Application of Schrodinger wave equation: Particle in a box

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 Continue reading “Application of Schrodinger wave equation: Particle in a box”