Clarification from the textbook_orthonormal

  • Thread starter terp.asessed
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In summary, the textbook explains that wave functions can be decomposed into simple wave patterns and that the coefficients for these wave patterns are always zero. It also states that orthogonal means that the sum of two orthogonal wave functions is zero, and that Fourier analysis can be used to decompose any function into a series of waves.
  • #1
terp.asessed
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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 cn such that:

ψ(x) = Σ (x = 0 to ∞) cn 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 ψn(x) and some mysterious coefficients an:

ψ(x) = Σ (x = 0 to ∞) an ψ(x), ψ(x) = 1/(2π)1/2 for n=0, ψ(x)= cos nx /π1/2 for n≠0.

The reason not to just use cos nx is that these ψn(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) Ψn(x)Ψm(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...
 
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  • #2
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.
 
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Thank you very much! I will check the links and invest time on the orthogonality!
 

Related to Clarification from the textbook_orthonormal

1. What does the term "orthonormal" mean in relation to a textbook?

The term "orthonormal" refers to a set of vectors in a vector space that are both orthogonal (perpendicular) to each other and normalized (have a length of 1). In the context of a textbook, this term is often used in linear algebra or geometry to describe a set of basis vectors that are both perpendicular and unit length.

2. How is orthonormality represented mathematically in a textbook?

In a textbook, orthonormality is represented using the dot product or inner product between vectors. If two vectors are orthogonal, their dot product is 0. If a vector is normalized, its magnitude or length is 1. Therefore, a set of orthonormal vectors will have a dot product of 0 and a magnitude of 1.

3. Why is orthonormality important in mathematics and science?

Orthonormality is important because it simplifies calculations and makes them more efficient. In linear algebra, orthonormal vectors form a basis for a vector space and can be used to find solutions to systems of equations. In physics, orthonormality is used to describe the relationship between different physical quantities in a coordinate system.

4. How can one determine if a set of vectors is orthonormal?

A set of vectors can be determined to be orthonormal by checking if they are both orthogonal and normalized. If the dot product between any two vectors is 0 and the magnitude of each vector is 1, then the set is orthonormal. Additionally, the Gram-Schmidt process can be used to transform a set of vectors into an orthonormal set.

5. What are some real-life applications of orthonormality?

Orthonormality has many practical applications in fields such as physics, engineering, and computer science. In physics, orthonormal basis vectors are used to describe the motion and forces acting on objects in a coordinate system. In engineering, orthonormal vectors are used to represent different physical quantities in a system or model. In computer graphics, orthonormality is used to create realistic 3D images by defining orthogonal and normalized light and color vectors.

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