Hyperbolic, parabolic, Schrodinger PDEs

Discussion in 'Physics & Math' started by Sacroiliac, Nov 5, 2003.

  1. Sacroiliac Registered Senior Member

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    73
    Ignoring initial conditions for a PDE and considering the solution to be in the form of a simple plane wave you get the following results.

    Hyperbolic

    u<sub>tt</sub> = c<sup>2</sup>u<sub>xx</sub>

    u(x,t) = A*cos(wt + kx)

    This is just the wave equation with a solution oscillating in time.

    Parabolic

    u<sub>t</sub> = a<sup>2</sup>u<sub>xx</sub>

    u(x,t) = Ae<sup>-a^2*k^2*t</sup>e<sup>ikx</sup>

    This is the diffusion equation and its solution describes a wave decaying in time.

    1. Is there an intuitive way to see why a PDE that is second order in time and second order in space causes oscillation, but one that is first order in time and second order in space causes a time decaying wave?

    2. How about the Schrodinger equation how would it behave under the above stipulations? It seems somewhat similar to the diffusion equation but its got that damn i in it.
     
    Last edited: Nov 5, 2003
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  3. ryans Come to see me about a dog hey Registered Senior Member

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    995
    Schrodinger's equation is first order in time and second order in space. For a potential free region of space the solutions are plane waves!!

    Not all PDE which are second order in time and space produce the solutions you suggest.

    This makes no sense at all, especially since you go on to show that the second equation has a solution which is in fact not in the form of a simple plane wave.

    The first solution is not totally correct either. There are an infinite number of solutions to this equation, the sum of any 2 solutions also being a solution.

    Keep working.
     
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  5. lethe Registered Senior Member

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    well, for ODEs, it is easy to see why this happens: the solution to the characteristic polynomial with odd degree always has a real part which gives you an exponential.

    i don t know if that qualifies as intuitive.... physically, you can think of the first derivative term as just a friction force that gets in the way of the harmonic oscillation.

    i think the picture is roughly the same for PDEs.

    and as you say, adding a complex number can change everything. if your characteristic polynomial does not have real valued coefficients, there is no longer a guarantee that an odd degree polynomial will have a real root. so the Schr&ouml;dinger equation is qualitatively different from the diffusion equation.

    that s the best explanation i can give, and i don t find it wholly convincing. take it or leave it.
     
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  7. lethe Registered Senior Member

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    did you even finish reading his post? he said as much in his post, and put forth a possible explanation of why.

    WTF?
    that solution is a plane wave if you allow for complex phase velocity.

    nowhere did he claim an exhaustive list of solutions, he only gave one solution. does that make that one solution incorrect?

    why don t you take this opportunity to brag about all the calculations you have done. i mean, god forbid you try to give some constructive input.
     
  8. Sacroiliac Registered Senior Member

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    73
    Re: Re: Hyperbolic, parabolic, Schrodinger PDEs

    I'll take it. And once again thanks for your clear and concise answers. The only sad thing is that I should have been able to figure it out for myself.
     
  9. Sacroiliac Registered Senior Member

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    73
    Oh and by the way ryans, I'm a division general manager for a very large corporation with over 1000 people reporting to me. But I wouldn't speak to the lowest level floor sweeper in the extremely rude manner you spoke to me.
     
  10. ryans Come to see me about a dog hey Registered Senior Member

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    995
    I wasn't being rude. Read my post without reading lethe's, and the tone will be seen to much more subdued than lethe would care to admit. I was simply pointing to some facts that I assume you misunderstood.
    I am sorry that you midunderstood my tone.

    BTW I couldn't give a rats what you do or who you are in charge of, the fact that the way your question was structured was very unintelligible and seemingly incorrect, propounded that response from me.
     
  11. Sacroiliac Registered Senior Member

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    73
    Well I see you're not being rude again huh? How old are you...15?

    Sorry James R, I don't normally let children's little rants annoy me, but in any case I'm done.
     
  12. lethe Registered Senior Member

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    he is older than i.
     
  13. thefountainhed Fully Realized Valued Senior Member

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    2,076
    Get the fuck outta here. No way.

    **examines pulse** conclusion, normal.

    I repeat: Get the fuck outta here. No way.
     
  14. ryans Come to see me about a dog hey Registered Senior Member

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    995
    Lethe, if I am older than you, it is not by much more than a year.
     
  15. lethe Registered Senior Member

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    2,009
    that is correct, less than a year (going by your DOB on your info page)
     
  16. ryans Come to see me about a dog hey Registered Senior Member

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    995
    And what are you up to these days lethe. i.e. trains of thought.
     
  17. lethe Registered Senior Member

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    2,009
    what do you mean? do you want to know what kind of physics i am doing these days?
     
  18. ryans Come to see me about a dog hey Registered Senior Member

    Messages:
    995
    Well I know that you were completing postgraduate work, contemplating do some work for that crackpot, and drifted of to the IAS for a bit. You were doing something to do with particle physics weren't you?
     
  19. lethe Registered Senior Member

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    2,009
    well, i went to the IAS for the summer, that was cool as hell. i met some real bigwigs, like e. witten, s. gubser, and d. gross. i think meeting the other upcoming theorists was much more valuable to me than meeting the current giants in the field though.

    i never finished the crackpot paper, but i got about halfway through it. it s a real hoot, i ll show it to you, if you want, you will get a real kick out of it. it s way the fuck out in left field.

    i guess i m doing particle physics. actually, what i m doing this semester is more like mathematical physics (which suits me just fine). right now, i m preparing a talk on classifying charged d-branes using k-theory, which is a sort of topological way of classifying fibre bundles. the talk is in about 2 weeks, and i regret to say that right now, i feel hopelessly ill-prepared.
     
  20. ryans Come to see me about a dog hey Registered Senior Member

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    995
    I would love to see that paper.

    I just gave a talk recently on the recent progress of my research, which I thought was shit. But since most of the people there didn't know what I was talking about anyway, I got positive feedback, which is good. But I go to a small uni in Australia. Where are you at? Big physics Dept.
     
  21. lethe Registered Senior Member

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    2,009
    attached.

    yeah, they are never as good as you imagine they will be in your head, i have found.

    i am at a large state university in the US, one which has a bridge program between mathematics and physics. for better or for worse, that means that most of the physics i do has to do with string theory. fortunately, there is enough mathematics in this area that i don t have to worry about it. but sometimes i worry.
     
    Last edited: Nov 13, 2003
  22. ryans Come to see me about a dog hey Registered Senior Member

    Messages:
    995
    I have started reading the paper.

    I am going to do the crackpot test to see if it qualifies, i.e. how many terms that are introduced in this paper.
    Thanks.
     
  23. lethe Registered Senior Member

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    yeah, lemme know what you think. i think it is more crackpotty than anything that ever appeared on sciforums, made all the worse because the author has approximately a high school level understanding of nuclear physics and GR.
     

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