Momentum operator -- Why do we use the plane wave solution?

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Discussion Overview

The discussion revolves around the use of plane wave solutions in quantum mechanics (QM) and their significance in deriving the momentum operator, as well as their importance in field theory and particle physics. Participants explore the conceptual and mathematical implications of using plane waves in these contexts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants propose that plane wave solutions are essential because any field in quantum field theory (QFT) can be expressed as a superposition of plane waves.
  • Others argue that there may be a deeper significance to plane waves beyond their practical utility, questioning whether they are merely the most physically appropriate choice.
  • One participant highlights the utility of decomposing fields into harmonic oscillators, noting that this approach allows for complete analytical solutions to the harmonic oscillator problem in QM.
  • Another participant expresses dissatisfaction with the justification that plane waves provide "nice solutions," suggesting that this reasoning feels inadequate from a fundamental QM perspective.
  • A later reply introduces a modern approach to deriving the momentum operator through the Lie algebra of Galilean transformations, suggesting an alternative to traditional methods involving plane waves.

Areas of Agreement / Disagreement

Participants express a mix of agreement and disagreement regarding the significance and justification for using plane wave solutions. While some acknowledge their utility, others seek deeper insights and express concerns about the foundational reasoning behind their use.

Contextual Notes

Some arguments presented regarding the significance of plane wave solutions are noted to lack structure or depth, indicating potential limitations in the discussion. The exploration of alternative derivations of the momentum operator suggests that multiple approaches exist, but the discussion does not resolve which is more valid or preferable.

zb23
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Why in order to derive the QM momentum operator we use the plane wave solution. Why later on in field theory and particle physics, the plane wave ansatz is so physically important?
 
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Because we can write any field in qft as a superposition of plane waves.
 
Yes, I agree, but still, is there some deeper meaning behind plane waves? Is it "just" most physically appropriate?
 
I think that it is quite nice tool when we can decompose any field into a set of harmonic oscillators. One of the reasons is that we can solve the harmonic oscillator problem in QM completely analytically. No perturbation theory or variations or any numerical method.
 
Yes, I understand that but aren't the things you said just consequences? It is like it bothers me in some fundamental QM way. I mean I can't be satisfied with the argument that it gives nice solutions. I mean, it is obviously correct considering the experiments in QM.
 
zb23 said:
I mean I can't be satisfied with the argument that it gives nice solutions.
That's the main basis for mathematical physics. Why represent force as a vector? Why is time a parameter in non-relativistic physics? Why have a spacetime manifold in relativity?
 
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Of course, I understand, I mean I am currently doing field theory in particle physics and some arguments for plane wave solutions, for me, lack some structure, therefore I asked...I just wanted to get some more insight into some basics...
 
zb23 said:
some arguments for plane wave solutions, for me, lack some structure
Can you give some specfic references to arguments that you find lacking?
 
zb23 said:
Why in order to derive the QM momentum operator we use the plane wave solution.
You don't have to do it in that old-fashioned way.

A more modern approach is to examine the Lie algebra of Galilean transformations as an abstract group. Then, representing that algebra in coordinate representation, implies that the operator of spatial translation, (##P_i = -i \hbar \partial_i##), must satisfy ##P = MV## (for a free particle), which is the usual expression for momentum.

For a detailed exposition of this, see Ballentine chapters 3 and 4.
 
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