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How to Solve a Two-Dimensional Particle in a Box

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.

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