One of the simplest and possibly most instructive problems in quantum theory is the particle in a box problem.
Our goal in this problem is to calculate the function which satisfies Schrodinger's Equation and consequently characterizes the probablity of finding the particle in question at any point x. The actual probability distribution for the particle will of course be .
A particle exists in a region of zero potential energy bounded by two regions where the potential energy goes to infinity. This is referred to as a square well. The particle is free to move within the well but has no possibility of moving into the region outside of the well
The Schrodinger Equation which we propose to use to solve this problem is:
However our results will be uneffected and our calculations will be made much simpler if we remove the time dependent factor from and from the Schrodinger Equation. The resulting equation is widely known as the Time Independent Schrodinger Equation and it used extensively in finding expectation (the most probable) values for particle postions, velocities and energy levels.
Note the constant 'E' on the right hand side of the equality. The selection of this particular letter was done with the fore knowledge that it does in fact represent the total energy of the particle, in this case it is the the sum of the potential and kinetic energies. For an explanation of how this equation was derived please visit the page entitled 'Time Independent Schrodinger Equation.'
Expressing the Time Independent Schrodinger Equation in one dimension, x, and setting V=0 within the interval (0,L) and V=infinity otherwise, gives us the following equation.
Within the Square Well this is the well known harmonic oscillator equation with boundary conditions that force to be zero at x=0 and at x=L. The solution can be any simple sine wave A.sin(x) beginning equal to 0 at x=0 and ending by going to zero at x=L. Note that cosine solutions are not applicable due to fact that any function B.cos(x) can only be made zero at x=0 by setting B=0.
The fact that =0 when x=L tell us that:
Here is a sketch of the first four solutions to the particle in a box problem. Note that the wave forms should all been aligned so that (x=0) = 0 but are separated vertically for ease of viewing..
So the general solution is , noting that is the probability of finding the particle at point x, integrating between x=0 and x=L will produce unity. We can use this fact to calculate the values of the coefficients An and determine that they are in fact equal to SQRT(2/L). So the general solutions to the quantum particle in a one dimensional square well problem generates the following relations:
For a discussion about what all of this means ... read more
