# Energy of a Particle in Three Dimensional Box

## Energy of a Particle in Three Dimensional (3D) Box

Let us consider a particle of mass m present in a three dimensional box. a, b and c be the side lengths along x, y and z directions. Potential energy is zero everywhere inside the box and hence the Schrodinger equation is given as-

This equation can be separated by putting Ψ

_{(x,y,z)}= X

_{x}Y

_{y}Z

_{z}

or simply, Ψ = X Y Z

Putting the value of Ψ in the above equation, we get-

Since this equation must be true for all values of the independent variables x,y,z. therefore, each term on LHS must be equal to a constant.

The general solution of this equation is given by-

X = Asin k

_{x}.x + Bcos k

_{x}.x

where, A and B are arbitrary constants.

Applying boundary conditions-

X = 0 at x = 0, then-

X = Asin k

_{x}.x

and from X = 0 at x = a, we have-

0 = A sink

_{x}.a

or, sink

_{x}.a = 0

or, k

_{x}.a = n

_{x}.π

or, k

_{x}= n

_{x}.π/a

Because the sine of angle is zero at any integral multiple of π. Hence, n

_{x}is an integer. So, we have-

In three dimentional box, the complete eigen function is given by-

Ψ = X Y Z

where, n

_{x}is an integer not excluding zero as this value of n

_{x}would make X

_{x}= 0 everywhere. The n

_{x}is the quantum number along x direction. Hence, the total kinetic energy (E) of the electron in three dimentional box is given by-

where, n

_{y}and n

_{z}are the quantum numbers along y and z axes respectively.

If the three dimentional box is cubical with side a, then-