Hydrogen-Like Energy Levels
For a one-electron Coulomb system with a heavy nucleus, the total energy of level n is -13.6 Z^2/n^2 eV. The magnitude is the binding energy required to remove the electron from that level.
Why this shows up in the exam
Finding atomic number from level energy · Comparing equal-energy levels in different ions · Locating an orbit from a stated energy
Learn the idea
Bound-state energy is negative and scales as minus Z squared over n squared. Zero energy means the electron is free at infinity. A bound orbit lies below zero; higher n levels are less negative and crowd toward the ionization limit.
🧠 Memory hook: Bound is below zero; bigger n climbs toward zero.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- E_n = -13.6 Z²/n² eV — hydrogen-like total energy
- binding energy = |E_n| — energy needed to ionize from level n
- E_infinity = 0 — continuum reference
How to approach it
- 1Identify n and Z
- 2Calculate the signed level energy
- 3Use magnitude only when the question asks binding or ionization energy
Common slip-ups that cost marks
- •Dropping the negative sign for total energy
- •Using Z rather than Z squared
- •Calling the ground-state magnitude the energy itself
🌟 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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