Ivancevic_Applied-Diff-Geom

1008 Applied Differential Geometry: A Modern Introductionor, equivalently,We also define the vector|ψ〉 H= e iHt |ψ, t〉 S.|q, t〉 H= e iHt |q〉 S,which is the Heisenberg version of the Schrödinger state |q〉. Then, we canequally well writeψ(q, t) = 〈q, t |ψ〉 H. (6.13)By completeness of states we can now write∫〈q f , t f |ψ〉 H= 〈q f , t f |q i , t i 〉 H〈q i , t i |ψ〉 Hdq i ,which with the definition of (6.13) becomes∫ψ(q f , t f ) = 〈q f , t f |q i , t i 〉 Hψ(q i , t i ) dq i .Comparing with (6.12), we getK(q f , t f ; q i , t i ) = 〈q f , t f |q i , t i 〉 H.Now, let us calculate the quantum–mechanics propagator〈〈q ′ , t ′ |q, t〉 H= q ′ |e −iH(t−t′) |q〉using the path–integral formalism that will incorporate the direct quantizationof the coordinates, without Hilbert space and Hermitian operators.The first step is to divide up the time interval into n + 1 tiny pieces:t l = lε + t with t ′ = (n + 1)ε + t. Then, by completeness, we can write(dropping the Heisenberg picture index H from now on)∫ ∫〈q ′ , t ′ |q, t〉 = dq 1 (t 1 )... dq n (t n ) 〈q ′ , t ′ |q n , t n 〉 ×× 〈q n , t n |q n−1 , t n−1 〉 ... 〈q 1 , t 1 |q, t〉 . (6.14)The integral ∫ dq 1 (t 1 )...dq n (t n ) is an integral over all possible paths, whichare not trajectories in the normal sense, since there is no requirement ofcontinuity, but rather Markov chains.Now, for small ε we can write〈〈q ′ , ε |q, 0〉 = q ′ |e −iεH(P,Q) |q〉 = δ(q ′ − q) − iε 〈q ′ |H(P, Q) |q〉 ,