Playground

Hydrogen energy level diagram with transition arrows. Click n to see photon wavelength for transitions to ground state.

Variables

SymbolNameSIDimensionRange
EnE_nEnergy of level noutput
Bound-state energy of the nth orbit
JM*L^2*T^-2-2.18e-18 – 0
nnPrincipal quantum number
Integer orbit index (1, 2, 3, ...)
dimensionless11 – 20

Deep dive

Derivation
Bohr postulated quantized angular momentum L = n*hbar. Combining Coulomb attraction F = e^2/(4*pi*epsilon_0*r^2) with circular motion m_e*v^2/r = F, and using L = m_e*v*r = n*hbar, solve for allowed radii r_n = n^2*a_0 (Bohr radius a_0 ≈ 5.29e-11 m). Total energy E_n = KE + PE = -m_e*e^4/(8*epsilon_0^2*h^2*n^2) ≈ -13.6 eV / n^2.
Experimental verification
Hydrogen emission spectrum (Balmer, Lyman, Paschen series) matches to 4+ significant figures. Franck-Hertz experiment (1914) directly demonstrated quantized atomic energy levels.
Common misconceptions
  • Electrons do NOT actually orbit in circles — this is a semiclassical approximation
  • The minus sign reflects that bound states are below the zero of free-electron energy
  • Bohr model fails for multi-electron atoms — full QM is required
Real-world applications
  • Astronomical spectroscopy (redshift measurements)
  • Hydrogen maser clocks
  • Plasma diagnostics
  • Derivation of Rydberg constant R_H

Worked examples

n=2 energy

Given:
n:
2
Find: E_2
Solution

E_2 = -13.6/4 = -3.4 eV = -5.45e-19 J

Balmer alpha (n=3 → n=2)

Given:
n_i:
3
n_f:
2
Find: lambda_photon
Solution

lambda ≈ 656.3 nm (red H-alpha line)

Scenarios

What if…
  • scenario:
    What if Z = 2 (He+)?
    answer:
    E_n = -13.6*Z^2/n^2 = -54.4/n^2 eV — ionization energy quadruples.
  • scenario:
    What if n → ∞?
    answer:
    E_n → 0, continuum states (ionized hydrogen).
  • scenario:
    What if the electron were heavier?
    answer:
    Muonic hydrogen has m ≈ 207*m_e — all energies scale by 207, Bohr radius shrinks by 207.
Limiting cases
  • condition:
    n = 1
    result:
    E_1 = -13.6 eV
    explanation:
    Ground state — the ionization energy of hydrogen.
  • condition:
    n = 2
    result:
    E_2 = -3.4 eV
    explanation:
    First excited state — source of Balmer series transitions.
  • condition:
    n → ∞
    result:
    E_n → 0
    explanation:
    Continuum limit — the electron is free.

Context

Niels Bohr · 1913

Bohr combined Rutherford's nuclear model with Planck quantization to explain hydrogen's emission spectrum.

Hook

Why does hydrogen glow only in specific colors?

Electrons in hydrogen occupy quantized orbits with discrete energies. What is the energy of the n=2 level?

Dimensions:
lhs:
E_n → [M*L^2*T^-2]
rhs:
[M*L^2*T^-2] / (dimensionless) → [M*L^2*T^-2]
check:
Both sides are energy. ✓
Validity: Valid for hydrogen and hydrogen-like ions (one electron). Fine structure, hyperfine splitting, and Lamb shift require relativistic/QED corrections.

Related formulas

Photon Energy (Planck Relation)de Broglie WavelengthTime-Dependent Schrödinger Equation