Energy of a Particle in One Dimensional Box

Energy of a Particle in One Dimensional Box

Let us consider a particle of mass m confined in a one dimensional box of length a along x axis. For value of x between 0 and a, the particle is completely free and hence, potential energy is taken as zero. At boundaries, the particle is constrained by an infinite potential wall over which there is no escape. Thus potential energy is infinite when x = a or 0.

Now applying one dimensional schrodinger equationto the particle in one domensional box, we get-

To have Ψ = 0, at x = 0, the cosine function must be vanish, a condition that requires B = 0. Thus,
Ψ = A sin kx
to have Ψ = 0 at x = a, we must have-
0 = A sin ka
If A = 0 then Ψ = 0 everywhere and this is not an acceptable solution.
Hence, sin ka = 0 = sin n𝝅
or, ka = n𝝅
or, k = n𝝅/a
or, k² = n²𝝅²/a²       ---Equation-1
where n is an integer having values 0,1,2,...
Hence, the wave function for a particle in one dimensional box is-

The constant A can be evaluated by applying normalization condition i.e.

This is the expression for kinetic energy of a particle in one dimensional box.

Energy of a Particle in Three Dimensional Box