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The eigenfunctions of a Hermitian operator are real. But consider a function $\psi(x)=e^{-\kappa x}$, $x\in\mathbb{R}$, where $\kappa$ is a real constant. Then, $$\hat p \psi(x)=-i\hbar \frac{d}{dx}e^{-\kappa x}=i\kappa \hbar \psi(x).$$ This gives a pure imaginary eigenvalue. Is it not a contradiction? Or am I missing some crucial point?

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Interesting question. Just pointing out the eigenvalues of a Hermitian operator are real, not the eigenfunctions. –  JeffDror Feb 12 at 13:48
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this function is unphysical –  Danu Feb 12 at 14:06
    
Unphysical doesn't mean that it is mathematically impossible. I believe that the right answer lies on the rigged hilbert space associated is $L^2(\mathbb{R})$, i have to check but I think that $\psi(x)=e^{-\kappa x}$ with real $\kappa$ don't lie there. Anyway, you don't need eigenvectors to define eigenvalues, and the spectral theorem, in this case, rules out non-real eigenvalues –  user23873 Jun 6 at 18:39

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What is your Hilbert space? In $L^2(\mathbb R)$ your eigenfunction would have infinite norm. If you dealt instead with a bounded set $L^2([a,b])$, your operator would not be Hermitian unless you impose suitable boundary conditions to discard boundary terms. These boundary conditions, however, would rule out your candidate eigenvector!

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Okay. I was guessing that while specifying an operator we must also specify its domain. Is that right? –  Roopam Feb 12 at 14:08
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Yes you must also specify the Hilbert space and the domain therein! –  Valter Moretti Feb 12 at 14:10
    
V. Moretti-But even $e^{ikx}$ is not a member of $L^2(-\infty,\infty)$. But this gives real eigenvalue for the same operator. Therefore, what I understood from your answer is that unless the domain is specified it is not guaranteed that the eigenvalues of the hermitian operator are real, it could be be anything, real or imaginary. –  Roopam Feb 12 at 14:21
    
(I rewrite, since it was unclear as I sent it from my mobile phone). Yes, $e^{ikx}$ is a generalized eigenvector like $\delta(x)$ for the position operator...For self-adjoint operators, reality condition holds for generalized eigenvectors, too, but it is more technical to prove. –  Valter Moretti Feb 12 at 18:57

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