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Lecture 6 - Phys 436 Modern Physics Lecture 6 Jan 22 2013...

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Phys 436 – Modern Physics Lecture 6 Jan. 22, 2013 Instructor: David Cooke Exchange forces Density of states 1
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What’s the three parIcle wavefuncIon? -­૒L/2 L/2 E n=1 n=2 n=3 Could write: ! = [ " 1 # ( x 1 ) " 1 $ ( x 2 ) % " 1 $ ( x 1 ) " 1 # ( x 2 )] " 2 # ( x 3 ) But this is symmetric under parIcle exchange (the x 3 term is singled out in the n=2 state.) We must subtract terms with P23 and P31 permutaIons to construct an asymmetric state So without solving the Shrödinger equaIon for this enormous soluIon, the symmetrizaIon postulate provides it for free. ! = " 1 # (x 1 ) " 1 $ (x 2 ) " 2 # (x 3 ) %" 1 $ (x 1 ) " 1 # (x 2 ) " 2 # (x 3 ) %" 1 # (x 1 ) " 1 $ (x 3 ) " 2 # (x 2 ) + " 1 $ (x 1 ) " 1 # (x 3 ) " 2 # (x 2 ) %" 1 # (x 3 ) " 1 $ (x 2 ) " 2 # (x 1 ) + " 1 $ (x 3 ) " 1 # (x 2 ) " 2 # (x 1 ) 2-­૒parIcle state 1-­૒parIcle state What’s the total energy of the system? Energy is MORE POSITIVE! i.e. like a repulsive interacIon…
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Exchange forces We’ve just seen that when two idenIcal parIcles arise, their symmetry under exchange induces a kind of repulsive (fermions) force. If the parIcles are bosons, the force is aaracIve. This type of non-­૒Newtonian force is called an exchange force. Let’s look at an example that illustrates this (from Griffiths): ! ( x 1 , x 2 ) = ! a ( x 1 ) ! b ( x 2 ) ! + ( x 1 , x 2 ) = 1 2 [ ! a ( x 1 ) ! b ( x 2 ) + ! b ( x 1 ) ! a ( x 2 )] ! " ( x 1 , x 2 ) = 1 2 [ ! a ( x 1 ) ! b ( x 2 ) " ! b ( x 1 ) ! a ( x 2 )] Composite two parIcle wavefuncIon Bosons Fermions We want the expectaIon value of the square of the separaIon distance between the 2 parIcles < ( x 1 ! x 2 ) 2 >=< x 1 2 > + < x 2 2 > ! 2 < x 1 x 2 >
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Case 1: disInguishable parIcles < x 1 2 >= x 1 2 | ! a ( x 1 ) | 2 dx 1 " | ! b ( x 2 ) | 2 dx 2 " =< x 2 > a the exp. value of 1 parIcle x^2 Similarly for < x 2 2 >= | !
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