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Homework Help: Wave function dimensions

  1. Jun 21, 2008 #1
    1. The problem statement, all variables and given/known data

    Does the wave function have a dimension? If it does, what are the dimensions for 1D and 2D box problems?Can you generalise this to n dimensions?


    2. Relevant equations



    3. The attempt at a solution

    Yes, it does have dimensions. For 1D box it's [tex] m^{-2} [tex], for 2D box it's [tex] m^{-1} [tex] thus for n dimensions it should be [tex] m^{n-3} [tex]. Is this correct?

    thanks
     
  2. jcsd
  3. Jun 21, 2008 #2
    I think it has no dimensions. p(r) = |\psi|^2 is the density of the probability of finding a practical in a specific location.

    But maybe i'm wrong
     
  4. Jun 21, 2008 #3
    a wave function inside a 1 dim box is
    [tex]\Psi= \sqrt{\frac{2}{L}}sin(\frac{px}{\hbar})[/tex]
    it appears it is per root meter, which is weird. but on the other hand, the probability in the interval dx is:
    [tex] |\Psi|^{2} dx = \frac{1}{\sqrt{meter}}^{2}*meter = 1 [/tex]
    which would make it dimensionless, which could make sense. ive never asked myself this.
     
    Last edited: Jun 21, 2008
  5. Jun 22, 2008 #4

    Mute

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    Homework Helper

    It's not dimensionless. If it were, then when you integrated its square modulus over all space you would not get a pure number. (Note to the second poster - the modulus squared of the wavefunction is not a probability, but a probability density, which had dimensions.). That is, since

    [tex]\int_{\rm all space} d\mathbf{r} \left|\psi(\mathbf{r})\right|^2 = 1[/tex]

    [itex]\psi(\mathbf{r})[/itex] must accordingly have the root of the inverse dimensions of [itex]d\mathbf{r}[/itex], which are length^{-1/2} for a 1D problem ([itex]d\mathbf{r} = dx[/itex]), length^{-1} for a 2D problem ([itex]d\mathbf{r} = dxdy[/itex]), etc.

    So, the original poster is correct about the wave functions having dimensions, but you got the dimensions incorrect.
     
    Last edited: Jun 22, 2008
  6. Jun 22, 2008 #5
    I stand corrected. although I did say |psy|^2 is the density of probability.
     
  7. Jun 22, 2008 #6
    Oh, I see. Thanks
     
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