Examples by System
Use system routes when the model is the fixed part of the problem and the method is still unclear. Each system below points to the pages where standard examples live and to reference cards for quick checks.
One-Dimensional Wave Mechanics
Section titled “One-Dimensional Wave Mechanics”One-dimensional systems are the best place to practice normalization, boundary conditions, matching, and probability current.
| System | Example Route | Method Practiced |
|---|---|---|
| Infinite well | Exercise Sets and Particle-in-a-Box Spectrum | normalization, orthogonality, discrete spectra |
| Delta interaction | Delta Function Potential | derivative jumps, bound-state scale setting |
| Potential step | Potential Step | current ratios and wave matching |
| Rectangular barrier | Rectangular Barrier Tunneling and WKB Barrier Tunneling | exact matching versus semiclassical tunneling |
Oscillators and Two-State Systems
Section titled “Oscillators and Two-State Systems”Oscillators and two-state models are the standard bridge from wave mechanics to algebraic quantum mechanics.
- Quantum Harmonic Oscillator anchors oscillator wavefunction examples.
- Number States is the natural starting point for energy-measurement examples.
- Harmonic Oscillator Ladder Operators gives compact identities after the method is understood.
- Two-State Hamiltonians develops finite-dimensional matrix examples in both matrix-entry and Pauli-vector form.
Angular and Central Systems
Section titled “Angular and Central Systems”Three-dimensional systems are where the measure and basis labels become the main source of mistakes.
| System | Example Route | Main Convention |
|---|---|---|
| Three-dimensional box | Three-Dimensional Box | Cartesian product normalization |
| Rigid rotor | Rigid Rotor | angular normalization on the sphere |
| Hydrogenic atom | Hydrogen Atom and Hydrogen Spectrum | radial function, angular function, and degeneracy labels |
| Landau levels | Landau Levels | gauge-dependent wavefunctions versus gauge-invariant degeneracy |
Spin and Composite Systems
Section titled “Spin and Composite Systems”For spin and tensor products, the system route is mostly a basis route. Fix the measurement axis or tensor-factor order before calculating.
- Spin-One-Half First Encounter gives the first physical two-level example.
- Spin-Half Hilbert Space gives the spinor convention.
- Two Spin-Half Particles gives the standard singlet-triplet example route.
- Bell States and Partial Trace anchor bipartite density-matrix examples.
Reference Companions
Section titled “Reference Companions”- Model Encyclopedia gives compact system definitions.
- Common Hamiltonians helps identify the operator before choosing an example.
- Examples by Method is better when the model is flexible but the technique is fixed.
References
Section titled “References”- R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
- D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
- L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory, 3rd ed., Pergamon, 1977.