Principles of Quantum Mechanics · Problem 21.18

Path Integrals—II — R. Shankar

Physics — Quantum Mechanics · path-integrals-functional-methods

Problem

The generating functional for a harmonic oscillator driven by an external source J(t)J(t) is Z[J]=∫exp⁡(iℏ∫−∞∞[m2x˙2−mω22x2+J(t)x(t)]dt)D[x(t)].Z[J] = \int \exp\left(\frac{i}{\hbar}\int_{-\infty}^{\infty} \left[\frac{m}{2}\dot{x}^2 - \frac{m\omega^2}{2}x^2 + J(t)x(t)\right]dt\right) \mathcal{D}[x(t)]. (a) Show that this can be written as Z[J]=Z[0]exp⁡(i2ℏ∫dt dt′ J(t)GF(t,t′)J(t′))Z[J] = Z[0] \exp\left(\frac{i}{2\hbar}\int dt\, dt'\, J(t) G_F(t,t') J(t')\right) where GFG_F is the Feynman propagator. (b) Use functional derivatives to recover the two-point function ⟨x(t)x(t′)⟩\langle x(t)x(t')\rangle.

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