To prove the validity of the perturbation expansion, let's start with a simple and obvious statement:
It seems intuitive that a variable x can be the sum of two terms containing two new variables. In fact, the same can be said of variable a1:
And, a2 can be the sum of two terms containing two new variables. We can repeat this exercise as many times as we like until we reach a(sub-n):
Using equations 2 through 5 we make a series of substitutions which transform equation 1 into the familiar perturbation expansion:
However, equation 6 is an exact solution which can't be had unless we know the value of a(sub-n). Let's assume we don't. Let's eliminate a(sub-n). If we do, we get an approximate solution:
Now, how do we apply this approximate series known as a perturbation series? Consider a function of x:
Let's assume this function of x is too hard to solve. So we re-write f(x) above to show the easy, solvable part and the seemingly unsolvable part:
Let's suppose the hard part is g(x). Wouldn't it be great if we could just get rid of it? Not permanently. Just for the time being. We do away with g(x) by multiplying it by epsilon, and epsilon has a zero limit.
We end up with equation 12 which is easy to solve. The solution can be found at 13. However, we didn't solve variable x--we solved what is called an unperturbed x or x0. We can now plug that solution into equation 7 and equation 8. Because x is approximately equal to x0 plus the rest of the series, we can make a substitution at equation 11 to get the following:
At 15 we set the equation to zero. The idea here is to solve the xi coefficients. This is often done by grouping terms on the basis of their epsilon powers. The equation is broken up into smaller, simpler equations that are easy to solve. At 18 below, the solved coefficients are plugged into the series and epsilon is set back to 1 (giving us back the equivalent to the missing hard part of the original equation). At 19 and 20 we simply add up the coefficients to get a close approximation of the true value of x.
Since the intent of this post was to do a general proof of perturbation theory, no specific problem-solving examples were provided. Click here to see an example of a specific problem solved using perturbation theory.