72 CHAPTER 5. MULTIPLE-PARTICLE SYSTEMS 5.6.1 Solution ident-a Question: Check that indeed any linear combination of the triplet states is unchanged under particle exchange. Answer: In the notations of the previous section, the most general linear combination of the triplet states takes the form: a 1 | 1 1 a + a 2 | 1 0 a + a 3 | 1 1 a or writing out their deFnitions as found there, a 1 ↑↑ + a 2 1 √ 2 ( ↑↓ + ↓↑ ) + a 3 ↓↓ . Exchanging the two particles involved means to interchange the order of each pair of arrows, since by deFnition the Frst arrow refers to particle 1 and the second to particle 2. A look at the expression above shows that such an interchange of order does absolutely nothing to these states. 5.6.2 Solution ident-b Question: Suppose the electrons of the hydrogen molecule are in the excited antisymmetric spatial state a [ ψ l ( vr 1 ) ψ r ( vr 2 ) − ψ r ( vr 1 ) ψ l ( vr 2 )] . In that case what can you say about the spin state?
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