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Lecture 5 part2

# Cle exchange 1 x1 1 x 2 2 x 3 1

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Unformatted text preview: electrons are non ­interac?ng Single par?cle wavefunc?ons are: ∞ n=2 n=1  ­L/2 L/2 !n = 2 n" x sin , n = 1, 2... L L ! 2 !2 n2 With eigenenergies: En = 2 mL2 If we put two electrons in the n=1 state, we have a symmetric spa?al wavefunc?on. ! ( x1 , x2 ) = ! ( x2 , x1 ) Therefore the spin part must be asymmetric (1 electron spin up, 1 spin down) for the composite wavefunc?on to be assymetric under exchange. ! = " 1# ( x1 )" 1\$ ( x2 ) % " 1\$ ( x1 )" 1# ( x2 ) Addi?on of another electron must occupy the next highest state, to avoid two fermions described by the same set of quantum numbers What’s the excited state wavefunc?on? … E ∞ n=3 n=2 n=1  ­L/2 L/2 Could write ! = [" 1# ( x1 )" 1\$ ( x2 ) % " 1\$ ( x1 )" 1# ( x2 )]" 2# ( x3 ) But this is symmetric under par?cle exchange ! = "1# ( x1 )"1\$ ( x 2 )" 2# ( x 3 ) % "1\$ ( x1 )"1# ( x 2 )" 2# ( x 3 )...
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