Fund Quantum Mechanics Lect & HW Solutions 86

Fund Quantum Mechanics Lect & HW Solutions 86 - 68...

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Unformatted text preview: 68 CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS of finding it somewhere with spin down is |Ψ− |2 d3 r. The sum of the two integrals must be one to express the fact that the probability of finding the particle somewhere, either with spin up or spin down, must be one, certainty. Compare that with the square norm of the wave function, which is by definition the inner product of the wave function with itself: Ψ|Ψ = Ψ+ ↑ + Ψ− ↓|Ψ+ ↑ + Ψ− ↓ = Ψ+ |Ψ+ + Ψ− |Ψ− and the final two inner products are by definition the two integrals above. Since their sum must be one, it follows that the norm of the wave function Ψ|Ψ must be one even if there is spin. 5.5.2.2 Solution complexsai-b Question: Show√ that if ψl and ψr are normalized spatial wave functions, then a combination like (ψl ↑ + ψr ↓) / 2 is a normalized wave function with spin. Answer: You have ψl ↑ + ψr ↓ ψl ↑ + ψr ↓ √ √ 2 2 = 1 √ 2 2 ψl ↑ + ψr ↓ |ψl ↑ + ψr ↓ , and multiplying out the inner product according to the rule spin-up components together and spin-down components together, 1 = ψl |ψl + ψr |ψr , 2 and since it is given that ψl and ψr are normalized 1 = (1 + 1) = 1. 2 5.5.3 Commutators including spin 5.5.3.1 Solution complexsac-a Question: Are not some commutators missing from the fundamental commutation relationship? For example, what is the commutator [Sy , Sx ]? Answer: Since the commutator is antisymmetric, [Sy , Sx ] is the negative of [Sx , Sy ], so it is −i¯ Sz . h ...
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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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