Q:

Good book on the history of Quantum Mechanics? 

Can anyone recommend a good book on the history of Quantum Mechanics, preferably one that is technical and not afraid to explain the maths (I did a degree in Physics many years ago) and also that explains the developments using the technical language of the time rather than using treatments as we know understand them.  If it also explained the input from experiments that would be great too. 

A:

There is an issue with the historical texts, that matrix mechanics is always glossed over, and if it is explained at all, it is explained completely incorrectly.  The main idea of matrix mechanics is that the off-diagonal matrix elements are the Fourier coefficients of the classical quantities in the Bohr orbits.  This idea is not stated anywhere with the exception of Heisenberg's paper, the Born Heisenberg Jordan "Dreimannarbeit", a certain quantum mechanics text I read twenty years ago (I can't remember the author, I suspect it was Born, but this didn't google for me), and Wikipedia.  The Wikipedia article on Matrix Mechanics was written to fill this demoralizing hole in the literature. 
To my mind, the development of matrix mechanics is the central event in the history of quantum mechanics.  Everything before is groping towards it, and everything after is only clarification and better calculations.  Matrix mechanics seems to have been only fully understood by the original authors, and a handful of others, like Pauli, Dirac, Bethe. 
Einstein and Schrodinger got to quantum mechanics by a completely different route, which allowed them to internalize the matrix methods.  They always thought of matrix elements in the only way they are presented in modern textbooks, as overlaps of wavefunctions with differential operators acting, not as best-possible quantum analogs of the Fourier coefficients of the periodic Bohr orbits. 
Having said that, here are some historical texts.  I think that the best histories are not written as histories at all, but as monographs.  My personal favorite for earliest quantum theory is

Ter Haar "The Old Quantum Theory"

This monograph explains the ideas of pre-Heisenberg quantum mechanics, as developed by Wien, Planck, Einstein, Bohr, Sommerfeld, Pauli and others.  It doesn't include Kramers and Heisenberg work. 
Aside from Wikipedia's matrix mechanics page, one of the only development of matrix mechanics in the literature which is comprehending seems to be:

Connes "Noncommutative Geometry"

The early chapters of this wonderful mostly unrelated mathematics text contains a presentation of the logical development of matrix mechanics which is 100% accurate in terms of the technical content, and very close to Heisenberg's original reasoning.  But it is not a history book. 
Once you get matrix mechanics, the original papers are pretty much enough to reconstruct what everyone was thinking, these are here: 

B. L. van der Waerden "Sources of Quantum Mechanics"

For the development of wave mechanics, there are many good sources, basically every quantum mechanics books.  Wave mechanics was pretty much understood by everybody, because it was in the style of the older generation.  Dirac's book gives a reasonable account of the formalism of quantum mechanics as best understood until Feynman and Schwinger revolutionized everything.  For this era, there is 

"QED and the Men Who Made It" by Schweber

which is best read with Schwinger's excellent collection of reprints

"Selected Papers on Quantum Electrodynamics" J. Schwinger

For me, the important parts are usually left out of all of these.  The path integral is only properly explained in the 1970s, by authors other than Feynman, although both Dyson and Mandelstam explained it well in the 1950s.  Feynman's presentations are pretty good, but they do not give the cleanest explanations of the relation to Wiener integrals or imaginary time.  Feynman certainly understood this stuff, it is just not best presented in his book or his articles (although they are required reading, of course). 
I forgot to add the last important formal development in quantum mechanics, the Grassmann integral.  To call this paper scandalously neglected is the understatement of the century.  Mandelstam is basically the only person who cites this paper! 

D.J. Candlin, Nuovo Cimento 4, 224 (1956)

This is the original source of all the Grassmann methods used by everyone today.  The author appears nowhere else in the theory literature (he might appear again as an experimentalist twenty years later, or maybe its someone else with the same name), but this paper is a great classic.  Similar converging work is due to Salam and collaborators, to Berezin in the Soviet Union, to Feynman and others, but nobody had the right idea except Candlin, and everyone copies him without attribution, until by the late 1960s, everyone is falsely attributing the results to Berezin's 1960's quantum mechanics textbook.  To be fair, Berezin went somewhat further, but Candlin's article is clearer and it is ten years earlier. 

A:

You might like Inward Bound by Abraham Pais. 
Author was a particle physicist.   The book is mostly a history of particle physics, but quantum mechanics is heavily intertwined.   Otherwise it meets your criteria perfectly. 

A:

Although it dates from 1966, and only just comes up to your requirement for mathematics, Max Jammer's "the conceptual development of quantum mechanics" is still a good read.  I went through Web of Science to look at whatever reviews there might that cited the Jammer, and found http://dx.doi.org/10.1119/1.15057, "RESOURCE LETTER HP-1 - HISTORY OF PHYSICS" (AMERICAN JOURNAL OF PHYSICS, Volume: 55, Issue: 8, Pages: 683-691, Published: AUG 1987). 
Hopefully it's fair use to include the relevant page here as a PNG,

