I am glad you brought up bra-ket notation. I have wasted a lot of time trying to understand how to formalize bra-ket notation. I kept seeing the words "C* algebra" and "rigged hilbert space" being thrown around. I recently learnt the full story:
- Von Neumann's paper "mathematical foundations of quantum mechanics"[http://alpha.math.uga.edu/~davide/The_Mathematical_Foundatio...] builds up hilbert space theory and solves problems without ever using dirac deltas. So if one wants to use bra-ket without dirac deltas, this is the formalism.
- To make dirac deltas themselves precise, forgetting the bra-ket context, one uses the language of distributions. So we view the dirac delta distribution as a linear functional which takes a function `f` and spits out the value `f(0)`, and all that jazz.
- To join distribution theory with Hilbert space theory, we construct rigged Hilbert spaces. This is the theory that carefully delineates from which subset of Hilbert space we can pick kets, and from which larger space of distribution we can pick bras, to allow the bra-ket formalism to continue to work.
This is the sort of thing that drives me nuts. I bought and tried to read Shankar, because I was told it is "rigorous". It's not. It casually uses dirac deltas and all sorts of "punning" with bra-kets with zero formalism.
Do you have a good recommendation for learning this heat and work perspective? I always found this confusing [eg. "adiabatic work", "infinitely slowly" and all that]. FWIW, I wanted to learn stat-mech for a closer look at entropy and information, which I felt I got with some LL.
I don't have the mathematical maturity to comment on the rigour of the bras and kets, but as to heat and work I was referring to thermodynamics in the sense of Carnot and co.
I picked Zemansky and Dittman off the shelf in the library and it's honestly really nice little book - the aim here is for intuition (particularly for experiment) while also making connections to the more theoretical perspective found in L&L (in my case this is because I can't be bothered to read 300 pages of formalism to get to the applications although YMMV). It is quite handwavey in the ways that you describe but I think that is the jazz that the great physicists played in order to get the result in the first place.
I learned stat mech from L&L, and had the same problem that I never really understood thermodynamics.
I've been working through Bohren and Albrecht's 'Atmospheric Thermodynamics' recently, which I've been really enjoying. They are much more concrete, and very opinionated about ditching notation and concepts that are unclear, such as the differentials that get bandied about in most other books.
I think the key problem is that most thermodynamics books try to develop the axiomatic theory in the abstract, as opposed to introducing real materials and building physical intuition with them first.
Isn't doing physics in a mathematically rigorous way very hard or close to impossible? I think Quantum Field Theory and the Standard Model remain non-rigorous because of their dependence on path integrals.
- Von Neumann's paper "mathematical foundations of quantum mechanics"[http://alpha.math.uga.edu/~davide/The_Mathematical_Foundatio...] builds up hilbert space theory and solves problems without ever using dirac deltas. So if one wants to use bra-ket without dirac deltas, this is the formalism.
- To make dirac deltas themselves precise, forgetting the bra-ket context, one uses the language of distributions. So we view the dirac delta distribution as a linear functional which takes a function `f` and spits out the value `f(0)`, and all that jazz.
- To join distribution theory with Hilbert space theory, we construct rigged Hilbert spaces. This is the theory that carefully delineates from which subset of Hilbert space we can pick kets, and from which larger space of distribution we can pick bras, to allow the bra-ket formalism to continue to work.
I found the reference "The role of rigged Hilbert spaces in QM"[https://arxiv.org/pdf/quant-ph/0502053.pdf] incredibly valuable.
This is the sort of thing that drives me nuts. I bought and tried to read Shankar, because I was told it is "rigorous". It's not. It casually uses dirac deltas and all sorts of "punning" with bra-kets with zero formalism.
Do you have a good recommendation for learning this heat and work perspective? I always found this confusing [eg. "adiabatic work", "infinitely slowly" and all that]. FWIW, I wanted to learn stat-mech for a closer look at entropy and information, which I felt I got with some LL.