In what cases is it wrong to treat Dirac delta function as an eigenvector?

You should never treat anything as an eigenvector if it doesn't belong to the vector space on which your operator is operating.
You may always treat something as an eigenvector if it does belong to the vector space on which your operator is operating, if it is nonzero, and it satisfies Tv=λv Tv=\lambda v where T T is your operator, v v is your vector and λ \lambda is some scalar.
That's it. That's all there is to it.
Now, sometimes you have an operator and you have something like a Dirac delta function that you really really want to use as an eigenvector of that operator, but you can't, because it's not in your vector space.
Your safest option is to find a better vector space, one that does contain delta functions, and extend your operator to act on this larger space, and all is good. Your less safe option is to reason informally, pretending that you've done the work without actually doing the work, and play around with expressions that aren't formally defined.
This is also fine if you have experience and common sense. Physicists tend to do these things more often than mathematicians. But whenever you're unsure, or run into trouble, remember that math will never fail you if you clearly define the objects you're working with and what they are supposed to do.
Specifically, for Dirac delta functions, the space you want is the space of distributions. Distributions are very nice and very natural objects that generalize ordinary functions. They form a perfectly well defined vector space, and many operators that come up in physics or pure math can be migrated to operators that act on spaces of distributions. If your operator is like that, do it: migrate it. If it's not, don't. If something is stopping your operator from acting on distributions, don't pretend that it does.
(In the question details, you said "Given that it's not in the Hilbert space". That statement doesn't make sense because there's no such thing as "the" Hilbert space. What Hilbert space? Furthermore, you seem to be thinking of the quantum mechanical formalism of position; this formalism works great even without introducing Dirac delta functions, simply by analyzing probabilities of finding the particle in any measurable set. But if you want to bring in the Dirac delta function, you can certainly do that.)

Answer by Alon Amit:

You should never treat anything as an eigenvector if it doesn't belong to the vector space on which your operator is operating.
You may always treat something as an eigenvector if it does belong to the vector space on which your operator is operating, if it is nonzero, and it satisfies [math]Tv=\lambda v[/math] where [math]T[/math] is your operator, [math]v[/math] is your vector and [math]\lambda[/math] is some scalar.
That's it. That's all there is to it.
Now, sometimes you have an operator and you have something like a Dirac delta function that you really really want to use as an eigenvector of that operator, but you can't, because it's not in your vector space.
Your safest option is to find a better vector space, one that does contain delta functions, and extend your operator to act on this larger space, and all is good. Your less safe option is to reason informally, pretending that you've done the work without actually doing the work, and play around with expressions that aren't formally defined.
This is also fine if you have experience and common sense. Physicists tend to do these things more often than mathematicians. But whenever you're unsure, or run into trouble, remember that math will never fail you if you clearly define the objects you're working with and what they are supposed to do.
Specifically, for Dirac delta functions, the space you want is the space of distributions. Distributions are very nice and very natural objects that generalize ordinary functions. They form a perfectly well defined vector space, and many operators that come up in physics or pure math can be migrated to operators that act on spaces of distributions. If your operator is like that, do it: migrate it. If it's not, don't. If something is stopping your operator from acting on distributions, don't pretend that it does.
(In the question details, you said "Given that it's not in the Hilbert space". That statement doesn't make sense because there's no such thing as "the" Hilbert space. What Hilbert space? Furthermore, you seem to be thinking of the quantum mechanical formalism of position; this formalism works great even without introducing Dirac delta functions, simply by analyzing probabilities of finding the particle in any measurable set. But if you want to bring in the Dirac delta function, you can certainly do that.)

In what cases is it wrong to treat Dirac delta function as an eigenvector?

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