q, p = e iqP e i p Q |0 = e i p q/2 e i( pQ qP ) |0 in .NET framework

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|q, p = e iqP e i p Q |0 = e i p q/2 e i( pQ qP ) |0
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(397)
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Since they are the standard ones |z , z = factor e
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i p q/2
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q+i p 2
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times the (important here!) phase
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, they have unit norm and they resolve the unity as well: |q, p q, p| dq d p/2 = I d
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q, p|q, p = 1 ,
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(398)
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R2 f
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Let us introduce, with the phase space variables (q, p), the (nonanalytical) symbol of the wave function like was de ned in (321): (q, p, t) = q, p| ( , t)
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(399)
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At this point, we should make clear the probabilistic interpretation of this object compared with (x, t) We know that | (x, t)|2 is the probability density of nding the particle at position x (at time t) On the other hand, the quantity | (q, p, t)|2 is the probability that the state | can be found in the state |q, p What is important
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is to be aware that in coherent state representation we can specify both values of the variables simultaneously at the same time In this Fock Bargmann representation, position and momentum operators are given by Qf = q + i , p P f = i , q (3100)
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and thus the coherent state representation of the Schr dinger equation with classical Hamiltonian H(q, p) is given by i (q, p, t)/ t = H( i / q, q + i / p ) (q, p, t) (3101)
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36 The Feynman Path Integral and Coherent States
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The solution to this form of Schr dinger s equation can be expressed in the form (q, p, t) = K (q, p, t; q , p , t ) (q , p , t ) dq d p /2 , (3102)
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where K (q, p, t; q , p , t ) denotes the propagator in the coherent state representation The propagator for the coherent state representation of Schr dinger s equation can also be given a formal phase space path integral form, namely, K (q, p, t; q , p , t ) = M exp{i [ p q H(q , p )] dt} Dq D p (3103)
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Despite the fact that this expression looks the same as (394), the pinned values and the lattice space formulations are different In the coherent state case, we have K (q, p, t; q , p , t ) =
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exp i
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( p l+1 + p l )(q l+1 q l )
H ~ exp
1 2 ( p l+1
+ p l ) + i 1 (q l+1 q l ), 1 (q l+1 + q l ) i 1 ( p l+1 p l ) 2 2 2
(1/4)
[( p l+1 p l )2 + (q l+1 q l )2 ]
dq l d p l /(2 )
(3104)
Observe that there are now the same number of p and q integrations in this expression Such a conclusion is fully in accord with the combination law as expressed in the coherent state representation, namely, K (q, p, t; q , p , t ) = K (q, p, t; q , p , t ) K (q , p , t ; q , p , t ) dq d p /2 (3105)
Daubechies and Klauder [30] gave a mathematical rigor to the expressions (3103) and (3104) by introducing a Brownian type regularization term They eventually proved the existence of the following limit: lim M exp i ~ exp 1 2 p q H(q , p ) du p
+q
Dq D p
= lim 2 e T /2
e i [ p dq H(q , p ) du] d (q , p ) W (3106)
= q, p| e i(t t )H |q , p = K (q, p, t; q , p , t ) ,
where the second line of (3106) is a mathematically rigorous formulation of the heuristic and formal rst line, and denotes the measure on continuous path W
3 The Standard Coherent States: the (Elementary) Mathematics
q (u), p (u), t u u u t, said to be a pinned Wiener measure Such a choice of regularization also justi es the choice of coherent states (397) and imposes the condition that H(q, p) be precisely the upper symbol of the quantum Hamiltonian H= H(q, p) |q, p q, p| dq d p/(2 )
(3107)
A suf cient set of technical assumptions ensuring the validity of this representation is given by 2 2 (a) H(q, p)2 e ( p +q ) dq d p < , for all > 0 , (b) H(q, p)4 e
( p 2 +q 2 )
dq d p < ,