Physics / core
Bound states and spectral structure
Solve and compare quantum bound-state problems using matching conditions, symmetry, radial equations, and controlled limiting behavior.
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Module 01 core
Bound states and spectral structure
Solve and compare quantum bound-state problems using matching conditions, symmetry, radial equations, and controlled limiting behavior.
- 01
Matching a finite square well
Determine bound energies by joining oscillatory interior solutions to normalizable exterior tails.
33 min - 02
Delta potentials and derivative discontinuities
Derive a contact-interaction matching condition and its unique attractive bound state without treating a distribution as an ordinary function.
32 min - 03
Double-well splitting and localized states
Separate stationary parity eigenstates from localized superpositions and derive tunnelling in an isolated two-state approximation.
34 min - 04
The radial equation and centrifugal barriers
Separate a central potential, choose the radial measure correctly, and identify origin conditions and angular-momentum barriers.
35 min - 05
Hydrogenic energies and degeneracy
Use the Coulomb spectrum with reduced mass and separate orbital multiplicity from spin and physical level splittings.
34 min - 06
Hydrogen wavefunctions and orbital nodes
Separate radial and angular zeros and distinguish probability per volume from probability per radial shell.
34 min - 07
Harmonic-oscillator ladder algebra
Obtain energies and matrix elements from positive norms and operator commutators rather than solving each eigenfunction separately.
35 min - 08
Coherent states and minimum uncertainty
Construct displaced oscillator ground states, follow their moving means, and distinguish coherent widths from general uncertainty saturation.
35 min - 09
Self-adjoint domains in one dimension
Use a half-line Hamiltonian to show how boundary conditions determine unitary evolution, reflection phases, and possible bound states.
38 min - 10
Counting bound states and approaching threshold
Use parity thresholds, nodal order, and admissible variational trials to count and diagnose weakly bound states.
37 min