Fund Quantum Mechanics Lect & HW Solutions 78

Fund Quantum Mechanics Lect & HW Solutions 78 -...

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60 CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS 5.2.2.2 Solution hmolb-a Question: When the protons are close to each other, the electrons do afect each other, and the wave Function above is no longer valid. But suppose you were given the true wave Function, and you were once again asked to draw the blob showing the probability oF ±nding electron 1 (using a plotting package, say). What would the big problem be? Answer: Since electron 2 now afects where electron 1 is, you would have to draw a diferent blob For every possible position oF electron 2, an impossible task. 5.2.3 The probability density 5.2.3.1 Solution hmolc-a Question: Suppose, given the wave Function ψ l ( vr 1 ) ψ r ( vr 2 ), that you Found an electron near the leFt proton. What electron would it probably be? Suppose you Found an electron at the
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Unformatted text preview: point halFway in between the protons. What electron would that likely be? Answer: The total probability oF ±nding electron 1 at a position vr is i | Ψ( vr,vr 2 ) | 2 d 3 vr 2 = | ψ l ( vr ) | 2 i | ψ r ( vr 2 ) | 2 d 3 vr 2 = | ψ l ( vr ) | 2 since ψ r is normalized. Similarly, the probability oF ±nding electron 2 at position vr is | ψ r ( vr ) | 2 . IF vr is close to the leFt proton, | ψ l ( vr ) | 2 is signi±cant, but | ψ r ( vr ) | 2 is small, so you are much more likely to ±nd electron 1 there than electron 2. But at the point halFway in between the protons, | ψ l ( vr ) | 2 and | ψ r ( vr ) | 2 are equal by symmetry, and you are just as likely to ±nd electron 1 there as electron 2. 5.2.4 States that share the electrons...
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