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Lecture 5 part2 - Phys 436 Modern Physics Lecture 5 Jan 18...

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Phys 436 – Modern Physics Lecture 5 Jan. 18, 2011 Indis?nguishable par?cles Fermions and Bosons Pauli exclusion principle Many non-­૒interac?ng electrons 1
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Indis?nguishable par?cles So far, we have considered systems where par?cles had some feature which made them dis?nguishable (mass, charge, spin) We must now consider what happens when two truly iden?cal par?cles are in a system. This is completely non-­૒classical. Dis?nguishable Indis?nguishable There is no way to tell which par?cle is which aRer the collision.
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Indis?nguishable par?cles If par?cles are iden?cal, the Hamiltonian must be symmetric to interchange of spa?al and spin coordinates of the par?cles (switching coordinates cannot change the overall energy of the system, otherwise you could tell them apart.) Create a permuta?on operator that switches the variables of par?cles i and j. [ P i , j , H ] = 0 If ! ( q 1 ,... q i ,... q j ,... q N ) is an eigenfunc?on of H with eigenvalue E, so is P i , j ! P i , j ! ( q 1 ,... q i ,... q j ,... q N ) = ! ( q 1 ,... q j ,... q i ,... q N ) Applying P twice brings us back to the ini?al configura?on: P i , j 2 = I Eigenvalues are therefore ! = ± 1
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Indis?nguishable par?cles The wavefunc?ons corresponding to eigenvalue +ε: P i , j ! ( q 1 ,... q i ,... q j ,... q N ) = ! ( q 1 ,... q j ,... q i ,... q N ) = ! ( q 1 ,... q i ,... q j ,... q N ) and are said to be symmetric under interchange. The wavefunc?ons corresponding to eigenvalue -­૒ε: P i , j ! ( q 1 ,... q i ,... q j ,... q N ) = ! ( q 1 ,... q j ,... q i ,... q N ) = " ! ( q 1 ,... q i ,... q j ,... q N ) and are said to be an+symmetric under interchange.
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