# Particle in 3D Rectangular Rigid Box

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## 6.2 Applications in 3-D for Cartesian coordinate system

**6.2.1 Particle in 3-D Rectangular potential**

Consider a particle of mass and energy constrained to move in a three-dimensional rectangular potential well having sides of length equal to parallel to the -axes respectively. Suppose there is no force acting on the particle in the box. The appropriate potential is for its position at a point given by

and outside the box

**Fig. Particle in 3-D rectangular box**

The total kinetic energy is given by

---------------(1)

is called as the Laplacian operator

For the motion of the particle inside the box the Schrödinger's time independent equation:

---------------(2)

It is a partial differential equation. It is assumed that the function can be written as a product of three functions, and each of which depends on only one of the coordinates respectively. The equation can be solved by the technique of separation of variables.

---------------(3)

The Hamiltonian is the sum of three independent one dimensional Hamiltonians in totally separate variables

.---------------(4)

---------------(5)

Substituting for the Schrödinger's equation in the separated variables is

---------------(6)

The terms are ordinary derivatives instead of the partial derivatives as each of the functions , is a function of one variable only. Dividing the equation by XYZ

,---------------(7)

Each term of this equation depends on a different variable , or and the three variables , and are independent. The term is a constant for a particular value of kinetic energy. Velocity of the particle is a vector quantity. Velocity and corresponding kinetic energy can be resolved in three components along three co-ordinate axes .

---------------(8)

The only way for the equation to remain valid for all of , and in the interval is for each term to be constant. Therefore with separation of variables, the Schrödinger's equation separates into three independent equations in respectively as follows

---------------(9)

These differential equations have the solution:

---------------(10a)

---------------(10b)

---------------(10c)

Applying the boundary conditions

---------------(10a)

---------------(10b)

---------------(11c)

The resulting x-components of eigenfunctions are of the form

The solution for components have the same form.
The total normalized eigenfunctions for the motion of the particle in the box are given by

---------------(12)

---------------(13)

The eigenfunctions are zero outside the box. The eigenvalues of energy are given by

---------------(14)

These eigenvalues of the energy are called as the energy levels of the particle. They form quantized or discrete energy spectrum.

The integers are called as the quantum numbers. They are required to specify each stationary state. If the sign of the quantum numbers is changed, then there is no change in the energy and in the wave function except for the minus sign. Hence all the stationary states are given by the positive integral values of .

** The value of any of the quantum numbers cannot be zero. If any one is taken as zero then . It implies that the particle does not exist in the box **

The lowest possible energy is obtained for . It is called as the ground state energy and it depends on the values of

The wave function corresponding to ground state energy is called as ground state wave function and denoted by

The excited state energy levels are obtained by substituting different positive integer values for and the corresponding eigen functions are labeled by these quantum numbers

**6.2.2 Particle in 3-D Cubic potential**