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Help with a extract of Schaum

  1. Jun 15, 2011 #1
    Hello,
    In http://img96.imageshack.us/img96/2825/sinttulobdd.png" [Broken] from Schaum's Quantum Mechanics, I don't get why they say that A2 (with ' ) must be zero; I think in that case [itex]\Phi_{II}(x)[/itex] will diverge if x=+infinite, and we don't want that function to diverge, do we?

    Thanks!
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
  3. Jun 15, 2011 #2

    vanhees71

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    With this step you simply fulfill the boundary condition that there should be incoming particles only from the left and not from the right.
     
  4. Jun 15, 2011 #3
    Yes, but it you put the condition that when x->infinite, the wave function must be zero, the it doesn't match!
    Maybe this condition that I say is just when E<V...
     
  5. Jun 16, 2011 #4

    vanhees71

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    If you are in the continuous part of the spectrum your "eigenstates" are not Hilbert-space vectors but generalized eigenstates. They belong to the dual of the nuclear space in the sense of the G'elfand triple (="rigged Hilbert space").

    The most simple example are the "eigenstates" of the momentum operator. These are (in position representation) just the plane waves (here for simplicity written for the one-dimensional motion of a particle)

    [tex]u_p(x)=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} x p).[/tex]

    This is not a square-integrable function. It can not be normalized to 1 in the usual sense but only to a [itex]\delta[/itex] distribution, according to

    [tex]\langle p_1|p_2 \rangle = \int_{-\infty}^{\infty} \mathrm{d} x \; u_{p_1}^*(x) u_{p_2}(x)=\delta(p_1-p_2).[/tex]
     
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