Find the error in the following derivation of the free-particle propagator sliced…

PHYSICS · Multi-part · Fermi University Physics · Quantum Mechanics · Quantum dynamics and its formulations · Quantum dynamics and its formulations

The Structure of Quantum Mechanics, Ch The path integral

Problem

Find the error in the following derivation of the free-particle propagator sliced into N=3N=3 equal steps of width ϵ=t/3\epsilon = t/3. Starting from the composition law and inserting resolutions of the identity, someone writesK(x,t;x0,0)=∫dx1 dx2 exp⁡ ⁣[im2ℏϵ((x1−x0)2+(x2−x1)2+(x−x2)2)],\begin{aligned}K(x,t;x_0,0)=\int dx_1\,dx_2\,\exp\!\bigg[&\frac{im}{2\hbar\epsilon}\Big((x_1-x_0)^2+(x_2-x_1)^2\\ &+(x-x_2)^2\Big)\bigg],\end{aligned}evaluates the two Gaussian integrals over x1x_1 and x2x_2 directly, and declares the resulting Gaussian in (x−x0)(x-x_0) to already be the exact propagator, with no further factors needed. ((a)) Identify precisely what has been left out of the sliced expression and explain why it cannot be dropped. ((b)) Supply the missing piece and complete the calculation, and confirm that the result agrees exactly with the one-slice propagator K=m/2πiℏt exp⁡[im(x−x0)2/2ℏt]K=\sqrt{m/2\pi i\hbar t}\,\exp[im(x-x_0)^2/2\hbar t] with t=3ϵt = 3\epsilon.

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