What is the significance of wave function ? derive the expression for energy eigen values for the free particle in one dimensional potential well.

#### Solution

Wave represents the propagation of a disturbance in a medium. A wave function which describes the behaviour of a matter wave as a function of position and time. It has no direct physical significance as it is not an observable quantity. However , the values of the wave function is related to the probability of finding the matter particle at a given point in space at a given time.

In classical physics it is known that

The intensity of radiation is directly proportional to the square of Amplitude of the electromagnetic wave.

In an analogy in quantum mechanics it can be written that.

The density of matter particle is directly proportional to the square of Amplitude of the matter wave.

An one dimensional potential well is a potential energy function mathematically given by.

V(x) = 0 at 0 ≤ x < L

= ∞ at x ≤ 0 and x ≥ L

The potential energy is zero inside the well and infinite at the boundaries. A particle trapped inside the infinitely high potential well can propagate along x-axis and gets reflected from the boundary walls at x = 0 and x = L, but can never leave the well. Such a state is called bound state.

With zero potential energy the particle behaves as a free particle inside the well. Therefore the schodinger equation reads.

`(ħ^2)/ (2m) .(d^2 Ψ(x))/( dx^2) = EΨ(x)`

`(d^2Ψ)/( dx^2) + (2mE)/(ħ^2 )= 0`

`(d^2Ψ) /(dx^2) + k^2E(x) = 0`

The behaviour of the particle describe by the solution of equation (+x) and the term Be-^{jkx }represents the motion in the backward (-x) direction. Here A and B are constants.

`Ψ(x)=Ae^(jkx) + Be^(-jkx)`