3. Ba­sic Ideas of Quan­tum Me­chan­ics

Ab­stract

In this chap­ter the ba­sic ideas of quan­tum me­chan­ics are de­scribed and then a ba­sic but very im­por­tant ex­am­ple is worked out.

Be­fore em­bark­ing on this chap­ter, do take note of the very sage ad­vice given by Richard Feyn­man, No­bel-prize win­ning pi­o­neer of rel­a­tivis­tic quan­tum me­chan­ics:

“Do not keep say­ing to your­self, if you can pos­si­bly avoid it, But how can it be like that? be­cause you will get down the drain, into a blind al­ley from which no­body has yet es­caped. No­body knows how it can be like that.” [Richard P. Feyn­man (1965) The Char­ac­ter of Phys­i­cal Law 129. BBC/Pen­guin]

“So do not take the lec­ture too se­ri­ously, ..., but just re­lax and en­joy it.” [ibid.]

And it may be un­cer­tain whether Niels Bohr, No­bel-prize win­ning pi­o­neer of early quan­tum me­chan­ics ac­tu­ally said it to Al­bert Ein­stein, and if so, ex­actly what he said, but it may be the san­est state­ment about quan­tum me­chan­ics of all:

Stop telling God what to do. [Neils Bohr, re­puted].

First of all, this chap­ter will throw out the clas­si­cal pic­ture of par­ti­cles with po­si­tions and ve­loc­i­ties. Com­pletely.

Quan­tum me­chan­ics sub­sti­tutes in­stead a func­tion called the wave func­tion that as­so­ciates a nu­mer­i­cal value with every pos­si­ble state of the uni­verse. If the uni­verse that you are con­sid­er­ing is just a sin­gle par­ti­cle, the wave func­tion of in­ter­est as­so­ciates a nu­mer­i­cal value with every pos­si­ble po­si­tion of that par­ti­cle, at every time.

The phys­i­cal mean­ing of the value of the wave func­tion, also called the quan­tum am­pli­tude, it­self is some­what hazy. It is just a com­plex num­ber. How­ever, the square mag­ni­tude of the num­ber has a clear mean­ing, first stated by Born: The square mag­ni­tude of the wave func­tion at a point is a mea­sure of the prob­a­bil­ity of find­ing the par­ti­cle at that point, if you con­duct such a search.

But if you do, watch out. Heisen­berg has shown that if you elim­i­nate the un­cer­tainty in the po­si­tion of a par­ti­cle, its lin­ear mo­men­tum ex­plodes. If the po­si­tion is pre­cise, the lin­ear mo­men­tum has in­fi­nite un­cer­tainty. The same thing also ap­plies in re­verse. Nei­ther po­si­tion nor lin­ear mo­men­tum can have an pre­cise value for a par­ti­cle. And usu­ally other quan­ti­ties like en­ergy do not ei­ther.

Which brings up the ques­tion what mean­ing to at­tach to such phys­i­cal quan­ti­ties. Quan­tum me­chan­ics an­swers that by as­so­ci­at­ing a sep­a­rate Her­mit­ian op­er­a­tor with every phys­i­cal quan­tity. The most im­por­tant ones will be de­scribed. These op­er­a­tors act on the wave func­tion. If, and only if, the wave func­tion is an eigen­func­tion of such a Her­mit­ian op­er­a­tor, only then does the cor­re­spond­ing phys­i­cal quan­tity have a def­i­nite value: the eigen­value. In all other cases the phys­i­cal quan­tity is un­cer­tain.

The most im­por­tant Her­mit­ian op­er­a­tor is called the Hamil­ton­ian. It is as­so­ci­ated with the to­tal en­ergy of the par­ti­cle. The eigen­val­ues of the Hamil­ton­ian de­scribe the only pos­si­ble val­ues that the to­tal en­ergy of the par­ti­cle can have.

The chap­ter will con­clude by an­a­lyz­ing a sim­ple quan­tum sys­tem in de­tail. It is a par­ti­cle stuck in a pipe of square cross sec­tion. While rel­a­tively sim­ple, this case de­scribes some of the quan­tum ef­fects en­coun­tered in nan­otech­nol­ogy. In later chap­ters, it will be found that this case also pro­vides a ba­sic model for such sys­tems as va­lence elec­trons in met­als, mol­e­cules in ideal gases, nu­cle­ons in nu­clei, and much more.