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Lecture 6

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 $ 2 ) " 2 # 3 ) 1 $ 1 ) " 1 # 2 ) " 2 # 3 ) 1 # 1 ) " 1 $ 3 ) " 2 # 2 ) + " 1 $ 1 ) " 1 # 3 ) " 2 # 2 ) 1 # 3 ) " 1 $ 2 ) " 2 # 1 ) + " 1 $ 3 ) " 1 # 2 ) " 2 # 1 ) 2-parIcle state 1-parIcle state What’s the total energy of the system? Energy is MORE POS±T±VE! 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. ±f 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 >= | a ( x 1 ) | 2 dx 1 " x 2 2 | b ( x 2 ) | 2 dx 2 " =< x 2 > b < x 1 x 2 >= x 1 | ( x 1 ) | 2 dx 1 " x 2 | ( x 2 ) | 2 dx 2 " =< x > a < x > b < ( x 1 ! x 2 ) 2 >=< x 1 2 > + < x 2 2 > ! 2 < x 1 x 2 > =< x 2 > a + < x 2 > b ! 2 < x > a < x > b
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Lecture 6 - Phys 436 Modern Physics Lecture 6 Jan 22 2013...

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