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<Moderator's note: this thread was split off from https://www.physicsforums.com/threads/eigenstates-eigenvalues.904774/>

Well, the unfortunate thing, pedagogically, is that in teaching about eigenfunctions and eigenvalues, the most obvious operators to use for examples are the position operator and the momentum operator. But those operators don't correspond to normalizable eigenfunctions. So you're left without any examples to motivate the idea of an eigenvalue and eigenfunction.

I suppose you can use matrices to motivate the ideas. But for wave functions, it's hard to see what's a simple example that doesn't have some problem or other. Maybe the harmonic oscillator wave function being an eigenfunction of energy?

The Dirac ##\delta## "function" is not a function but a distribution (in the sense of "generalized function"). It should be forbidden to all textbook writers to call it a function!

A pure state is described by a square integrable function, and neither the ##\delta## distribution nor the plane-wave ##\exp(\mathrm{i} \vec{k} \cdot \vec{x})## are square integrable. So these "generalized eigenfunctions" of position or momentum, respectively, do not represent pure states of a particle.

Well, the unfortunate thing, pedagogically, is that in teaching about eigenfunctions and eigenvalues, the most obvious operators to use for examples are the position operator and the momentum operator. But those operators don't correspond to normalizable eigenfunctions. So you're left without any examples to motivate the idea of an eigenvalue and eigenfunction.

I suppose you can use matrices to motivate the ideas. But for wave functions, it's hard to see what's a simple example that doesn't have some problem or other. Maybe the harmonic oscillator wave function being an eigenfunction of energy?

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