- #1

terp.asessed

- 127

- 3

## 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) = sin

^{2}x (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 sin*

^{2}x = 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, c

_{0}= 1/2 c

_{2}= -1/2, and all the other coefficients are zero (

*why, the rest are zero? Why isn't c*)

_{1}= -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...