# A rigorous treatment of distributions in quantum mechanics

In many introductory courses to quantum mechanics, we see $\delta$-functions all over the place. For example when expressing an arbitrary wave function $\psi(x)$ in the basis of eigenfunctions of the position operator $\hat x$ as $$\psi(x) = \int\mathrm d\xi\, \delta(x-\xi)\, \psi(\xi).$$ In bra-ket notation this corresponds to $$\left|\psi\right\rangle = \int\mathrm d\xi\,\left|\,\xi\,\right\rangle\!\left\langle\,\xi\,\middle|\,\psi\,\right\rangle,$$ where $\left|\,\xi\,\right\rangle$ is the state corresponding to the wavefunction $x\mapsto\delta(x-\xi)$. Now the $\delta$-function is really not a function, but a distribution, that's defined by how it acts on test-functions, i.e. $\delta[\varphi] = \varphi(0)$.

Do you know of an introductory text on quantum mechanics that stresses this point and uses the language of distributions properly, avoding any functions with seamingly infinite peaks?

Perhaps you are starting by the wrong end. Your concern seems to be related in the first term with the totally misleading notation of integrals in quantum mechanics, and this is more related with the spectral theorem than with distributions itself. Distributions only appear in Quantum mechanics when certain operators has empty spectrum in the usual Hilber space. Then, you need to consider a bigger underlying space.

So, for the integral notation and interpretation you should start with a pure mathematical book like Walter Rudin "Functional Analysis". There is nothing here related with physics since this "projection-measure integrals" pertains to the world of pure mathematics and spectral theorem. This is a totally rigourous pure mathematics book, so be equipped with a strong will to read in from begginign to end,

Once you have the mathematical background and feel totally comfortable with the integrals of quantum mechanics as (which is nothing more tan spectral thorem) you can move onto distributions in quantum mechanics, which are developed in the context of Gelfand triplets. An excellent reference is "The role of the rigged Hilbert space in Quantum Mechanics" from Rafael de la Madrid. It is freely available in the web.

• This is probably because you do not need distributions so much to do basic QM in a rigorous way ;-). However if you look at the entry "generalized eigenvector" on the index, you will find some reference to $\delta$ states of physicists (see page 124 and section 6.6 in particular) Commented Jul 15, 2014 at 15:29