Light Quanta · Section 8

Energy is discrete.
Rates scale with power.

Why does increasing light intensity release more electrons without increasing their individual energy, while increasing frequency increases electron energy without requiring higher intensity?

Read Section 8 of Einstein’s 1905 paper →

LQ-08Status: accepted | Step: 0 | Run: _R_6av5uiv5b_/run/1

Millikan (1916) Overlay

Incident Power P_opt (mW)1.00 mW

Frequency ν (THz)600.0 THz

Work Function Φ (eV)2.20 eV

Quantum Efficiency η10.0 %

Collector Potential U_c (V)0.00 V

Single-Quantum Energy Conservation Ladder

hν = 2.481 eV | Φ = 2.20 eV (ν_0 = 532.0 THz)

Stopping Potential vs. Light Frequency: V_s(ν)

Universal theoretical slope h/e = 4.136 × 10^-15 V·s

Historical Validation: Millikan (1916) Sodium

Empirical Fit Slope: 4.1270e-15 V·s ± 3.85e-19 | Theoretical (h/e): 4.1357e-15 V·s

Current-Voltage Characteristic: I(U_c)

Saturation current I_sat = 40.30 μA

Accepted Laboratory Snapshot (Instance Telemetry)

Quantity ID	Status	Value / Result	Unit	Owner ID
incidentPower	value	1.0000e-3	W	lq08.acceptedInputs
frequency	value	6.0000e+14	Hz	lq08.acceptedInputs
workFunction	value	2.2000e+0	eV	lq08.acceptedInputs
quantumEfficiency	value	1.0000e-1	1	lq08.acceptedInputs
collectorPotential	value	0.0000e+0	V	lq08.acceptedInputs
quantumEnergy	value	3.9756e-19	J	photoelectric.quantumEnergy
thresholdFrequency	value	5.3196e+14	Hz	photoelectric.thresholdFrequency
maxKineticEnergy	value	4.5085e-20	J	photoelectric.kMax
stoppingPotentialMagnitude	value	2.8140e-1	V	photoelectric.stoppingPotentialMagnitude
quantumRate	value	2.5153e+15	s^-1	photoelectric.quantumRate
emissionRate	value	2.5153e+14	s^-1	photoelectric.emissionRate
photocurrent	value	4.0300e-5	A	photoelectric.photocurrent

Historical Readout: Einstein 1905 §8 Order-of-Magnitude Check

What was neglected: Einstein sets P' = 0 as a deliberate neglect of escape work for order-of-magnitude comparison against Lenard's spark-potential observations, not as a physical prediction for a named metal.

What it is not: This is not a prediction for any named metal; any real substance has P' > 0, so its stopping potential at this frequency is lower by exactly the amount the work function contributes.

Representation A (Printed Form):

Π = (R · β · ν) / E = 4.3385 V (ca. 4,3 Volt)

Slope: 4.2121e-15 V·s (modern h/e = 4.1357e-15 V·s)

Live Hypothetical Comparison:

ν = 600.0 THz → hν = 2.481401 eV

Hypothetical Φ = 2.2 eV → V_s = 0.281401 V (hypothetical)

Discovery Mode: Predict Before Interacting

Select an inquiry to test your deductive understanding of light-quantum mechanics:

Limits of this Reference Model (Not Modeled)

This reference owner implements Einstein’s 1905 single-quantum absorption and escape energy relations. The following physical regimes require higher-order quantum optics or microscopic surface physics and are explicitly not modeled:

Real-material electron energy distributions and yields
Contact potentials and surface states
Space charge
Reflection losses
Emission angles
Multi-photon or thermionic emission
The timing of individual emissions
Energy transfer models beyond the declared complete or partial cases
Any claim that the moving marks depict photons

Compare two setups side by side in your reading. Each laboratory has its own settings, stepwise state and accepted results.

The Physical Argument

The Single-Quantum Energy Conservation Law

In §8 of his 1905 paper, Einstein applies the light-quantum hypothesis to the generation of cathode rays by light (the photoelectric effect). If monochromatic light consists of energy quanta of magnitude

R\beta\nu = h\nu

, an absorbed quantum transfers its entire energy to a single electron in the cathode.

\Pi E = R\beta\nu - P

In modern notation, writing

\Pi

as the stopping potential

V_s

E

as the elementary charge

e

, and

P

as the surface escape work

\Phi

e V_s = h\nu - \Phi \implies V_s = \frac{h}{e}\nu - \frac{\Phi}{e}

Two Qualitative Predictions Classical Waves Cannot Explain

Intensity Invariance of Electron Energy: Increasing the radiant power of the incident light increases the photon flux (
$\dot{N} = P_{\text{light}}/(h\nu)$
), yielding proportionally more photoelectrons per second, but leaves the kinetic energy of every single ejected electron completely unchanged.
Linear Frequency Dependence and Threshold Cutoff: The maximum kinetic energy and retarding potential depend exclusively and linearly on the light frequency
$\nu$
. Below the threshold frequency
$\nu_0 = \Phi/h$
, no electrons can escape the cathode surface regardless of how intense the illumination is.

Einstein's 1905 Historical Check

At the time of writing in 1905, quantitative photoelectric data was scarce. Philipp Lenard had observed in 1902 that spark potentials reached several volts under ultraviolet arc illumination. Einstein calculated that for UV light of wavelength 290 nm (

\nu \approx 1.03\times 10^{15}\text{ Hz}

), neglecting surface escape work, the expected potential is:

V_s \approx \frac{h\nu}{e} \approx 4.3\text{ Volts}

This matched the order of magnitude of Lenard's sparks, providing the first numerical plausibility test of the light-quantum hypothesis.

Universal Slope: Robert Millikan's 1916 Precision Validation

Over a decade later, Robert Millikan undertook exhaustive vacuum experiments on freshly cut alkali metals (sodium, potassium, lithium) to test Einstein's linear equation. While initially skeptical of light quanta, Millikan's 1916 data confirmed with remarkable precision that every metal exhibits the exact same slope

\frac{dV_s}{d\nu} = \frac{h}{e}

, varying only in the horizontal threshold cutoff intercept

\nu_0

Epistemic Boundary: Deductive Consequences vs. Empirical Proof

A simulator programmed with an energy threshold does not prove that nature has a threshold; it demonstrates the deductive consequences of single-quantum energy exchange and surface escape work (

E_q = h\nu

W = \Phi

). Independent experiments (such as Millikan’s 1916 precision dataset and later single-photon anti-bunching measurements) test whether those assumptions describe the physical world.

When

\nu < \nu_0

, the stopping potential is strictly not applicable (a typed non-value), never zero, because no emitted photoelectrons exist to be retarded. Millikan 1916 is later historical evidence on the timeline, never an axiom of the 1905 derivation.

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