It turns out that this book precedes In Search of the Big Bang, but never mind. You'll just have to do a bit of time travel.
In the beginning, Gribbin explores (a small part of) the development of physics the period from Newton to the end of the nineteenth century, and describes the problems and "magic" rules that were dreamed up to fit experimental results (the ultraviolet catastrophe sounds particularly spectacular). He moves on to Planck's statistical handling of blackbody radiation, Einstein's explanation of the photoelectric effect, Rutherford's discovery of the nucleus and the Bohr atomic model (which includes a section amusingly entitled "Chemistry Explained").
In part two, new characters are introduced - de Broglie, Pauli, Heisenberg, Schrödinger and Dirac. There's no mathematics to follow here, although Gribbin does refer to matrix mechanics, Hamiltonians, and wavefunctions. He also cautions readers on the Bohr model: it's a familiar enough picture of the atom, just like the solar system, but it's not really like that. This part wraps up with a tour of the (more or less) everyday objects that we take for granted - such as lasers, microwave ovens, and electronics - that rely on quantum mechanics.
Part three devotes itself to more philosophical questions. Gribbin presents the Copenhagen interpretation with understanable discomfort: no matter how you look at it, it just doesn't seem right, yet Gribbin points out that the numerous attempts to overcome it merely strengthen it when scrutinised. The final chapter presents the Many Worlds interpretation, which Gribbin prefers, though the perpetual and seemingly infinite splitting into alternative realities seems at least as implausible as the Copenhagen viewpoint to me. An epilogue discusses briefly some of the work on Grand Unification that is focused on in the Big Bang volume.
The book is, for a physics graduate, surprisingly un-technical. Certainly Big Bang seems much more demanding. I think there are at least two effects at work. There is no half-way house for the mathematics of quantum mechanics: either you must omit it completely, or you must go into some depth. In astrophysics many of the arguments can be technical but made in a more qualitative way. The other effect may be Gribbin's own participatory universe: he links quantum mechanics (in particular the Many Worlds interpretation) to science fiction, and perhaps this volume was written in part for that general audience.
One interesting feature of reading these two books in the 'wrong' order was that I noticed an inconsistency: in this volume Gribbin is clear that the number of leptons and hadrons in the universe is conserved, whereas in Big Bang Gribbin goes on to explain how the strong nuclear force can transform particles between these two families. It's a small point and can be forgiven.
[and in case you were wondering, the wavefunction of the cat is 2-1/2(|dead> + |alive>)]