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.