Quantization Beyond the Coulomb Atom
For a circular orbit under a stated central force F(r), solve m v^2/r = F(r) together with mvr = n hbar. For a uniform magnetic field perpendicular to motion, qvB supplies the centripetal force and yields r_n proportional to sqrt(n).
Why this shows up in the exam
Quantized magnetic circular motion · Power-law central-potential problems · Rigid-rotor and particle-in-a-box comparisons
Learn the idea
Bohr-Sommerfeld style quantization can determine scales for other circular forces, but Coulomb-specific formulas must not be copied. The rule mvr = n hbar supplies one relation; the actual force law supplies the other. Change the force, and the powers of n in radius and energy change.
🧠 Memory hook: Quantization stays; the force law decides the scaling.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- m v²/r = F(r) — circular-orbit dynamical condition
- m v r = n hbar — semiclassical angular-momentum quantization
- r_n = sqrt(n hbar/(|q|B)) — quantized circular radius in a uniform perpendicular magnetic field under the stated model
- E_n = n² hbar²/(2I) — rigid-rotor energy under Bohr angular-momentum quantization
How to approach it
- 1Write the given force law
- 2Combine it with mvr = n hbar
- 3Derive the requested scaling before inserting values
Common slip-ups that cost marks
- •Using r_n = a₀ n²/Z outside a Coulomb field
- •Forgetting the force-balance equation
- •Presenting an old semiclassical model as exact modern quantum mechanics
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
In hydrogen, an electron transitions from n = 2 to n = 1. Using E_n = -13.6/n^2 eV, find the emitted photon energy.
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