N.1 Why this book?

With the cur­rent em­pha­sis on nan­otech­nol­ogy, quan­tum me­chan­ics is be­com­ing in­creas­ingly es­sen­tial to en­gi­neer­ing stu­dents. Yet, the typ­i­cal quan­tum me­chan­ics texts for physics stu­dents are not writ­ten in a style that most en­gi­neer­ing stu­dents would likely feel com­fort­able with. Fur­ther­more, an en­gi­neer­ing ed­u­ca­tion pro­vides very lit­tle real ex­po­sure to mod­ern physics, and in­tro­duc­tory quan­tum me­chan­ics books do lit­tle to fill in the gaps. The em­pha­sis tends to be on the com­pu­ta­tion of spe­cific ex­am­ples, rather than on dis­cus­sion of the broad pic­ture. Un­der­grad­u­ate physics stu­dents may have the lux­ury of years of fur­ther courses to pick up a wide physics back­ground, en­gi­neer­ing grad­u­ate stu­dents not re­ally. In ad­di­tion, the cov­er­age of typ­i­cal in­tro­duc­tory quan­tum me­chan­ics books does not em­pha­size un­der­stand­ing of the larger-scale quan­tum sys­tem that a den­sity func­tional com­pu­ta­tion, say, would be used for.

Hence this book, writ­ten by an en­gi­neer for en­gi­neers. As an en­gi­neer­ing pro­fes­sor with an en­gi­neer­ing back­ground, this is the book I wish I would have had when I started learn­ing real quan­tum me­chan­ics a few years ago. The rea­son I like this book is not be­cause I wrote it; the rea­son I wrote this book is be­cause I like it.

This book is not a pop­u­lar ex­po­si­tion: quan­tum me­chan­ics can only be de­scribed prop­erly in the terms of math­e­mat­ics; sug­gest­ing any­thing else is crazy. But the as­sumed back­ground in this book is just ba­sic un­der­grad­u­ate cal­cu­lus and physics as taken by all en­gi­neer­ing un­der­grad­u­ates. There is no in­ten­tion to teach stu­dents pro­fi­ciency in the clever ma­nip­u­la­tion of the math­e­mat­i­cal ma­chin­ery of quan­tum me­chan­ics. For those en­gi­neer­ing grad­u­ate stu­dents who may have for­got­ten some of their un­der­grad­u­ate cal­cu­lus by now, there are some quick and dirty re­minders in the no­ta­tions. For those stu­dents who may have for­got­ten some of the de­tails of their un­der­grad­u­ate physics, frankly, I am not sure whether it makes much of a dif­fer­ence. The ideas of quan­tum me­chan­ics are that dif­fer­ent from con­ven­tional physics. But the gen­eral ideas of clas­si­cal physics are as­sumed to be known. I see no rea­son why a bright un­der­grad­u­ate stu­dent, hav­ing fin­ished cal­cu­lus and physics, should not be able to un­der­stand this book. A cer­tain ma­tu­rity might help, though. There are a lot of ideas to ab­sorb.

My ini­tial goal was to write some­thing that would “read like a mys­tery novel.” Some­thing a reader would not be able to put down un­til she had fin­ished it. Ob­vi­ously, this goal was un­re­al­is­tic. I am far from a pro­fes­sional writer, and this is quan­tum me­chan­ics, af­ter all, not a mur­der mys­tery. But I have been told that this book is very well writ­ten, so maybe there is some­thing to be said for aim­ing high.

To pre­vent the reader from get­ting bogged down in math­e­mat­i­cal de­tails, I mostly avoid non­triv­ial de­riva­tions in the text. In­stead I have put the out­lines of these de­riva­tions in notes at the end of this doc­u­ment: per­son­ally, I en­joy check­ing the cor­rect­ness of the math­e­mat­i­cal ex­po­si­tion, and I would not want to rob my stu­dents of the op­por­tu­nity to do so too. In fact, the cho­sen ap­proach al­lows a lot of de­tailed de­riva­tions to be given that are skipped in other texts to re­duce dis­trac­tions. Some ex­am­ples are the har­monic os­cil­la­tor, or­bital an­gu­lar mo­men­tum, and ra­dial hy­dro­gen wave func­tions, Hund’s first rule, and ro­ta­tion of an­gu­lar mo­men­tum. And then there are ex­ten­sive de­riva­tions of ma­te­r­ial not even in­cluded in other in­tro­duc­tory quan­tum texts.

