5.1.2 So­lu­tion com­plex-b

Ques­tion:

Show that for a sim­ple prod­uct wave func­tion as in the pre­vi­ous ques­tion, the rel­a­tive prob­a­bil­i­ties of find­ing par­ti­cle 1 near a po­si­tion ${\skew0\vec r}_a$ ver­sus find­ing it near an­other po­si­tion ${\skew0\vec r}_b$ 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.

An­swer:

The prob­a­bil­ity of find­ing par­ti­cle 1 within a vicin­ity ${ \rm d}^3{\skew0\vec r}_1$ of ${\skew0\vec r}_a$ and par­ti­cle 2 within a vicin­ity ${ \rm d}^3{\skew0\vec r}_2$ of ${\skew0\vec r}_2$ is:

$$\psi_1({\skew0\vec r}_a)^*\psi_2({\skew0\vec r}_2)^* \psi_1(... ...c r}_2) { \rm d}^3 {\skew0\vec r}_1{\rm d}^3 {\skew0\vec r}_2$$

while the cor­re­spond­ing prob­a­bil­ity of find­ing par­ti­cle 1 within a vicin­ity ${ \rm d}^3{\skew0\vec r}_1$ of ${\skew0\vec r}_b$ and par­ti­cle 2 within a vicin­ity ${ \rm d}^3{\skew0\vec r}_2$ of ${\skew0\vec r}_2$ is:

$$\psi_1({\skew0\vec r}_b)^*\psi_2({\skew0\vec r}_2)^* \psi_1(... ... r}_2) { \rm d}^3 {\skew0\vec r}_1{\rm d}^3 {\skew0\vec r}_2.$$

Tak­ing the ra­tio of the two prob­a­bil­i­ties, the chances of find­ing par­ti­cle 1 at ${\skew0\vec r}_a$ ver­sus find­ing it at ${\skew0\vec r}_b$ are the same wher­ever par­ti­cle 2 is likely to be found.