Principles of Quantum Mechanics · Problem 21.2

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals

Problem

Derive the free-particle propagator G0(x,x′;t)G_0(x, x'; t) by summing over paths, i.e., by evaluating G0(x,x′;t)=∫x(0)=x′x(t)=xeiS[x(t′)]/ℏD[x(t′)]G_0(x, x'; t) = \int_{x(0)=x'}^{x(t)=x} e^{iS[x(t')]/\hbar} \mathcal{D}[x(t')] directly. Break the time interval [0,t][0, t] into NN equal pieces and show that in the limit N→∞N \to \infty, you recover the standard result G0(x,x′;t)=m2πiℏtexp⁡(im(x−x′)22ℏt).G_0(x, x'; t) = \sqrt{\frac{m}{2\pi i \hbar t}} \exp\left(\frac{im(x-x')^2}{2\hbar t}\right).

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