Stokes's Rule and the Single-Quantum Energy Budget
Why the frequency of emitted fluorescent light cannot exceed that of the exciting light under elementary quantum transformation, and how Einstein deduced the exact conditions for exceptions.
Light Quanta · §7 Fluorescence & Stokes's Rule
Fluorescence Energy Budget & Stokes's Rule
How single-quantum energy conservation hν₁ = hν₂ + E_other explains Stokes's rule (ν₂ ≤ ν₁) and correctly predicts multi-quantum and thermal deviation conditions.
Predict Mode · Energy Conservation
Can fluorescent emission occur at higher frequency than the exciting light (ν₂ > ν₁) under single-quantum absorption?
Predict Mode · Weak-Illumination Linearity
How does the emission rate behave as the incident light becomes extremely weak?
Elementary Quantum Energy LedgerAllowed by Budget
Stokes's Rule (§7)
Verdict: Allowed under Stokes's rule: emitted quantum energy does not exceed absorbed quantum energy (nu2 <= nu1).
Spectral Bands & False-Color Legend
Wavelength λ = c / ν
Ultraviolet (UV)
> 789 THz (< 380 nm)
Visible Spectrum
400–789 THz (380–750 nm)
Infrared (IR)
< 400 THz (> 750 nm)
Weak-Illumination Photon Rates (Zero Threshold)Yield Y = 0.50
Absorbed Rate Ṅ₁:1.7755e+12 s⁻¹
Emitted Rate Ṅ₂:8.8776e+11 s⁻¹
Emitted Power:0.5000 μW
Heat Dissipated:0.5000 μW
Interactive Energy & Parameter Controls
Calculated Energy Ledger & Transition Quantities
Physical Quantity
Symbol
Calculated Value
Physical Meaning
Budget Verdict
Verdict
Allowed
Allowed under Stokes's rule: emitted quantum energy does not exceed absorbed quantum energy (nu2 <= nu1).
Maximum Allowed Frequency
ν₂,max
850.00 THz
Upper frequency bound for emitted light
Absorbed Quantum Energy
hν₁
3.5153 eV
Energy of one exciting light quantum
Emitted Quantum Energy
hν₂
3.5153 eV
Energy of candidate emitted light quantum
Non-Optical Dissipation (Heat)
E_other
0.0000 eV
Energy transferred to thermal modes of medium
Energy Deficit
ΔE
0.0000 eV
Zero (Conserved)
The Single-Quantum Energy Budget in Einstein 1905 §7
In 1852, George Gabriel Stokes formulated the empirical rule that fluorescent light always has a lower frequency (longer wavelength) than the light that excited it. In §7 of his 1905 paper, Einstein showed that this rule is an immediate consequence of the light-quantum hypothesis:
“If monochromatic light of frequency ν₁ is transformed into light of frequency ν₂ by photoluminescence, and if the process occurs such that one absorbed quantum is converted into one emitted quantum plus non-optical energy... then the energy of the emitted quantum cannot be greater than that of the exciting one.”
Rather than stating Stokes's rule as an unbreakable law, Einstein explicitly deduced the physical conditions under which anti-Stokes emission (ν₂ > ν₁) can occur:
Deviation Case 1 (Multi-quantum absorption): If the elementary process involves the simultaneous absorption of k light quanta, the available energy is k hν₁, permitting emission up to ν₂ ≤ k ν₁.
Deviation Case 2 (Non-Wien exciting radiation): If the incident light is not in the Wien regime (where the light-quantum volume law was derived), single-quantum behavior is not guaranteed.