In order to find the energy eigen values and normalized eigen functions we apply the method of Box with periodic boundary conditions.

Consider two wave functions 𝜓(x) and u(x). The momentum operator (Px = -iℏ∂/∂x) for the finite space interval -l/2 ⩽ x ⩽ l/2 implies that

< 𝜓(x) l Px l u(x) > = < Px 𝜓(x) l u(x)>

Let us choose this constant as unity then

u(L/2) = u(-L/2) ...ii

This equation implies that the periodicity condition. Thus the boundary condition for box normalization is thus can be written as

u(x) = u(x + L); L being periodic ....iii

We are going to deal with the free particle. Let us consider

u(x) = Ae^iknL ....iv

Then u(x + L) = u(x) e^iknL ....v

Using the boundary condition given in equation iii we have

e^iknL = 1

e^iknL = e^i2𝜋n

kn = 2𝜋n / l .....vi

For the free particle with V= 0 we have the Schrodinger wave equation

d²u(x)/ dx² + 2mE/ ℏ² u(x) = 0

kn² = 2mEn/ ℏ²

En = ℏ² kn² / 2m

Substituting the value of kn² from eqn vi we find

En = 4𝜋²n² ℏ² / 2mL² ....vii

Let us find the normalization constant A in

u_n(x) = Ae^iknx

This is the box normalization with periodic boundary condition and L be the period. The box normalization refers to the process of ensuring that the wave functions are properly normalized within the confines of the box. Normalization ensures that the total probability of finding the particle within the box is unity ∫∣ψ(x)∣²dx=1. Thus, the normalized eigen function of free particle can be written as

The orthonormality of three normalized eigen function is given by

Solving for the energy eigenvalues and normalized eigenfunctions of a free particle within a box allows us to understand the allowed energy states and spatial distribution of the particle's probability density within the confined region, providing essential information about its quantum behaviour.

This note is a part of the Physics Repository.

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