Distributions (e.g., Dirac Delta): confused and unhappy [closed]

Question

I am sorry that the following set of questions is very fuzzy and ill informed.

I am a trained mathematician and now studying an undergraduate theoretical physics course. We use distributions. I have no clue of the How and Why, which makes me unhappy.

So far I have some tiny idea about the physical intuition; and I have read the definition (and some properties) in http://en.wikipedia.org/wiki/Distribution_(mathematics) .

Very fuzzy questions

Sorry for the vagueness. If you do not want to be bothered but would kindly be willing to answer more concrete questions please jump to the next section.

1. Distributions make most sense to me in the following mindset: We have a situation A too complicated to calculate, we approximate it as B using step functions or delta distributions or similar stuff, which we can solve. Then we somehow claim that this solution (e.g., equations of motion) of B is close to the solution of the original problem A. Question: Is there any good argument for this claim?
2. Continuing the previous question: My (very poor) physical intuition would rather be: a delta distribution is the limit of stuff which has integral 1 and smaller and smaller support. The derivatives have no role in this intuition, which seems incompatible with even the most basic mathematical contents of distributions: The topology of the test function space uses (higher order) derivatives. Question: Is the simple intuition just nonsense? Is there a better working intuition?
3. Specific rant: It makes me particularly uneasy that some QM texts (cf.\, for instance, the introduction of Cohen-Tannoudji et al's) often use as an argument in a proof that wave functions have to be continuous, bizarrely arguing that non-continuous things make no physical sense because of measurement precision(!), while in the same proof happily using non-continuous potentials, distributions (and, of course, ignore the fact that the basic operators are non-continuous). Question: Is there any valid point in Cohen-Tannoudji et al's argument (that wave functions have to be continuous, even when appearing in solution for artificial problems that where constructed using distributions)? Which one?
4. I am completely lost about which calculus concepts/constructions/arguments are still OK with distributions and which ones are not. Well, obviously you can differentiate, and you can generally not multiply. But this leaves open the vast area of vector calculus, where it is not at all clear to me what is OK and what is not. I see two ways to rectify this: Either study mathematical physics for 5 years and then arrive at a very complicated mathematical understanding of simple stuff that physicists already knew anyway; or just continue to stumble through the horrors of distributions, without a clue, and somehow get used to it and learn simply by example what you can do and what not. Both sounds depressing. Question: Is there any better way to learn about the practical use of distributions in a consistent way, including some mathematical justification?

Slightly less fuzzy questions

1. (Classical) functions that are not locally integrable, such as $\frac{1}{x}$, can not be interpreted as distributions. Is that correct?
2. In particular, the formula $\Delta\frac{f(r)}r=-4 \pi f(0) \delta^{(3)}(\vec x)+\frac{f''(r)}r$ does not seem to make much sense if $\frac{f''(r)}{r}$ is not integrable?
3. Speaking of which: $\Delta\frac{1}r=-4 \pi \delta^{(3)}(\vec x)$. What does this even mean? (It seems that $\frac{1}r$ is not a distribution, how can I differentiate it?) And, once it is clarified what it means: How is it proven?

Thank you for your patience.

Answer 1

First of all, I would like to recommend the following book. It is the best introduction to distributions -and many other topics- I have ever seen, insisting on intuition and applications in physics, but the mathematical presentation should be rigorous enough to give you an entry point into the literature. As for your questions:

1. The simplest example is that of a problem with two very different length scales, e.g. the electric field created by a number of charged objects, much smaller than the typical distance between them (and hence than the distance over which the electric field varies). We can then treat these objects as point charges. One can justify this approach a posteriori, by going to a more detailed description of the system and showing that the solutions are close enough.
2. This is indeed a good way of seeing the Dirac delta. The trick with differential operators is to shift them from the distribution onto the test function by partial integration. If you are uncomfortable with this, you can also go back to the limiting process that you allude to and perform the partial integration with "nice" functions that approach the distribution.
3. Discontinuous wave functions would have infinite momentum (although they do seem relevant for some class of peculiarly "spiky" potentials). In complement H1, Cohen-Tannoudji & al. show that the wave function remains continuous at a step potential by taking the limit of smooth potentials. I cannot find the place where they invoke measurement precision.
4. I think a good idea would be to start from a physicist's point of view, hence the reference above.

The final questions:

As to $\frac{1}{x}$, see the answer by @5891user; you can define it using Cauchy's principal value and show that it has the expected propeties, e.g. that $\text{pv} \frac{1}{x} \cdot x = \mathbb{1}$ (the identity). Thus, your last three questions pose no particular difficulty apart from acquiring an intuitive understanding of these equations (which can be a challenge).