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Homework Help: Free particle wave equation

  1. Aug 28, 2005 #1
    for a free particle, the wave equation is a superposition of plane waves,

    [tex]\psi(x,0)= \int_{-\infty}^{\infty}g(k)\exp(ikx)dk[/tex]
    and
    [tex]g(k)= \int_{-\infty}^{\infty}\psi(0,0)\exp(-ikx)dx[/tex]

    one is the fourier transform of the other. some cases to solve this is when we assume a small delta k, so g(k) behaves like a pulse "delta" function.

    is there any more general case we can solve this?

    i have been thinking hard if we have definite periodic x, say from 0 up to [itex]2\pi L[/itex], is it solvable?

    what would be the (periodic) eigen energy function (if it is)?
     
  2. jcsd
  3. Aug 29, 2005 #2

    vanesch

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    The above pair is, as you say, a Fourier transform pair. Any "nice" function can be written that way, so the above is not a "wave equation" or something, it is a general way of writing a function.
    The quantum state of a single scalar particle is described by just such a nice function, called the wave function. At any time, it can be (almost) any function. However, what the wave equation (not written here) gives you, is how this wavefunction AT A CERTAIN TIME t0 will change into the wavefunction at another time t1. This equation will be different according to the situation at hand (free particle, particle in a potential...).

    cheers,
    Patrick.
     
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