Fund Quantum Mechanics Lect & HW Solutions 75

Fund Quantum Mechanics Lect & HW Solutions 75 -...

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Unformatted text preview: Chapter 5 Multiple-Particle Systems 5.1 5.1.1 Wave Function for Multiple Particles Solution complex-a Question: A simple form that a six-dimensional wave function can take is a product of two three-dimensional 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 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 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 57 ...
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