How do you normalize this wave function?

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Discussion Overview

The discussion revolves around the normalization of wave functions in the context of quantum mechanics, specifically for a Hamiltonian involving a Dirac delta potential. Participants explore the properties of eigen wave functions, their orthogonality, and the implications of singular potentials on normalization.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant questions how to determine the normalization factor ##N_k## for the wave functions associated with the Hamiltonian involving a delta potential.
  • Another participant notes that wave functions like ##e^{ikx}## cannot be normalized in the traditional sense in position space, referencing properties of Fourier transforms.
  • Some participants assert that while the wave functions are not normalized to unity, they can be normalized to a Dirac delta function, which is the goal of the inquiry.
  • Concerns are raised about the interpretation of singular Hamiltonians and the need for renormalization to make physical sense of such systems.
  • There is a discussion about the orthogonality of wave functions and the mathematical challenges in proving this for continuous spectra, with references to spectral projections and distinct eigenvalues.
  • One participant suggests using a regularization technique to handle integrals involving complex exponentials, indicating that this approach may help in resolving normalization issues.
  • Another participant mentions that the terminology used in the context of normalization may not be technically correct, but is commonly accepted in educational materials.

Areas of Agreement / Disagreement

Participants express differing views on the normalization of wave functions, with some agreeing that normalization to a Dirac delta function is possible, while others emphasize the limitations and challenges posed by singular potentials. The discussion remains unresolved regarding the exact methods to achieve normalization and the implications of these methods.

Contextual Notes

Participants highlight limitations in the treatment of singular Hamiltonians and the need for renormalization, particularly in higher dimensions. The discussion also touches on the mathematical subtleties involved in proving orthogonality for continuous spectra.

  • #31
Isaac0427 said:
Griffiths states for an integral like that (specifically with showing that the integral of exp(ikx) from negative infinity to infinity is zero), although it does not converge, you can replace infinity with L (which I see you have done), and take the average value of the integral as L goes to infinity. This takes you to zero. Thus (based on Griffith’s text— and he acknowledges that it would drive a mathematician insane) I would conclude that the previously mentioned sine integral would be equal to zero when k=k’ (which, of course, is the result we want).

Really many thanks. But I don’t think that strategy works here. That average would not yield zero for the integral.
 
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