Fund Quantum Mechanics Lect & HW Solutions 76

Fund Quantum Mechanics Lect & HW Solutions 76 - 2 ....

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58 CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS 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 diference ±or electron 1 where electron 2 is likely to be and vice-versa. To handle such situations, usually sums o± product wave ±unctions are used. However, ±or some cases, like ±or the helium atom, a single product wave ±unction is a per±ectly acceptable ²rst approximation. Real-li±e electrons are crowded together around attracting nuclei and learn to live with each other. Answer: The probability o± ²nding particle 1 within a vicinity d 3 vr 1 vr a and particle 2 within a vicinity d 3 vr 2 vr 2 is: ψ 1 ( vr a ) ψ 2 ( vr 2 ) ψ 1 ( vr a ) ψ 2 ( vr 2 ) d 3 vr 1 d 3 vr 2 while the corresponding probability o± ²nding particle 1 within a vicinity d 3 vr 1 vr b and particle 2 within a vicinity d 3 vr 2 vr 2 is: ψ 1 ( vr b ) ψ 2 ( vr 2 ) ψ 1 ( vr b ) ψ 2 ( vr 2 ) d 3 vr 1 d 3 vr
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Unformatted text preview: 2 . Taking the ratio o± the two probabilities, the chances o± ²nding particle 1 at vr a versus ²nding it at vr b are the same wherever particle 2 is likely to be ±ound. 5.2 The Hydrogen Molecule 5.2.1 The Hamiltonian 5.2.1.1 Solution hmola-a Question: Veri±y that the repulsive potential between the electrons is in²nitely large when the electrons are at the same position. Note: You might there±ore think that the wave ±unction needs to be zero at the locations in six-dimensional space where vr 1 = vr 2 . Some authors re±er to that as a “Coulomb hole.” But the truth is that in quantum mechanics, electrons are smeared out due to uncertainty. That causes electron 1 to “see electron 2 at all sides”, and vice-versa, and they do there±ore not encounter any unusually large potential when the wave ±unction is nonzero at vr 1 = vr 2 . In...
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This note was uploaded on 01/06/2012 for the course PHY 3604 taught by Professor Dr.danielarenas during the Fall '11 term at UNF.

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