projecting Psi(x) onto momentum basis

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I haven’t given up learning QM yet. I almost did give up. Ha not really! I’ve done the math behind QM in school so why would I quit? 

Okay, Mr susskind in his Modern Physics lecture ( http://youtu.be/LBFBQr_xKEM?t=34m54s ) does something really cute using Dirac notation, like:

<x|k> = e^(ikx)/sqrt(2pi) is the projection (inner product) of a vector k (momentum) onto a position basis.

Then he does something like

Psi^hat( k ) = <k|Psi>

                  = integral { <k|x> <x | Psi> } dx }

                  = 1/sqrt(2pi)  integral {  e^(-ikx) Psi(x) } dx }

It’s called a fourier transform!

If you done any graphics you may have used Spherical Harmonics for lighting ( where you reconstruct a lighting function by projecting the coefficients of a lighting function). It’s the same idea. Spherical Harmonics are the solutions to the Laplacian equation (http://en.wikipedia.org/wiki/Laplace’s_equation).

sh guide.

expected value

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Something just blew my mind!

See (1.15) the forces acting on N particles is just the first moment of the mass function, or center of mass. Total force on the body of particles is just 2nd derivative of the center of mass! It’s the same as a single particle (it’s just all the forces acting on that particle).

In QM for an Observable A of an Operator A, <A> is the expected value the eigen function psi ( when the psi function is an eigen vector of the operator A)! This is the inner product <psi(x)|A|psi(x)>. (http://en.wikipedia.org/wiki/Expectation_value_(quantum_mechanics)). Say A is random event then <A> is just saying E[A(x)]!

Like I was saying you can apply operators, for example the position operator is just xPsi(x) = lambda psi(x) where psi(x) is the eigen vector of the operator. The expected value <psi(x)|A|psi(x)> = a just the projection of operator A applied to Psi(x) onto Psi(x)*! There are many connections to different mathematics here, I’m still trying to wrap my head around it all. Sometimes I would convince my self of something then later forget about it. I don’t write it down, perhaps I should. Better yet, code it!

It’s crazy and cool how everything is connected.

Why do we care about stuff like the position operator xPsi(x)? Because we want to know where the particle may be ( if we draw a picture at some x ), how fast something is going, etc.

Next I’m going to write some code to visualize some of this.

Reference:

http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html

BTW: Apsi(x) = lambda psi(x) implies different sets of eigen vectors forms a spanning set. That is basis of QM are subspaces.

 

 

 

 

 

 

 

how to observe x?

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In the Modern Physics lecture video #3 he goes over <x|psi> (Braket notion) for the inner product of the vector space of real-valued complex linear functions. This effectively read x. x is the basis vector representing the function that maps x into it’s eigenvector basis. psi is the mapping that maps x into the vector space. Well I”m still puzzling over it. I just know the basis is under the operator xpsi(x).

Makes sense to me. It’s the projection of some function psi(x) onto the basis vector x, linear algebra stuff.

How to get the basis? The basis is dirac function which are the eigenvector of the operator xPsi(x). That is xPsi(x) = lambda Psi(x). Dirac functions are like impulse functions. Every Dirac function is orthogonal to each other so they form a basis. Thus if the operator is xPsi(x) then projecting onto a basis is just <x|psi>

And the inner product (the projection thing) for square- integrable real-valued complex function is defined as:

http://en.wikipedia.org/wiki/Square-integrable

In classical mechanics once you know x you automatically know velocity.

modern physics video

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I’m watching https://www.youtube.com/watch?v=epzh76hNl8I
It’s very informative. Especially watch the 1st one if you are confused about experiments in QM.

Braket notion is just vector space algebra. |A> is just a column vector in a vector space. Braket is defined as the dot product of the conjugate of a vector B with A. = B1A1 +…+ AmBn.

You can express vector |A> as a linear combination of basis vectors:

Ax|ex> + Ax|ey| + … = |A>

One can talk about operators say B':

So B’|A> would just be matrix column vector multiplication.

B’1.A + B’2.A + … = B’|A>

….

Watch the 1st video it’s very informative.

LoL I’m still here and QM

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Yeah I’m still on my break from gamedev. I know I started working on going back to Linux for my project but somehow I got side tracked. Let’s see what happened? Hmmm… I installed windows and I have been playing a lot of games.

My current plan for the game is I’m going to port my game from Ogre to Unreal Engine. The whole sub. thing is just too damn good to pass up. I will get it started soon LOL. Probably after I finish Divinity Original Sin.

Anyway, lately I’m finally beginning to wrap my head around QM. I was read the EPR paper and I actually understood it! I may post some stuff regarding QM in a future post! http://www.nat.vu.nl/~wimu/Pictures/EPR-paper.pdf

Normally I have hard time understanding QM due to Dirac formalism but in the ERP paper where it’s just mostly D.E and orthogonal functions it was very clear to me.

This week’s progress so far

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Image

This week I copied over (and cleaned up) GraphicsController from project zombie. In the screen above I’m running HDRBackground glsl shader. Converting all my HLSL shaders to GLSL will be a painful process.

I’m building my project on Ubuntu 13.10 using CMAKE, eclipse CDT, Ninja, and Clang.

I tried using Codeblocks from one of the ubuntu PPA but it was extremely unstable. Constantly freezing and such. I gave up.

I will starting posting source code soon. It’s on github.