582-chapter8 - 8 Coherent State Path Integral Quantization...

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Unformatted text preview: 8 Coherent State Path Integral Quantization of Quantum Field Theory 8.1 Coherent states and path integral quantization. 8.1.1 Coherent States Let us consider a Hilbert space spanned by a complete set of harmonic oscillator states {| n )} , with n = 0 ,..., . Let a and a be a pair of creation and annihi- lation operators acting on that Hilbert space, and satisfying the commutation relations bracketleftbig a, a bracketrightbig = 1 , bracketleftbig a , a bracketrightbig = 0 , [ a, a ] = 0 (1) These operators generate the harmonic oscillators states {| n )} in the usual way, | n ) = 1 n ! ( a ) n | ) (2) a | ) = 0 (3) where | ) is the vacuum state of the oscillator. Let us denote by | z ) the coherent state | z ) = e z a | ) (4) ( z | = ( | e z a (5) where z is an arbitrary complex number and z is the complex conjugate. The coherent state | z ) has the defining property of being a wave packet with opti- mal spread, i.e., the Heisenberg uncertainty inequality is an equality for these coherent states. How does a act on the coherent state | z ) ? a | z ) = summationdisplay n =0 z n n ! a ( a ) n | ) (6) Since bracketleftBig a, ( a ) n bracketrightBig = n ( a ) n 1 (7) we get a | z ) = summationdisplay n =0 z n n ! parenleftBigbracketleftBig a, ( a ) n bracketrightBig + ( a ) n a parenrightBig | ) (8) Thus, we find a | z ) = summationdisplay n =0 z n n ! n ( a ) n 1 | ) z | z ) (9) Therefore | z ) is a right eigenvector of a and z is the (right) eigenvalue. 1 Likewise we get a | z ) = a summationdisplay n =0 z n n ! ( a ) n | ) = summationdisplay n =0 z n n ! ( a ) n +1 | ) = summationdisplay n =0 ( n + 1) z n ( n + 1)! ( a ) n +1 | ) = summationdisplay n =1 n z n 1 n ! ( a ) n | ) (10) Thus, a | z ) = z | z ) (11) Another quantity of interest is the overlap of two coherent states, ( z | z ) , ( z | z ) = ( | e z a e z a | ) (12) We will calculate this matrix element using the Baker-Hausdorff formulas e A e B = e A + B + 1 2 bracketleftBig A, B bracketrightBig = e bracketleftBig A, B bracketrightBig e B e A (13) which holds provided the commutator bracketleftBig A, B bracketrightBig is a c-number, i.e., it is propor- tional to the identity operator. Since bracketleftbig a, a bracketrightbig = 1, we find ( z | z ) = e zz ( | e z a e z a | ) (14) But e z a | ) = | ) (15) and ( | e z a = ( | (16) Hence we get ( z | z ) = e zz (17) An arbitrary state | ) of this Hilbert space can be expanded in the harmonic oscillator basis states {| n )} , | ) = summationdisplay n =0 n n !...
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This note was uploaded on 09/20/2010 for the course PHYS 582 taught by Professor Leigh during the Fall '08 term at University of Illinois, Urbana Champaign.

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582-chapter8 - 8 Coherent State Path Integral Quantization...

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