This script can be pasted into the Eigenmath app.

 -- Feynman and Hibbs problem 4-1 x = (x1,x2,x3) eta = (eta1,eta2,eta3) -- Directional derivative (unnormalized) D(f) = dot(eta,d(f,x)) -- Gaussian integral G(f) = do( f = f (2 pi i hbar epsilon / m)^(3/2), f = eval(f, eta1^2, i hbar epsilon / m), f = eval(f, eta2^2, i hbar epsilon / m), f = eval(f, eta3^2, i hbar epsilon / m), f = eval(f, eta1, 0), f = eval(f, eta2, 0), f = eval(f, eta3, 0), f) -- return f "Verify equation (13)" B = G(psi()) C = psi() (2 pi i hbar epsilon / m)^(3/2) check(B == C) "ok" "Verify equation (14)" B = G(D(psi())) check(B == 0) "ok" "Verify equation (15)" B = G(D(D(psi()))) C = i hbar epsilon / m * (2 pi i hbar epsilon / m)^(3/2) * (d(psi(),x1,x1) + d(psi(),x2,x2) + d(psi(),x3,x3)) check(B == C) "ok" "Verify equation (16)" f = psi() + D(psi()) + 1/2 D(D(psi())) B = G(f) B = B (1 - i epsilon / hbar V) B = eval(B,epsilon^(7/2),0) -- discard epsilon^2 B = B / (2 pi i hbar epsilon / m)^(3/2) -- cancel A B = B - psi() -- cancel psi B = B / epsilon -- cancel epsilon C = i hbar / (2 m) * (d(psi(),x1,x1) + d(psi(),x2,x2) + d(psi(),x3,x3)) + (-1) i / hbar V psi() check(B == C) "ok" "Compare with Feynman and Hibbs equation (4.13)" C = -hbar^2 / (2 m) * (d(psi(),x1,x1) + d(psi(),x2,x2) + d(psi(),x3,x3)) + V psi() C = -i / hbar C check(B == C) "ok"