Practice & Projects

Make problem solving, computation, and research-entry practice reusable across learning routes.

Projects available now

These projects connect the lessons to an independent check. Each linked page supplies the relevant reasoning, executable code or analytic tasks, inputs, expected results, and exercises with solutions.

Project	Preparation	Main test
Three-site Toda	Hamilton’s equations, Lax matrices and conserved quantities	Compare time integration with an independent QR solution; measure convergence and errors that invariants miss
Finite-chain Bethe states	One- and two-magnon amplitudes and ring equations	Build the Hamiltonian in two ways and test actual normalized eigenvectors
Commuting charges and the XXX Hamiltonian	Tensor products, the rational R-matrix and monodromy	Compare transfer identities and analytic differentiation against independently built short-chain operators
An exact spin correlation	Normalized Bethe matrix elements and basic Python	Compare a complete spectral sum with direct evolution; detect lost weight, normalization and Fourier-sign errors
KdV convergence laboratory	The travelling pulse, conserved integrals and basic Python	Refine time, spatial resolution and domain separately against an exact line profile
Two constructions of a KdV soliton	Travelling-wave ODE and one-pole reconstruction	Match field, speed, eigenvalue and normalization; identify an inconsistent construction
Two-soliton collision	Exact collision and phase shifts, the one-pulse laboratory and basic Python	Evolve interacting pulses, compare the full field, and separate numerical error from finite-time overlap
A singular Bethe state	The XXX Hamiltonian, monodromy blocks and polynomial limits	Reproduce a published finite result, distinguish root and state residuals, and diagnose loss of numerical precision
Finite-ring TASEP dynamics	Markov generators, stationary current and basic Python	Compare fixed-time trajectory samples with deterministic evolution, reconstruct a Bethe mode, and detect direction and sampling mistakes

Use solutions to diagnose a missing step

Begin with a calculation you can predict by hand. Record the equation, initial or boundary data and error measure before running code. Compare against the stated result, then change one setting and explain the consequence. A small residual has meaning only for the equation and norm that produced it.

The lesson sequences provide shorter guided, independent and changed-setting tasks: Toda, the XXX chain, KdV, and TASEP. For elementary Hamiltonian practice, test a coordinate change in the bracket bridge and match initial data in the oscillator lesson. For quantum preparation, construct tensor matrices and test a symmetry-breaking perturbation in the linear algebra bridge.

Open a hint before a full solution when you need one intermediate step. After reading a solution, redo the calculation with the lesson’s altered data.

For differential-equation practice, the phase-portrait bridge asks you to check an initial-value solution, calculate a logistic trajectory and distinguish finite-time blowup from nonuniqueness.

The remaining chapter map describes future problem collections, laboratories and projects. It does not imply that the unlinked tasks are already available.