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Thread: Question: Fourier Transform (Solid State Physics)

  1. #1 Question: Fourier Transform (Solid State Physics) 
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    How can the Fourier Transform of this function be drawn?:
    f(x) = 3cos(5x) + 2sin(3y)

    I suppose that the first step is to convert this into complex exponentials
    f(x) = (3/2)exp(-5ix) + (3/2)exp(5ix) + iexp(-3iy) - iexp(3iy)
    And then take the sum over all the allowed k values, assuming that k=(2*pi*p)/(2i) where "p" is an integer and "a" is the lattice constant? But, I'm not quite sure why I should do that; it seems like it would make the equation more confusing.

    Up until this point, the periodic function has just been converted into a different form. How do I make this into a Fourier series? Is it just a matter of scooting a sigma in there, or what?
    And, in the end, what will it look like? Will it look like a bunch of delta functions, or will it actually have a curve associated with it?

    FYI: In our class, the ultimate point of learning to do these Fourier Transforms is so we can convert reciprocal-space crystal lattices to real-space crystal lattices (handy in x-ray diffraction) or vice-verse. Or, at least, that's my best guess.


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  3. #2 Re: Question: Fourier Transform (Solid State Physics) 
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    Quote Originally Posted by redwire321
    How can the Fourier Transform of this function be drawn?:
    f(x) = 3cos(5x) + 2sin(3y)
    You have a problem here. The usual Fourier is only defined for functions that are absolutely integrable. Neither cos nor sin are and they do not have a Fourier transform in the usual sense.

    Once can define a Fourier transform for these functions using the theory of Schwartz distributions. The Fourier transforms are expressed in terms of Dirac delta functions. Drawing them depends on how you represent delta functions graphically.


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  4. #3  
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    As I understand, delta functions are graphically represented by vertical arrows on a horizontal graph. The heights vary depending on the Fourier coefficients. Would there be an infinite series of these then, on the graph of the FT of this function? If so, how do you find the Fourier coefficients?
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  5. #4  
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    Quote Originally Posted by redwire321
    As I understand, delta functions are graphically represented by vertical arrows on a horizontal graph. The heights vary depending on the Fourier coefficients. Would there be an infinite series of these then, on the graph of the FT of this function? If so, how do you find the Fourier coefficients?
    no

    Fourrier coefficients pertain to the Fourier series, which applies to functions defined on the circle group.

    I thought you were dealing with the Fourier transorm on the real line.

    What is your application ?
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  6. #5  
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    The problem is for my introduction to solid state physics class. I'm no physicist, I'm a chemistry major. Some of this math stuff is really baffling because I haven't had to work with it on such a high level before. The only class that came close was physical chemistry. Ugh.

    Anyway, the application is for crystal diffraction, which is introduced later on. As I've been told, the representation of the electron density of the crystal is a convolution of delta functions (representing the points in the crystal lattice) and a Gaussian (representing the electron density of a single atom).

    However, this problem doesn't seem to have to do with any of that, and it's just a straight-up math problem. *shrug* I'd be grateful for any advice you have to offer.
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