Physics / advanced
Controlled approximations in quantum dynamics
Choose and test perturbative, variational, semiclassical, and adiabatic approximations by identifying small parameters and diagnosing their failure.
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Module 01 advanced
Controlled approximations in quantum dynamics
Choose and test perturbative, variational, semiclassical, and adiabatic approximations by identifying small parameters and diagnosing their failure.
- 01
Small couplings must be compared with energy gaps
Derive stationary corrections and diagnose when a seemingly small perturbation strongly mixes states.
28 min - 02
Diagonalize inside a degenerate subspace first
Construct the correct zeroth-order combinations and connect near degeneracy with avoided crossings.
28 min - 03
Trial states give upper bounds with a precise scope
Use the Rayleigh quotient, optimize a Gaussian, and distinguish an energy bound from an error certificate.
28 min - 04
Connect semiclassical waves across a simple turning point
Match a decaying forbidden-region solution to its allowed-region phase and diagnose the WKB breakdown scale.
28 min - 05
Endpoint phases quantize a closed orbit
Derive the half-integer shift for two soft turning points and separate it from hard-wall quantization.
28 min - 06
Tunneling splits a doublet without creating a decay rate
Compare forbidden-region actions, coherent doublet oscillation, and Euclidean instanton estimates.
28 min - 07
A weak drive accumulates a complex transition amplitude
Use the interaction picture and finite-time interference while keeping time ordering and perturbative limits explicit.
28 min - 08
A transition rate emerges in a restricted continuum window
Derive the golden rule from a finite-time peak and check density, normalization, and depletion assumptions.
28 min - 09
Slow following requires a smooth path and an isolated eigenstate
Distinguish an adiabatic theorem from a local rate diagnostic and track the state rather than fixed basis populations.
28 min - 10
A crossing sweep separates diabatic survival from ground-state following
Apply the Landau–Zener formula with explicit slope, coupling, endpoints, and final-state labels.
28 min