finite-square-well-bound-states
Finite Square Well and Bound States in Quantum Mechanics (Schrödinger). Theoretical foundations, governing equations, analytical problem-solving methods, and typical archetypes.
- License 许可
- MIT
- 最近更新
- 2026-10-17
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- ★ 8
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01 前言 Front Matter
- name
- finite-square-well-bound-states
- metadata
- { "id": "qm.schrodinger.finite_well", "domain": "Quantum Mechanics", "subdomain": "Schrödinger", "difficulty": "4/5" }
- description
- Finite Square Well and Bound States in Quantum Mechanics (Schrödinger). Theoretical foundations, governing equations, analytical problem-solving methods, and typical archetypes.
02 资源文件 1
- 参考文档metadata.json1.1 KB
03 SKILL.md 正文
Finite Square Well and Bound States
ID: qm.schrodinger.finite_well
Domain: Quantum Mechanics → Schrödinger
Difficulty: 4/5
Prerequisites
- qm.schrodinger.particle_in_box
Core Concepts
- finite well
- evanescent tails
- bound vs scattering states
- transcendental eigenvalue equation
Key Equations
$$ \tan(ka) = \frac{\kappa}{k}\ \text{(even)} $$ $$ k^2 + \kappa^2 = \frac{2mV_0}{\hbar^2} $$
Methods
- Solve matching conditions at the walls
- Solve the transcendental equation graphically/numerically
- Identify the number of bound states from V_0 a^2
Typical Problem Types
- Number of bound states in a well
- Penetration depth of the tail
- Deuteron as a shallow bound state
Common Pitfalls
- Assuming infinite-well wavefunctions
- Forgetting the exponentially decaying tail
Related Skills
- qm.schrodinger.tunneling
- qm.approx.wkb
References
- Griffiths QM Ch.2
- Cohen-Tannoudji Ch.2
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