How to Solve a Two-Dimensional Particle in a Box

The solution of a two-dimensional "Particle in a box" can be determined by solving the Schrodinger equation.

Classical mechanics suggests that sub-atomic particles such as electrons can be tracked, and their absolute position and momentum can be known. Quantum mechanics is a subject that was developed in the early to mid-1900s. It has demonstrated that particles can also be described as waves, and knowing the position leaves an uncertainty in the momentum. The "Particle in a box" is a common problem in quantum mechanics and involves finding the wave function of electrons that are placed within an energy well.

Instructions

1. Write down the Schrodinger equation for two dimensions. The Schrodinger equation is a key equation in quantum mechanical problems. It takes the form:

   -h^2/2m (d2Psi/dx^2 + d2Psi/dy^2) = E Psi

2. Separate the variables. The wave-function psi can be written as a product of two functions:

   Psi(x,y)=X(x)Y(y)

   Substituting this into the Schrodinger equation leads to two equations, one for x and one for y:

   -h^2/2m (d2X/dx^2 )= ExX

   -h^2/2m (d2Y/dx^2 )= EyY

   These are differential functions that have well-known solutions.

3. Write down the solutions to the two differential equations. The solutions are:

   Xnx = SQRT (2/Lx) sin (npix/L)

   Yny = SQRT (2/Ly) sin (npiy/L)

   Psi(x,y)=X(x)Y(y)

   Psi(x,y)= SQRT (2/Lx) sin (npix/L) * SQRT (2/Ly) sin (npiy/L)

   That equation is the general solution to the two-dimensional particle in a box.

References

• EVERY SCIENCE: The Particle in A Two-Dimensional Box