How to Normalize a Wave Function in Maple

In mathematics, a wave is, more or less, what the average person would think: a curve that continually changes direction, appearing much like a wave on the shore of a beach. What might not be so clear to the average person is that most mathematicians and physicists do not care about the amplitude, or size, of the wave. In wave analysis, mathematicians attach the most emphasis to the shape of the wave. In order to compare different waves, mathematicians first normalize all waves so that the waves are of comparable size. Mathematically, normalizing waves can be difficult, but with the powerful mathematics software package Maple, even those with lighter mathematics backgrounds can normalize a wave function.

Instructions

1. Write the wave function into Maple. The wave function should have at least two variables inside: an unknown constant and the independent variable. Call the unknown constant “A” and the independent variable “x.” In Maple, write the name of the function (use “f” if you have no special name), followed by a colon, an equals sign, the function itself and a semicolon. For example, if your wave function is "A*sin(n*pi*x/L)," then type “f:=A*sin(n*pi*x/L);".

2. Integrate the square of the wave function with respect to x. Perform the integration in Maple with the “int” function. Inside the function, place the wave function, followed by a comma and the variable with which you are integrating. Finish the command with a semicolon. Specifically, type “int(f^2,x);” and hit “enter.” The solution will appear. For the wave function given above, the output "A^2*L/2" will appear.

3. Solve the equation in which the solution to the integral is set equal to one. Solve for the unknown constant, “A.” Use Maple’s “solve” function to perform this algebra calculation. In “solve,” place the equation to solve, followed by a comma and the variable that you want to solve for. Specifically, type “solve(A^2*L/2=1,A);” to solve for “A.” The solution will appear, which in this case will be "sqrt(2/L)," with “sqrt” indicating the square-root function.

4. Place the numerical value for the unknown constant into the original wave equation. Rewrite the original wave equation, using the value just found for “A” in place of A. Thus, for the example, the normalized wave is "sqrt(2/L)*sin(n*pi*x/L)."

References

• “Physics with Maple”; Frank Wang; 2006
• “NIST Handbook of Mathematical Functions”; Frank W. J. Olver, et. al.; 2010