chapter  6
12 Pages

Path Integrals in Quantum Mechanics

An important ingredient in deriving the path integral expression will be the completeness relations

1ˆ = ∫

dx |x >< x| and 1ˆ = ∫

dp |p >< p|. (6.5)

For arbitrary N we can use this to write

< x′|e−i HT/h¯ |x > = < x′| (e−i HT/Nh¯)N |x > =

∫ · · ·

∫ < x′|e−i HT/Nh¯ |xN−1 > dxN−1

< xN−1|e−i HT/Nh¯ |xN−2 > dxN−2 < xN−2| · · · · · · e−i HT/Nh¯ |x2 > dx2 < x2|e−i HT/Nh¯ |x1 > dx1 < x1|e−i HT/Nh¯ |x > . (6.6)

A in

We will now use the so-called Trotter formula

e−i( A+B)/N = e−i A/Ne−i B/N(1 +O(N−2)) (6.7) for two operators A and B. This can be seen by expanding the exponents, and the error term is actually of the form [A, B]/N2. (One can also use the Campbell-Baker-Hausdorff formula, which will be introduced later). With the Hamiltonian of Equation (6.1) this can be used to write for N → ∞

By inserting the completeness relation for the momentum we can eliminate the operators

< xi+1|e−i HT/Nh¯ |xi > = ∫

dpi < xi+1|pi >< pi |e−i HT/Nh¯ |xi >

≈ ∫

dpi < xi+1|pi >< pi |e−i pˆ2T/2mNh¯e−iV(xˆ)T/Nh¯ |xi >

= ∫

dpi < xi+1|pi >< pi |e−i p2i T/2mNh¯e−iV(xi )T/Nh¯ |xi >

= ∫

2πh¯ e−i

] T/Nh¯

. (6.9)

This can be done for each matrix element occurring in Equation (6.6). Writing t = T/N, xN = x′ and x0 = x we find

< x′|e−i HT/h¯ |x > = lim N→∞

∫ · · ·

∫ N−1∏ i=1

dp j

× N−1∏ i=0

< xi+1|pi >< pi |e−i Ht/h¯ |xi >

= lim N→∞

∫ dp0 2πh¯

∫ dxi dpi

2πh¯

× exp [

it h¯

( pi (xi+1 − xi )

t − p

2m − V(xi )

)] . (6.10)

It is important to observe that there is one more p integration than the number of x integrations.