Exactly Solvable Models
Exactly solvable models are controlled laboratories. They make the assumptions, Hilbert space, Hamiltonian, spectrum, eigenstates, and limiting behavior explicit enough that other pages can link to them instead of rederiving the same facts.
Exact solvability is always conditional. A model may be exactly solvable only after choosing a dimension, boundary condition, gauge, spin convention, or idealization. The purpose of these cards is to state those choices compactly and point to the canonical derivations.
| Model | Main lesson | Canonical home |
|---|---|---|
| Free Particle | Continuous spectra, momentum eigenstates, wave packets | Free Particle |
| Harmonic Oscillator | Ladder operators, zero-point energy, Gaussian ground state | Quantum Harmonic Oscillator |
| Hydrogen Atom | Coulomb spectrum, central potentials, orbital structure | Hydrogen Atom |
| Landau-Level System | Magnetic quantization and degeneracy | Landau Levels |
| Pöschl–Teller Potential | Solvable wells, shape invariance, reflectionless scattering | this reference card |
How to Use These Cards
Section titled “How to Use These Cards”Use the model card to identify conventions before using a formula. Then follow the canonical page for derivations, normalization, pictures, and exercises. If a card links to a Hamiltonian card, the Hamiltonian card is the compact operator reference; the model card is the physical interpretation and taxonomy.
Common Mistakes
Section titled “Common Mistakes”- Treating the word “exact” as independent of boundary conditions or domains.
- Using a spectrum after changing dimensionality, gauge, or spin content.
- Forgetting that exactly solvable toy models are often starting points for approximations, not descriptions of every real system.
- Duplicating a derivation in a model card instead of linking to the canonical page.
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.