Topics#

Key Concepts#

• Propagator \(K(x,t;x',t') = \langle x\vert\hat{U}(t,t')\vert x'\rangle\) — kernel that evolves any wavefunction.

• Composition (semigroup) property — the engine of time slicing.

• Slice propagator \(K_{\delta t}\) — phase fixed by \(S_{\text{slice}}\), amplitude fixed by self-consistency at \(\delta t = 0\).

• Path integral as a limit — \(\int\mathcal{D}[x]\,\mathrm{e}^{\mathrm{i}S[x]/\hbar}\) defined as \(N\to\infty\) of the slicing.

• Schrödinger equation — derived from one slice; emerges as the local content of phase = action.

• Free propagator \(K_{\text{free}} = (m/2\pi\mathrm{i}\hbar t)^{d/2}\,\mathrm{e}^{\mathrm{i}S_{\text{cl}}/\hbar}\) — phase = classical action, exactly.

Learning Objectives#

• Define the propagator and explain how it evolves wavefunctions and composes over intermediate times.

• Write the path integral as the \(N\to\infty\) limit of time-sliced integrals; identify what \(\mathcal{D}[x]\) means.

• Derive the Schrödinger equation from the one-slice path integral and explain how the requirement at \(\delta t = 0\) fixes the slice normalization.

• Compute the free-particle propagator from a superposition of plane waves and identify its phase as the classical action.

• Articulate the two conceptual closures: (i) the slice formula was already exact for the free particle, (ii) the propagator picture restores the classical particle that a single plane wave seems to lose.

Project: Quantum Revivals and the Talbot Effect#

Objective: Study quantum carpets—spatiotemporal patterns of probability density—arising from the free particle propagator, connect to the Talbot effect in optics, and uncover the hidden periodic structure beneath quantum interference.

Background: The free particle propagator exhibits a striking phenomenon: at rational times \(t = 2\pi M/(\hbar k^2)\), the probability density returns to a periodic pattern (quantum revival), while at irrational times it fills space densely (ergodic-like behavior). This is closely related to the Talbot effect in classical optics: when coherent light passes through a periodic grating, the intensity pattern repeats periodically in the paraxial direction even though diffraction causes spreading. The quantum analog reveals how periodic boundary conditions and rational multiples of \(\hbar\) create exact revivals—a frontier in understanding coherence and control in quantum systems.

Suggested Approach:

• Compute the free particle propagator \(K(x,t; x_0, 0) = \sqrt{m/(2\pi \mathrm{i}\hbar t)} \mathrm{e}^{\mathrm{i}m(x-x_0)^2/(2\hbar t)}\) and visualize \(|K|^2\) as a 2D “carpet” with space on the x-axis and time on the y-axis.

• Initialize a localized Gaussian wavepacket and evolve it under the free particle Hamiltonian; track \(|\psi(x,t)|^2\).

• Identify quantum revivals: plot \(|\psi(x,t)|^2\) at times \(t_n = 2\pi M_n/(\hbar k^2)\) for various integers \(M_n\). Describe the periodic structure.

• Introduce a periodic initial condition (e.g., a Bloch state or a periodic superposition) and study how the carpet pattern changes; compare to the Talbot effect.

• Compute Loschmidt echo or fidelity to quantify how closely the carpet returns to its initial structure at revival times.

• Extend to a weakly perturbed system (add a small potential \(V(x)\)) and explore how revivals degrade—a probe of decoherence.

Expected Deliverable: A research report (5–10 pages) with code and visualizations. Include: (i) numerical computation and visualization of quantum carpets, (ii) demonstration of quantum revivals and their connection to Talbot periodicity, (iii) analysis of the period as a function of \(\hbar\) and initial momentum, (iv) comparison of quantum vs. classical optics Talbot effects, (v) investigation of how perturbations destroy revivals, with implications for quantum control and decoherence.

Key References: J. H. Hannay & M. V. Berry, Physica D 17, 385 (1985) [quantum revivals]; F. Haake et al., Phys. Rev. A 47, 2506 (1993); M. Berry & M. Keating, J. Phys. A 23, 4839 (1990) [Talbot effect].