Consider one dimensional closed box of width L. A particle of mass ‘m’ is moving in a one-dimensional region along X-axis specified by the limits x=0 and x=L as shown in fig. The potential energy of particle inside the box is zero and infinity elsewhere.

I.e Potential energy V(x) is of the form

V(x) = {o; if o<x<L

∞: elsewhere

The one-dimensional time independent Schrodinger wave equation is given by

d²ψ/dx²+ 2m/Ћ²[E-V] ψ=0                                             (1)

Here we have changed partial derivatives in to exact because equation now contains only one variable i.e x-Co-ordinate. Inside the box V(x) =0

Therefore   the Schrodinger equation in this region becomes

d²/ψ/dx²+ 2m/Ћ²Eψ=0

Or                 d²ψ/dx²+ K²ψ=0                                          (2)

Where                       k=    2mE/Ћ² (3)

K is called the Propagation constant of the wave associated with particle and it has dimensions reciprocal of length.

The general solution of eq (2) is

Ψ=A sin Kx + B cos K x                                   (4)

Where A and B are arbitrary conditions and these will be determined by the boundary conditions.

(ii) Boundary Conditions

The particle will always remain inside the box because of infinite potential barrier at the walls. So the probability of finding the particle outside the box is zero i.e.ψx=0 outside the box.

We know that the wave function must be continuous at the boundaries of potential well at x=0 and x=L, i.e.

Ψ(x)=0 at x=0                                            (5)

Ψ(x)=0 at x= L                                           (6)

These equations are known as Boundary conditions.

(iii) Determination of Energy of Particle

Apply Boundary condition of eq.(5) to eq.(4)

0=A sin (X*0) +B cos (K*0)

0= 0+B*1

B=0                                                             (7)

Therefore eq.(4) becomes

Ψ(x) = A sin Kx                                    (8)

Applying the boundary condition of eq.(6) to eq.(8) ,we have

0=A sin KL

Sin KL=0

KL=nπ

K=nπ/L                                                                       (9)

Where    n= 1, 2, 3 – – –

A Cannot be zero in eq. (9) because then both A and B would be zero. This will give a zero wave function every where which means particle is not inside the box.

Wave functions. Substitute the value of K from eq. (9) in eq. (8) to get

Ψ(x)=A sin(nπ/Lx)

As the wave function depends on quantum number π so we write it ψ_n. Thus

Ψ_n=A sin (nπx/L)0<x<L

This is the wave function or eigen function of the particle in a box.

Ψ_n=0    outside the box

Energy value or Eigen value of particle in a box: Put this value of K from equation (9) in eq. (3)

nπ/L = 2m E/Ћ²

Squaring both sides

n²π²/L²=2mE/Ћ²

E=n²π²Ћ²/2mL²

Where n= 1, 2, 3… Is called the Quantum number

As E depends on n, we shall denote the energy of particle ar E_n. Thus

E_n= n²π²Ћ²/2mL² (10)

This is the eigen value or energy value of the particle in a box.