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Unformatted text preview: Lecture 8 52 Dec 4, 2002 7.4 Bosons and Fermions Initially, the Pauli principle was just that, a postulate that two indistinguishable electrons (i.e. electrons with precisely the same quantum numbers) could not cohabitate. Together with the system of allowed energy levels in an atom, it explained the gradual filling of higher and higher atomic levels with in- creasing number of atomic electrons. Without the Pauli principle, nothing would prevent electrons from all congregating in the lowest energy state, making life in an atom exceedingly dull, and prevent- ing the wide variety of chemical properties of the elements. After discovery of the spin quantum num- ber, and the discovery of the Dirac equation, it was quickly realized that the Pauli principle could be generalized to: no two indistinguishable fermions (i.e. half-integer spin particles) can cohabitate in the same quantum state. Equivalently, and even more general: the wave function of a system of indistin- guishable fermions is antisymmetric under interchange (switch in position ) of any two fermions. Consider the electron. Any electron is truly indistinguishable from any other, like identical cars with- out license plates and VIN numbers. Consider two such indistinguishable particles, one of them with a wave function a ( r ), the other with b ( r ). Here, the subscript labels the quantum state that the electron is in. The overall two-particle wave function may be written as ( r 1 , r 2 ) = a ( r 1 ) b ( r 2 ), where the la- bels 1, 2 refer to the position of the first and second electron respectively. This follows directly from the fact that probabilities multiply: Prob( r 1 , r 2 ) = Prob(electron in state a at r 1 ) Prob(electron in state b at r 2 ) | ( r 1 , r 2 )| 2 = | a ( r 1 ) | 2 | b ( r 2 ) | 2 ( r 1 , r 2 ) = a ( r 1 ) b ( r 2 ) If the two switch place, no observable effects should ensue, i.e. ( r 1 , r 2 ) = ( r 2 , r 1 ). However, with the two-particle wave function as defined above, one quickly notes a problem: ( r 1 , r 2 ) = a ( r 1 ) b ( r 2 ) a ( r 2 ) b ( r 1 ) = ( r 2 , r 1 ) The way out is to change the two-particle wave function so that it explicitly incorporates indistin- guishability: ( ) 1 2 1 2 1 2 1 2 2 1 ( , ) ( ) ( ) ( ) ( ) ( , ) ( , ) a b b a C = = r r r r r r r r r r (I.137) where C is some normalization constant. Depending on the sign in (I.137), the wave function changes sign under interchange of the two particles, i.e. it is symmetric or antisymmetric under particle inter- change. These two possibilities are fundamental: it turns out that systems with identical fermions (half- integer spin particles, e.g. electrons) must be antisymmetric under interchange of any pair of fermions, whereas the wave function describing a system of indistinguishable bosons (integer spin particles) is symmetric under interchange of any pair of them: 1 1 1...
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- Fall '01