Why Can't Free-Particle Wave Functions Be Normalized Over Their Entire Range?

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Discussion Overview

The discussion revolves around the normalization of free-particle wave functions over their entire range of motion, specifically addressing the implications of using plane waves and the nature of wave packets in quantum mechanics. The scope includes theoretical considerations and conceptual clarifications related to quantum mechanics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • Some participants assert that normalizing free-particle wave functions over the entire range from minus infinity to plus infinity leads to divergence in the integral of |u(x)|².
  • Others propose that while plane waves like exp(ikx) are not normalizable, wave packets constructed from these plane waves can be normalizable but do not represent eigenstates of the Hamiltonian.
  • A participant questions the justification for the behavior of wave functions at infinity, particularly in relation to particles in a square potential barrier, suggesting a parallel to the free particle case.
  • There is acknowledgment of the assumption that "free particle" implies being an eigenstate of the free Hamiltonian, which some participants clarify may not necessarily hold true.

Areas of Agreement / Disagreement

Participants generally agree on the divergence issue related to normalizing free-particle wave functions, but there are competing views regarding the nature of wave packets and their relationship to eigenstates of the Hamiltonian. The discussion remains unresolved regarding the implications of wave functions at infinity.

Contextual Notes

Limitations include the dependence on definitions of "free particle" and "normalizable wave functions," as well as unresolved questions about the behavior of wave functions at infinity in various potential scenarios.

alimehrani
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why it is not possible to normalize the free-particle wawe functions over the whole range of motion of the particles?
 
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The whole range of motion is from minus infinity to plus infinity (no restricted range, otherwise the particle would not be free).
The free particle is described by a plane wave u(x) = exp(ikx).
The normalization means to integrate |u(x)|² = 1; the integral will certainly divergy
 
One should add that exp(ikx) is obviously not the only possible form for the wavefunction of a free particle. Rather the wavefunction may have almost any form, especially normalizable.
However, there are no normalizable wavefunctions which are eigenfunctions of the Hamiltonian.
 
Let me see if I understood you correctly.

You propose to use (nearly) arbitrary wave packets constructed from plane waves (Fourier modes) and let these wave packets evolve in time using the free Hamiltonian. That means you construct normalizable wave functions, but they are no longer eigenstates of the free Hamiltonian.

Yes, of course you are right. I assumed that "free particle" means "eigenstate of the free Hamiltonian H", but that need not be the case.

@alimehrani: what was your intention?
 
tom.stoer said:
Let me see if I understood you correctly.

You propose to use (nearly) arbitrary wave packets constructed from plane waves (Fourier modes) and let these wave packets evolve in time using the free Hamiltonian. That means you construct normalizable wave functions, but they are no longer eigenstates of the free Hamiltonian.

Yes, of course you are right. I assumed that "free particle" means "eigenstate of the free Hamiltonian H", but that need not be the case.

@alimehrani: what was your intention?

yes
yes
exactly
thank you very much
i think you are very good teacher
 
thank tom.stoer and others
you greatly help me
 
this problem exist also about a particle in a square potential barrier!?
what is the justification here?

another question:
we assume that wave function is zero at infinity while we couldn't observe such a thing in the cases of free particle and particle in a square potential barrier?
 

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