Chapter 2 of 'The Physics of Quantum Mechanics' by James Binney and David Skinner is a veritable tour de force. The authors grab you by the scruff of the neck and drag you through a torrent of concepts and equations:
The Hamiltonian operator; Hermitian operators; Hermitian adjoints; commutators; the time-dependent Schrödinger equation; the time-independent Schrödinger equation; Ehrenfest's theorem; stationary states; the position representation; the wavefunction of a particle; the Hamiltonian of a particle; the canonical commutation relation; canonically conjugate observables; the de Broglie wavelength; the uncertainty principle; the correspondence principle; probability current; and the virial theorem - all that and more on just 22 pages.
Is it possible to fully understand all this if you only have 10 or 15 minutes every other weekend to do so? It's certainly not easy.
There are a number of places where I wish the authors could slow down a bit and explain their reasoning in more detail.
To give a few examples:
On page 32, the authors derive Ehrenfest's theorem in two short lines. The first line looks like the application of the product rule of differentiation even though the three terms to which it is applied were previously defined as functions: what we have in the equation is the function of a function of a function rather than a product made up of three factors.
It is only when you insert the definitions of the three functions into the equation that, after some calculations, the expression reduces to a product to which the product rule can be applied, but this isn't mentioned. The authors seem to imply that any steps missing in their derivation are too trivial to write down.
Subsequently, on page 33, they speak of the expectation value of an 'operator' when they appear to mean the expectation value of an 'observable'. Indeed, the basis for the confusion was laid a few pages earlier when they chose to use the same symbol for observables and their associated operators.
On page 34 the authors introduce the 'position representation'. The equations that are presented are said to have been formed 'by analogy' with the energy representation. It's clear that the analogy isn't perfect, in other words that there are differences from the energy representation, but these are not explained.
For example, what the authors refer to as 'probability amplitudes' in the position representation in fact appear to be what could be termed 'probability density amplitudes', but this is never made explicit. Basis kets are said to be the states "in which the particle is definitely at x". So apparently, while previously the authors just looked at vector spaces of finite or at most infinite but countable dimension, we are now dealing with a vector space of infinite and uncountable dimension. Again not worth a mention to the authors.
Of course, presumably this book is primarily addressed to students on a university physics course. Such students do not just read textbooks but attend lectures and seminars where the material presented in the book is explained to them face to face and they have opportunities to ask questions.
Still, it might be helpful to signpost passages where the explanation provided is rather abbreviated by using phrases such as "it can be shown that…." rather than implying that it's obvious how a result was obtained.
I still like this book as the explanations given are mostly clear and the order of presentation seems logical. But the going is getting tougher.
To familiarise myself a bit more with the concepts introduced in Chapter 2, over the next few weeks and months I will try to solve as many of the problems presented at the end of the chapter as possible.
