Clarification from the textbook_orthonormal

[terp.asessed]

Homework Statement

Well, rather the problem statement, this is rather sentences from the textbook on quantum mechanic waves, that I'd like to understand...

"Wave patterns, no matter how complicated, can always be written as a sum of simple wave patterns. For example, the UN-normalized (how is this possible?) wave function ψ(x) = sin²x (0<x<2π) can be decomposed into 2 parts, one a constant and the other with wavelength 2π/2 = pi:

ψ(x) = 1/2 - cos2x/2 (I understand this is based on double angle formula from trignometry, where sin²x = 1/2 - cos2x/2)
More generally, the wave function can be decomposed into COMPONENTS corresponding to a constant pattern plus ALL possible wavelength of the form 2π/n with n an integer. That is, we can find coefficients c_n such that:

ψ(x) = Σ (x = 0 to ∞) c_n cos nx

From ψ(x) example above, c₀ = 1/2 c₂ = -1/2, and all the other coefficients are zero (why, the rest are zero? Why isn't c₁ = -1/2? )

The general formula for these coefficients valid for ANY function ψ(x) can be made. The first step is to imagine writing up ψ(x) in terms of the basis function ψ_n(x) and some mysterious coefficients a_n:

ψ(x) = Σ (x = 0 to ∞) a_n ψ(x), ψ(x) = 1/(2π)^1/2 for n=0, ψ(x)= cos nx /π^1/2 for n≠0.

The reason not to just use cos nx is that these ψ_n(x) functions are ORTHOGONAL. That is they are all normalized and orthogonal to one another over the interval (0, 2π).

Could someone pls explain the LAST sentences? I am trying to figure out about the ORTHOGONALITY in this example, but don't understand at all...I've been trying to understand the last sentence for past TWO days by proving the function, without any success.

Homework Equations

given above

The Attempt at a Solution

I am not sure how to progress from the given example, but what I know for sure is that, according to the next page on the textbook is that:

∫(x = 0 to 2π) Ψ_n(x)Ψ_m(x) dx = 1 for n = m, and = 0 for n ≠ m. So, I am trying to figure out how the textbook made this HUGE jump from the given example to this orthonormality the next page. Someone please explain how the textbook has done so...D-; I am so lost...

 

[BvU]

A lot of questions, I can try a few...
Unnormalized wavefunctions are an intermeditate result: they are usually solutions of the Schroedinger equation that can be good for all kinds of considerations (energy levels, etc.). But they have to be normalized before they can serve as a representation of a state for which measurable quantities are desired.

The decomposition of $\sin^2$ in the example is what it is: just an example to illustrate the power of Fourier transforms, on which you want to read up. http://www.engr.uconn.edu/~lanbo/G377FFTYC.pdf, there, everywhere. Useful in lots of branches of lots of sciences, and beyond.

Simple trig got you sin2x = 1/2 - cos2x/2 and that all other coefficients are zero follows from $\int \cos mx \; \cos nx \; dx = 0$ for $m\ne n$, $\pi$ for $m=n$. You'll have to check it. It's the core of the orthogonality claim, the underpinning of Fourier analysis.
Your question on c₁ = -1/2 : I suspect a mistake, perhaps in the bounds: from $[0,2\pi]$ if you make a drawing, you can see from symmetry that it is zero.

Orthogonal means $\int \psi_m^*\psi_n=0$ unless m=n (and then it should be > 0). Orthonormal is orthogonal plus $\int \psi_n^*\psi_n=1\$ (that's where the $1\over \sqrt \pi$ comes in).

The nice thing about this decomposition is that you can decompose every function with some period into a Fourier series. And this goes even further: you have a differential equation, find multiple solutions, turn them into an orthonormal basis... but now I am making this huge jump too.

My advice: if this textbook isn't satisfactory, go to the library (do they still exist? -- otherwise google) and find a few you like better. The concept is crucial, so the investment is really worth it.

 

[terp.asessed]

Thank you very much! I will check the links and invest time on the orthogonality!