N.12 Second quantization in other books

The approach to second quantization followed in this book is quite different from what you will find in other basic quantum mechanics or advanced physics books. This book simply sticks to its guns. Right at the beginning, this book said that observable properties are the eigenvalues of Hermitian operators. And that these act on particle wave functions. These same rules are then used to quantize the electromagnetic field.

What other books do is write down various classical wave solutions to Maxwell’s equations. Then these books reach deep inside these messy equations, cross out certain coefficients, and scribble in new ones. The new ones have operators in them and undetermined coefficients. The undetermined coefficients are then determined by examining the energy of the wave and comparing it with a harmonic oscillator, as analyzed using quantum field theory.

This book, however, greatly dislikes writing down classical solutions. A general student may not be familiar with these solutions. Or have long forgotten them. And it seems quite doubtful that even physics students are really familiar with the messy electric and magnetic multipole fields of classical electromagnetics. The approach in this book is to skip classical physics and give a self-contained and reasonable quantum derivation wherever possible. (Which means almost always.)

This book detests reaching into the middle of equations known to be wrong, and then crossing out things and writting in new things, all the while waving your hands a lot. The method of science is to make certain fundamental assumptions and then take them to their logical conclusion, whatever it may be. Not messing around until you get something that seems the right answer. And a book on science should showcase the methods of science.

Then there is the problem that the classical waves are inherently time-dependent. The Schrö­din­ger approach, however, is to put the time dependence in the wave function. For good reasons. That means that starting from the classical waves, you have two options, both ugly. You can suddenly switch to the Heisenberg representation, which is what everybody does. Or you can try to unextract the time dependence and put it on an explicit wave function.

And things get even uglier because the entire approach depends essentially on a deep familiarity with a different problem; the quantum-field description of the harmonic oscillator.

In fact, it may be noted that in early versions, this book did really try to give an understandable description of second quantization using the usual approach. The result was an impenetrable mess.