How to Solve a Two-Dimensional Particle in a Box
- 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.
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