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Maybe it seems so easy, but it is not!

How can we obtain the normalization constant $N$ for a set of eigenfunctions with different domains?

For example, we have

$\psi_{1}=N(f_{1}e^{-\kappa x}+g_{1}e^{\kappa x}),\hspace{1cm}x\in[0,1],\hspace{.2cm} \nonumber \\ \psi_{2}=N(f_{2}e^{-\kappa x}+g_{2}e^{\kappa x}),\hspace{1cm}x\in[0,1], \hspace{.2cm} \nonumber \\ \psi_{3}=N(f_{3}e^{-\kappa x}+g_{3}e^{\kappa x}),\hspace{1cm}x\in[-1,0], \\ \psi_{4}=N(f_{4}e^{-\kappa x}+g_{4}e^{\kappa x}),\hspace{1cm}x\in[-1,0].\nonumber$.

we can normalize each wavefunction by the integral $\int^{x2}_{x1}\psi^{*}\psi dx=1$, but that way, the other eigenfunctions are not normalized to one!

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closed as unclear what you're asking by John Rennie, Kyle Kanos, Jon Custer, Aaron Stevens, Cosmas Zachos Aug 14 at 14:16

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    $\begingroup$ why do you want to normalize the individual pieces? $\endgroup$ – Wolphram jonny Aug 12 at 16:34
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    $\begingroup$ Very confused by "other eigenfunctions" The definition has to include the space on which they exist and any boundary conditions. For example are these functions required to vanish at the end points, or be periodic? If k is the mode index then your normalization might be a function of k and hence should work for all eigenfunctions (provided they are defined in a meaningful way). Why, for example, are some of your functions defined on one space while other on another space? $\endgroup$ – ggcg Aug 12 at 16:45
  • $\begingroup$ boundary conditions have been included in the coefficients. $\endgroup$ – Baran Aug 12 at 17:03
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    $\begingroup$ Here is what everyone is asking you, I think: Are your wave functions living into different superselection domains, two different problems disjoint from each other, or do they somehow "talk to each other"? Is, e.g., $\psi_1$ also defined in [-1,0], except vanishes there now -- but can leak in, in future? $\endgroup$ – Cosmas Zachos Aug 12 at 21:19
  • $\begingroup$ Thanks, Cosmas. No, the wavefunctions can not leak in. They just live in the mentioned domain. $\endgroup$ – Baran Aug 12 at 22:47
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If you're doing some sort of scattering problem like in this setting (maybe there's a smoothly varying potential over some range $[a,b]$, a delta function potential at some other point $x=c$ etc.) the normalization of the individual wavefunctions is pretty much arbitrary. The point is that you will be gluing together eigenfunctions that live on different domains at the point where the domains meet - here, at $x=0$. The choice of normalization drops out of any physical quantities (transmission or reflection coefficients etc.).

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