I’m somewhat opposed to the way in which David Griffiths approaches quantum mechanics in his popular undergraduate text. I’d prefer more foundational linear algebra stuff rather than jumping into the Schroedinger equation right away. Nevertheless, given that he is taking this approach, it’s as good an attempt at it as I can imagine. Crystal clear explanations, which is great praise for any explanation of QM phenomena. Knows exactly how to play to people’s intuitions. I particularly like the occasional long paragraphs in which he explains in conversational language the significance of the previous mathematical result. Terseness may be elegant but verbosity does help in getting one’s head around the physical aspects of the situation. There’s also the occasional witty aside. An example:
To the layman, the philosopher, or the classical physicist, a statement of the form “this particles doesn’t have a well-defined position”… sounds vague, incompetent, or (worst of all) profound.
Griffiths, Introduction to Quantum Mechanics, 2nd Ed, Section 4.4.1
Writing about the indistinguishability of particles:
It’s not just that we don’t happen to know which electron is which; God doesn’t know which is which, because there is no such thing as “this” or “that” electron; all we can legitimately speak about is “an” electron.
On my first reading of that it struck me as an assertion based on some assumptions about the nature of omniscience. That would have been so if Griffiths had meant to just say that it is impossible for us in principle to ever identify individual electrons. But, the word God catching my eye, I returned for a closer read and realised he said there is no such thing as an individual electron (this reads wrongly, but I don’t know how better to phrase it — Griffiths’ way is probably best).
Which got me wondering how, if at all, knowledge which is in principle impossible for us to attain differs from knowledge which cannot exist (of which, according to Griffiths, the identity of an electron is an example). How do the equations tell us that the identity of an electron falls under the latter category? Does it even make sense to make such a distinction?
“In principle impossible to know” still leaves the possibility that it is some limitation inherent in humans (or conscious beings that we might expect to occur in the universe) which prevents them from identifying individual electrons. But if there is no such thing as “this” or “that” electron, then the identity lies in reality itself and is not a consequence of the limitations of conscious beings. So the “no such thing” hypothesis goes deeper. But I see no obvious method by which we could distinguish between “no such thing” and a human limitation. It seems to me that they are saying it’s not a human limitation because it’s a consequence of the equations that it is outright impossible to know which electron is which, no matter how accurately we measure anything. But to make that inference one must assume that the mathematics does not itself reflect human limitations. It could well be that we include peculiarly human limitations in our mathematical system, or in our translation of the physical situation into mathematics. And within these limitations the mathematics produces a result telling us electrons will always seem identical to us.
And then I realise that the indistinguishability of electrons is the basis for the so-called exchange force which accounts for basic molecular structures. It is implausible that a human limitation in mathematical understanding could fortuitously create such a successful explanation. So much for my blathering.
The answer, then, seems to be that there is a distinction between knowledge which is in principle impossible to know and knowledge which does not exist. And it is a distinction we can bring out by empirically testing models which include the putative knowledge against models which do not. If it really is just a human limitation, then the model which includes the putative knowledge should come out tops.
