding the electron is different is different regions.

For hydrogon atom, Schrodinger Wave Equation gives the wave-function of the electron with energy =-2.18 × 10

^{-11}ergs situated at a distance 'r'.

A zero value of ψ

^{2}at a fixed point shows that the chances of getting electrons at that point are zero and a high value of ψ

^{2}at a point indicates the greater probability of finding or locating the electron at that point. Thus the electron is just like a diffused cloud rather than a discrete individual. To distinguish this from Bohr's orbit the name 'Atomic Orbit' is attached with these clouds. In the atomic orbital, there is a maximum probability of finding electron in a particular volume element at a given distance and direction from the nucleus. The boundaries of such an orbital are not very distinct because there always remains a finite, even if small probability of finding electron relatively distant from the nucleus.

**Summary**

Wave function ψ represents the amplitude of electron wave i.e. probability amplitude and has no physical significance indivisually because it may be positive, negative or imaginary. ψ

^{2}is known as probability density and determines the probability of finding an electron at a point within small volume (d𝜏) (atom). This means that if:

1. ψ

^{2}is zero, the probability of finding an electron at that point is negligible.

2. ψ

^{2}is high, the probability of finding an electron is high i.e. electron is present at that place for a long time.

3. ψ

^{2}is low. the probability of finding an electron is low i.e. electron is present at that place for a shorter time.

## Why does wave function ψ have no physical significance ?

The wave function ψ associated with a moving particle is not an observable quantity and does not have any physical meaning indivisually. It is a complex quantity. The complex wave function can be represented as ψ(x, y, z,*t*) = a + ib and its complex conjugate as 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) = a

^{2}+ b

^{2}is a real quantity.

However, this can represent the probability density of locating the particle at a place in a particular point 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 density.

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