I have what is probably a silly question about one step that is made in the derivation of the ground state solution for a quantum harmonic oscillator. My textbook gives no explanation, only the solution, so I could not get an answer from that. The following link shows how one can obtain the solution: http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc3.html#c1 What I do not understand is the bit that is described with ''For this to be a solution to the Schrodinger equation for all values of x, the coefficients of each power of x must be equal. That gives us a method for fitting the boundary conditions in the differential equation. Setting the coefficients of the square of x equal to each other...'' Basically, I do not see why the those terms should be equal/ what it means. So I was wondering if anyone could explain what it means or why that step is made?
2 Answers
At the risk of being overly simple, and this not really being a physics concept (but a very important concept, nonetheless, for doing physics): The set of functions which make up a polynomial, $$\left\{x^0, x^1, x^2, x^3, x^4, ... \right\},$$ form a set of linearly independent functions. What that means is that you can't take any single one (or a pair, for that matter) and use a combination of the remaining others to make an exact copy of that one (or pair) which works for all values of $x$. In other words $$x^2=a+bx+dx^3 + ex^4 + gx^5 + ...$$ is an impossible equation if it's required to be true for any value of $x$.
So, with an equation like $$(ax^2-5)x= d+fx+gx^2+hx^3,$$ for all $x$, then $d=0$, $f=-5$, $g=0$, and $h=a$.
The solution must satisfy the Schrödinger equation for every $x$. The equation after the ansatz follows the shape \begin{align} (-\alpha+\alpha^2 x^2) f(x) = (b x^2 +c)f(x) \end{align} Now in order to be true for every $x$ the polynomial function prefactors of $f(x)$ must be the same.
