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. 220.127.116.11 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 = 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,
ψl |ψl + ψr |ψr ,
and since it is given that ψl and ψr are normalized
= (1 + 1) = 1.
2 5.5.3 Commutators including spin 18.104.22.168 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 .
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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.
- Fall '11