While typ­i­cal physics texts jump back and for­ward from is­sue to is­sue, I thought that would just be dis­tract­ing for my au­di­ence. In­stead, I try to fol­low a con­sis­tent ap­proach, with as cen­tral theme the method of sep­a­ra­tion-of-vari­ables, a method that most me­chan­i­cal grad­u­ate stu­dents have seen be­fore al­ready. It is ex­plained in de­tail any­way. To cut down on the is­sues to be men­tally ab­sorbed at any given time, I pur­posely avoid bring­ing up new is­sues un­til I re­ally need them. Such a just-in-time learn­ing ap­proach also im­me­di­ately an­swers the ques­tion why the new is­sue is rel­e­vant, and how it fits into the grand scheme of things.

The de­sire to keep it straight­for­ward is the main rea­son that top­ics such as Cleb­sch-Gor­dan co­ef­fi­cients (ex­cept for the un­avoid­able in­tro­duc­tion of sin­glet and triplet states) and Pauli spin ma­tri­ces have been shoved out of the way to a fi­nal chap­ter. My feel­ing is, if I can give my stu­dents a solid un­der­stand­ing of the ba­sics of quan­tum me­chan­ics, they should be in a good po­si­tion to learn more about in­di­vid­ual is­sues by them­selves when they need them. On the other hand, if they feel com­pletely lost in all the dif­fer­ent de­tails, they are not likely to learn the ba­sics ei­ther.

That does not mean the cov­er­age is in­com­plete. All top­ics that are con­ven­tion­ally cov­ered in ba­sic quan­tum me­chan­ics courses are present in some form. Some are cov­ered in much greater depth. And there is a lot of ma­te­r­ial that is not usu­ally cov­ered. I in­clude sig­nif­i­cant qual­i­ta­tive dis­cus­sion of atomic and chem­i­cal prop­er­ties, Pauli re­pul­sion, the prop­er­ties of solids, Bragg re­flec­tion, and elec­tro­mag­net­ism, since many en­gi­neers do not have much back­ground on them and not much time to pick it up. The dis­cus­sion of ther­mal physics is much more elab­o­rate than you will find in other books on quan­tum me­chan­ics. It in­cludes all the es­sen­tials of a ba­sic course on clas­si­cal ther­mo­dy­nam­ics, in ad­di­tion to the quan­tum sta­tis­tics. I feel one can­not be sep­a­rated from the other, es­pe­cially with re­spect to the sec­ond law. While me­chan­i­cal en­gi­neer­ing stu­dents will surely have had a course in ba­sic ther­mo­dy­nam­ics be­fore, a re­fresher can­not hurt. Un­like other books, this book also con­tains a chap­ter on nu­mer­i­cal pro­ce­dures, cur­rently in­clud­ing de­tailed dis­cus­sions of the Born-Op­pen­heimer ap­prox­i­ma­tion, the vari­a­tional method, and the Hartree-Fock method. Hope­fully, this chap­ter will even­tu­ally be com­pleted with a sec­tion on den­sity-func­tional the­ory. (The Lennard-Jones model is cov­ered ear­lier in the sec­tion on mol­e­c­u­lar solids.) The mo­ti­va­tion for in­clud­ing nu­mer­i­cal meth­ods in a ba­sic ex­po­si­tion is the feel­ing that af­ter a cen­tury of work, much of what can be done an­a­lyt­i­cally in quan­tum me­chan­ics has been done. That the great­est scope for fu­ture ad­vances is in the de­vel­op­ment of im­proved nu­mer­i­cal meth­ods.

Knowl­edge­able read­ers may note that I try to stay clear of ab­stract math­e­mat­ics when it is not needed. For ex­am­ple, I try to go slow on the more ab­stract vec­tor no­ta­tion per­me­at­ing quan­tum me­chan­ics, usu­ally phras­ing such is­sues in terms of a spe­cific ba­sis. Ab­stract no­ta­tion may seem to be com­pletely gen­eral and beau­ti­ful to a math­e­mati­cian, but I do not think it is go­ing to be in­tu­itive to a typ­i­cal en­gi­neer. The dis­cus­sion of sys­tems with mul­ti­ple par­ti­cles is cen­tered around the phys­i­cal ex­am­ple of the hy­dro­gen mol­e­cule, rather than par­ti­cles in boxes. The dis­cus­sion of solids in chap­ter 10 avoids the highly ab­stract Dirac comb (delta func­tions) math­e­mat­i­cal model in fa­vor of a phys­i­cal dis­cus­sion of more re­al­is­tic one-di­men­sion­al crys­tals. The Lennard-Jones po­ten­tial is de­rived for two atoms in­stead of har­monic os­cil­la­tors.

