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-

Energy of a Particle in One Dimensional Box