lecture_14_07b

lecture_14_07b - Other atoms with more e- ,s In QM: system...

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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 1 Other atoms with more  e -  ‘s In QM: system of particles described by single wave function for all the classical position variables e.g. x 1 ( t ) and x 2 ( t ) Y( x 1 , x 2 , t ) stationary equation for 2 identical particles: - h 2 2 m 2 x 1 2 y ( x 1 , x 2 ) - h 2 2 m 2 x 2 2 y ( x 1 , x 2 ) + U ( x 1 , x 2 ) y ( x 1 , x 2 ) = Ey ( x 1 , x 2 ) But which is particle 1 and which is particle 2?
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 2 Indistinguishability QM: identical elementary particles are indistinguishable (all electrons are identical, all photons, all quarks of fixed color and charge,…) Probability density: y ( x 1 , x 2 ) 2 = y ( x 2 , x 1 ) 2 So y ( x 1 , x 2 ) = + y ( x 2 , x 1 ) symmetric under interchange or y ( x 1 , x 2 ) = - y ( x 2 , x 1 ) anti- symmetric under interchange x 1 , x 2 can represent pair of 3-vectors and spins Only 1 state. For latter can not have two particles at same x Profound consequences! Two kinds of particles : bosons - satisfy Bose-Einstein statistics - symmetric fermions - satisfy Fermi-Dirac statistics - anti-symmetric
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein 3 Identical particles For 2 identical non-interacting particles U(x 1 ,x 2 ) = U(x 1 )+U(x 2 ) and Schrödinger equation separates so ψ (x 1 ,x 2 ) = (x 1 ) (x 2 ) e.g. 1 dim. box y n 1 ( x 1 ) y n 2 ( x 2 ) = A sin n 1 p 2 L x 1 sin n 2 p 2 L x 2 E = E n 1 + E n 2 = h 2 k 1 2 2 m + h 2 k 2 2 2 m Symmetrizing: y S o r A ( x 1 , x 2 ) = A y n 1 ( x 1 ) y n 2 ( x 2 ) y n 1 ( x 2 ) y n 2 ( x 1 ) ( ) For anti -symmetric case can not have n 1 =n 2
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05/13/09 Physics 13 - Fall 07 - G.R. Goldstein
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This note was uploaded on 03/27/2008 for the course PHY 13 taught by Professor Garyr.goldstein during the Fall '04 term at Tufts.

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lecture_14_07b - Other atoms with more e- ,s In QM: system...

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