Energy of a Particle in Three Dimensional Box

Energy of a Particle in Three Dimensional Box

Energy of a Particle in Three Dimensional Box

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

Energy of a Particle in 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-
Energy of a Particle in Three Dimensional Box
This equation can be separated by putting Ψ(x,y,z) = Xx Yy Zz
or simply, Ψ = X Y Z
Putting the value of Ψ in the above equation, we get-
Energy of a Particle in Three Dimensional Box
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.
Energy of a Particle in Three Dimensional Box
The general solution of this equation is given by-
X = Asin kx.x + Bcos kx.x
where, A and B are arbitrary constants.
Applying boundary conditions-
X = 0 at x = 0, then-
X = Asin kx.x
and from X = 0 at x = a, we have-
0 = A sinkx.a
or, sinkx.a = 0
or, kx.a = nx
or, kx = nx.π/a
Because the sine of angle is zero at any integral multiple of π. Hence, nx is an integer. So, we have-
Energy of a Particle in Three Dimensional Box
In three dimentional box, the complete eigen function is given by-
Ψ = X Y Z
Energy of a Particle in Three Dimensional Box
where, nx is an integer not excluding zero as this value of nx would make Xx = 0 everywhere. The nx is the quantum number along x direction. Hence, the total kinetic energy (E) of the electron in three dimentional box is given by-
Energy of a Particle in Three Dimensional Box
where, ny and nz 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 Three Dimensional Box

Energy of a Particle in One Dimensional Box