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Unformatted text preview: Chapter 5
MultipleParticle Systems
5.1 5.1.1 Wave Function for Multiple Particles Solution complexa Question: A simple form that a sixdimensional wave function can take is a product of two
threedimensional ones, as in ψ (r1 , r2 ) = ψ1 (r1 )ψ2 (r2 ). Show that if ψ1 and ψ2 are normalized,
then so is ψ .
Answer: This is a direct consequence of the fact that integrals can be factored if their
integrands can be and the limits of integration are independent of the other variable:
2 all r1 5.1.2 all r 2 ψ1 (r1 )ψ2 (r2 ) d3 r1 d3 r2 = 2 all r 1 ψ1 (r1 ) d3 r1 2 all r 2 ψ2 (r2 ) d3 r2 = 1 Solution complexb Question: Show that for a simple product wave function as in the previous question, the
relative probabilities of ﬁnding particle 1 near a position ra versus ﬁnding it near another
position rb 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
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 Fall '11
 Dr.DanielArenas
 mechanics

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