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#c1What 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?
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