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Very stupid question

  1. Jul 17, 2011 #1
    I study mathematics and am in my sixth year, but...
    I have a very elementary question:

    I stumbled upon it while learning for quantum mechanics. But it's nothing new, it's happening to me all the time: I get confused by things like this!

    Observe the following facts:
    Suppose we deal with the space of wave functions over the real line.
    The wave function [tex]\psi(x)[/tex] is a complex scalar. Take for example [tex]\psi(x)=e^{ikx}[/tex] (not normalizable, don't need it)
    The derivative [tex]\psi'(x)=ike^{ikx}[/tex] has 1/length as it's unit.
    Integrating that over some interval yields a scalar. [tex]\psi(b)-\psi(a)[/tex]
    The probability density [tex]\psi*(x)\psi(x)[/tex] is a scalar.
    Integrating this over the real line (length) gives 1. A scalar...
    Shouldn't such an operation yield a length? Am I stupid?

    I am not joking. For me this is a mystery.

    Thanks for your answers.
  2. jcsd
  3. Jul 17, 2011 #2
    When you integrate the probability density (scalar) with respect to length, the answer will have the same dimension (scalar x length).
  4. Jul 17, 2011 #3


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    I can't get what you want to calculate and it seems for me that it hardly makes sense, but obvious mistake you did is that [itex]\int_a^b\psi^{*}(x)\psi(x) dx[/itex] is not equal 1, but [itex]b-a[/itex]
  5. Jul 17, 2011 #4
    Sorry, I was talking about integrating over the whole (1-dimensional) space here (which has the physical dimension of length). And the total probability (of an physical wave function, not the example I used) should be 1.
    But I see that my example e^ikx is not an example for a real wave function. And an actual wave function (in 1-space) actually has dimension sqrt(length), as for example the Gaussian wave packet:

    [tex]\psi(x)=\frac{1}{\sqrt{\sqrt{2\pi}\sigma}}[/tex] with sigma being a length.

    So my question was useless and came from wrong presumptions.
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