The book tries to be as con­sis­tent as pos­si­ble. Elec­trons are grey tones at the ini­tial in­tro­duc­tion of par­ti­cles, and so they stay through the rest of the book. Nu­clei are red dots. Oc­cu­pied quan­tum states are red, empty ones grey. That of course re­quired all fig­ures to be cus­tom made. They are not in­tended to be fancy but con­sis­tent and clear. I also try to stay con­sis­tent in no­ta­tions through­out the book, as much as is pos­si­ble with­out de­vi­at­ing too much from es­tab­lished us­age.

When I de­rive the first quan­tum eigen­func­tions, for a pipe and for the har­monic os­cil­la­tor, I make sure to em­pha­size that they are not sup­posed to look like any­thing that we told them be­fore. It is only nat­ural for stu­dents to want to re­late what we told them be­fore about the mo­tion to the com­pletely dif­fer­ent story we are telling them now. So it should be clar­i­fied that (1) no, they are not go­ing crazy, and (2) yes, we will even­tu­ally ex­plain how what they learned be­fore fits into the grand scheme of things.

An­other dif­fer­ence of ap­proach in this book is the way it treats clas­si­cal physics con­cepts that the stu­dents are likely un­aware about, such as canon­i­cal mo­men­tum, mag­netic di­pole mo­ments, Lar­mor pre­ces­sion, and Maxwell’s equa­tions. They are largely “de­rived“ in quan­tum terms, with no ap­peal to clas­si­cal physics. I see no need to rub in the stu­dent's lack of knowl­edge of spe­cial­ized ar­eas of clas­si­cal physics if a sat­is­fac­tory quan­tum de­riva­tion is read­ily given.

This book is not in­tended to be an ex­er­cise in math­e­mat­i­cal skills. Re­view ques­tions are tar­geted to­wards un­der­stand­ing the ideas, with the math­e­mat­ics as sim­ple as pos­si­ble. I also try to keep the math­e­mat­ics in suc­ces­sive ques­tions uni­form, to re­duce the al­ge­braic ef­fort re­quired. There is an ab­solute epi­demic out there of quan­tum texts that claim that “the only way to learn quan­tum me­chan­ics is to do the ex­er­cises,” and then those ex­er­cises turn out to be, by and large, elab­o­rate ex­er­cises in in­te­gra­tion and lin­ear al­ge­bra that take ex­ces­sive time and have noth­ing to do with quan­tum me­chan­ics. Or worse, they are of­ten ba­sic the­ory. (Lazy au­thors that claim that ba­sic the­ory is an ex­er­cise avoid hav­ing to cover that ma­te­r­ial them­selves and also avoid hav­ing to come up with a real ex­er­cise.) Yes, I too did waste a lot of time with these. And then, when you are done, the an­swer teaches you noth­ing be­cause you are un­sure whether there might not be an al­ge­braic er­ror in your end­less mass of al­ge­bra, and even if there is no mis­take, there is no hint that it means what you think it means. All that your work has earned you is a 75/25 chance or worse that you now know some­thing that is not true. Not in this book.

Fi­nally, this doc­u­ment faces the very real con­cep­tual prob­lems of quan­tum me­chan­ics head-on, in­clud­ing the col­lapse of the wave func­tion, the in­de­ter­mi­nacy, the non­lo­cal­ity, and the sym­metriza­tion re­quire­ments. The usual ap­proach, and the way I was taught quan­tum me­chan­ics, is to shove all these prob­lems un­der the ta­ble in fa­vor of a good sound­ing, but upon ex­am­i­na­tion self-con­tra­dic­tory and su­per­fi­cial story. Such su­per­fi­cial­ity put me off solidly when they taught me quan­tum me­chan­ics, cul­mi­nat­ing in the un­for­get­table mo­ment when the pro­fes­sor told us, se­ri­ously, that the wave func­tion had to be sym­met­ric with re­spect to ex­change of bosons be­cause they are all truly the same, and then, when I was pop­ping my eyes back in, con­tin­ued to tell us that the wave func­tion is not sym­met­ric when fermi­ons are ex­changed, which are all truly the same. I would not do the same to my own stu­dents. And I re­ally do not see this pro­fes­sor as an ex­cep­tion. Other in­tro­duc­tions to the ideas of quan­tum me­chan­ics that I have seen left me sim­i­larly un­happy on this point. One thing that re­ally bugs me, none had a solid dis­cus­sion of the many worlds in­ter­pre­ta­tion. This is ob­vi­ously not be­cause the re­sults would be in­cor­rect, (they have not been con­tra­dicted for half a cen­tury,) but sim­ply be­cause the teach­ers just do not like these re­sults. I do not like the re­sults my­self, but bas­ing teach­ing on what the teacher would like to be true rather on what the ev­i­dence in­di­cates is true re­mains ab­solutely un­ac­cept­able in my book.