### 5.1 Wave Func­tion for Mul­ti­ple Par­ti­cles

While a sin­gle par­ti­cle is de­scribed by a wave func­tion , a sys­tem of two par­ti­cles, call them 1 and 2, is de­scribed by a wave func­tion

 (5.1)

de­pend­ing on both par­ti­cle po­si­tions. The value of gives the prob­a­bil­ity of si­mul­ta­ne­ously find­ing par­ti­cle 1 within a vicin­ity of and par­ti­cle 2 within a vicin­ity of .

The wave func­tion must be nor­mal­ized to ex­press that the elec­trons must be some­where:

 (5.2)

where the sub­script 6 of the in­ner prod­uct is just a re­minder that the in­te­gra­tion is over all six scalar po­si­tion co­or­di­nates of .

The un­der­ly­ing idea of in­creas­ing sys­tem size is “every pos­si­ble com­bi­na­tion:” al­low for every pos­si­ble com­bi­na­tion of state for par­ti­cle 1 and state for par­ti­cle 2. For ex­am­ple, in one di­men­sion, all pos­si­ble po­si­tions of par­ti­cle 1 geo­met­ri­cally form an -​axis. Sim­i­larly all pos­si­ble po­si­tions of par­ti­cle 2 form an -​axis. If every pos­si­ble po­si­tion is sep­a­rately com­bined with every pos­si­ble po­si­tion , the re­sult is an -plane of pos­si­ble po­si­tions of the com­bined sys­tem.

Sim­i­larly, in three di­men­sions the three-di­men­sion­al space of po­si­tions com­bines with the three-di­men­sion­al space of po­si­tions into a six-di­men­sion­al space hav­ing all pos­si­ble com­bi­na­tions of val­ues for with all pos­si­ble val­ues for .

The in­crease in the num­ber of di­men­sions when the sys­tem size in­creases is a ma­jor prac­ti­cal prob­lem for quan­tum me­chan­ics. For ex­am­ple, a sin­gle ar­senic atom has 33 elec­trons, and each elec­tron has 3 po­si­tion co­or­di­nates. It fol­lows that the wave func­tion is a func­tion of 99 scalar vari­ables. (Not even count­ing the nu­cleus, spin, etcetera.) In a brute-force nu­mer­i­cal so­lu­tion of the wave func­tion, maybe you could re­strict each po­si­tion co­or­di­nate to only ten com­pu­ta­tional val­ues, if no very high ac­cu­racy is de­sired. Even then, val­ues at 10 dif­fer­ent com­bined po­si­tions must be stored, re­quir­ing maybe 10 Gi­ga­bytes of stor­age. To do a sin­gle mul­ti­pli­ca­tion on each of those those num­bers within a few years would re­quire a com­puter with a speed of 10 gi­gaflops. No need to take any of that ar­senic to be long dead be­fore an an­swer is ob­tained. (Imag­ine what it would take to com­pute a mi­cro­gram of ar­senic in­stead of an atom.) Ob­vi­ously, more clever nu­mer­i­cal pro­ce­dures are needed.

Some­times the prob­lem size can be re­duced. In par­tic­u­lar, the prob­lem for a two-par­ti­cle sys­tem like the pro­ton-elec­tron hy­dro­gen atom can be re­duced to that of a sin­gle par­ti­cle us­ing the con­cept of re­duced mass. That is shown in ad­den­dum {A.5}.

Key Points
To de­scribe mul­ti­ple-par­ti­cle sys­tems, just keep adding more in­de­pen­dent vari­ables to the wave func­tion.

Un­for­tu­nately, this makes many-par­ti­cle prob­lems im­pos­si­ble to solve by brute force.

5.1 Re­view Ques­tions
1.

A sim­ple form that a six-di­men­sion­al wave func­tion can take is a prod­uct of two three-di­men­sion­al ones, as in . Show that if and are nor­mal­ized, then so is .

2.

Show that for a sim­ple prod­uct wave func­tion as in the pre­vi­ous ques­tion, the rel­a­tive prob­a­bil­i­ties of find­ing par­ti­cle 1 near a po­si­tion ver­sus find­ing it near an­other po­si­tion is the same re­gard­less where par­ti­cle 2 is. (Or rather, where par­ti­cle 2 is likely to be found.)

Note: This is the rea­son that a sim­ple prod­uct wave func­tion is called un­cor­re­lated. For par­ti­cles that in­ter­act with each other, an un­cor­re­lated wave func­tion is of­ten not a good ap­prox­i­ma­tion. For ex­am­ple, two elec­trons re­pel each other. All else be­ing the same, the elec­trons would rather be at po­si­tions where the other elec­tron is nowhere close. As a re­sult, it re­ally makes a dif­fer­ence for elec­tron 1 where elec­tron 2 is likely to be and vice-versa. To han­dle such sit­u­a­tions, usu­ally sums of prod­uct wave func­tions are used. How­ever, for some cases, like for the he­lium atom, a sin­gle prod­uct wave func­tion is a per­fectly ac­cept­able first ap­prox­i­ma­tion. Real-life elec­trons are crowded to­gether around at­tract­ing nu­clei and learn to live with each other.