Quantum Experiments · I · Observations and models · Chapter 02
How do spectra and photoelectric measurements reveal energy scales?
When monochromatic light of frequency \(f\) illuminates a clean metal cathode in vacuum, electrons are liberated into the gap facing a collector electrode. Two optical controls act on completely different observables: varying the optical frequency \(f\) changes the energy quantum \(hf\) delivered per absorption event, whereas varying the optical intensity at fixed \(f\) changes the photon arrival rate and therefore the magnitude of the photocurrent. Applying a variable retarding voltage \(V\) between cathode and collector opposes the emitted electrons; the stopping potential \(V_s\) is the bias at which even the fastest photoelectrons, with maximum kinetic energy \(K_{\max}\), are turned back just before reaching the collector.
Two complementary experiments probe discrete energy transfer in matter from different angles. In optical emission and absorption spectroscopy, a dilute atomic gas emits or absorbs light only at sharp resonant frequencies \(f_{if}\), mapping out differences between internal atomic energy levels. In the Franck–Hertz collision experiment, electrons accelerated through a low-pressure mercury or neon vapor undergo strictly elastic collisions until their kinetic energy reaches a discrete atomic excitation threshold \(\Delta E\), producing periodic drops in collected current as a function of accelerating voltage. Together, photoemission, optical spectra, and inelastic electron scattering constrain the same underlying principle of quantized energy exchange.
The interactive diagram below represents photoemission as an energy-balance schematic: incident packets of energy \(hf\) overcome a step barrier of height \(\Phi\) to populate a kinetic-energy gauge \(K_{\max}\). It is not a spatial simulation of electron wave packets propagating inside a crystal lattice. In a physical phototube, surface patch potentials, contact-potential offsets between dissimilar electrodes, thermal Fermi–Dirac tailing at temperature \(T\), and bulk scattering all broaden the onset of current near \(V_s\).
Visual animation · Section 2.1
Photoemission, line spectra, and inelastic collisions
In Einstein’s single-quantum photoelectric model, an electron near the Fermi level absorbs one energy quantum \(hf\) and pays a minimum energy cost \(\Phi\)—the material’s work function—to escape the metal surface. Whenever \(hf\ge\Phi\), the maximum kinetic energy of the emitted photoelectrons obeys the linear relation
\[K_{\max} = hf - \Phi, \qquad eV_s = K_{\max} = hf - \Phi,\]
where \(h\) is Planck’s constant, \(e\) is the elementary charge, and \(f_0=\Phi/h\) is the threshold frequency below which single-photon emission from states at or below the zero-temperature Fermi level cannot occur.
Measuring energies in electron-volts (\(\mathrm{eV}\)) makes stopping-potential arithmetic transparent because an electron traversing a potential difference of \(1\text{ V}\) gains or loses exactly \(1\text{ eV}\) of kinetic energy. For a cathode with work function \(\Phi=2\text{ eV}\) illuminated by photons of energy \(hf=3\text{ eV}\), the fastest emitted electrons emerge with \(K_{\max}=3\text{ eV}-2\text{ eV}=1\text{ eV}\), and a retarding bias of \(V_s=1\text{ V}\) halts the photocurrent. Doubling the optical intensity in the linear single-photon regime doubles the emission rate without altering \(K_{\max}\) or \(V_s\).
In atomic spectroscopy, each observed line frequency \(f_{if}\) measures the separation between two stationary energy levels \(E_i\) and \(E_f\) via Bohr’s frequency condition \(hf_{if}=|E_i-E_f|\). Because every spectral line reports a difference between two levels, shifting all energy eigenvalues by a common constant \(E_0\) leaves the entire spectrum unchanged; establishing an absolute energy scale requires choosing a reference, such as the ionization continuum \(E=0\). Similarly, in Franck–Hertz curves, the spacing between successive current minima reflects the lowest inelastic excitation threshold together with the mean free path and contact potential of the tube.
Visual animation · Section 2.2
Linear stopping potentials and level differences
Observing a sharp frequency threshold \(f_0=\Phi/h\) and a universal slope \(dV_s/df=h/e\) confirms that energy is transferred between the optical field of frequency \(f\) and the electrons in discrete units of \(hf\). Historically, this linear energy balance motivated the photon concept, yet semiclassical perturbation theory—treating the electromagnetic wave classically while quantizing the electronic states of the detector—also reproduces the resonant transition rate proportional to intensity and the energy conservation condition \(K_{\max}=hf-\Phi\). Definitive proof that the electromagnetic field itself is quantized requires correlation measurements such as the single-photon antibunching tests in Chapter 6.
Extracting \(h/e\) and \(\Phi\) in the laboratory requires subtracting leakage and reverse photoemission from the collector, measuring \(V_s\) across multiple monochromatic wavelengths, and fitting the linear slope \(V_s(f)\) with propagated error bars rather than computing \(h\) from a single frequency setting. In optical spectroscopy, Doppler and instrumental resolution widths must likewise be deconvolved before relating a spectral linewidth \(\Delta f\) to a radiative lifetime.
In our interactive lab, the work function is fixed at \(\Phi=2\text{ eV}\) while the slider sweeps the photon energy \(hf\) from \(0.5\text{ eV}\) to \(5.0\text{ eV}\). The model intentionally omits two-photon absorption under intense pulsed lasers, thermal tailing above the Fermi level, and space-charge saturation so that the elementary piecewise-linear cutoff \(K_{\max}=\max(0,hf-\Phi)\) can be examined directly.
Visual animation · Section 2.3
Energy quantization versus field quantization
Sweep the photon energy \(hf\) across the threshold \(\Phi=2\text{ eV}\). Verify that \(K_{\max}=0\) for \(hf<2\text{ eV}\), and predict the stopping potential \(V_s\) at \(hf=2.5\text{ eV}\) and \(hf=4.0\text{ eV}\) before inspecting the readout.
Interactive model · Idealized predictions
On a small screen, swipe the diagram to inspect its components. Moving marks illustrate the model; they are not measured trajectories.
Two atomic emission lines have photon energies \(hf_{1}=2.0\text{ eV}\) and \(hf_{2}=3.0\text{ eV}\) and terminate on the exact same lower level \(E_g\). What is the energy separation between the two excited upper levels \(E_1\) and \(E_2\)?
From \(E_1-E_g=2.0\text{ eV}\) and \(E_2-E_g=3.0\text{ eV}\), subtracting the two equations cancels the common lower level \(E_g\) and gives \(E_2-E_1=3.0\text{ eV}-2.0\text{ eV}=1.0\text{ eV}\). Note that these two lines determine the relative spacing \(E_2-E_1\) without fixing the absolute energy \(E_g\) relative to the ionization threshold.
For a metal with work function \(\Phi=2.0\text{ eV}\) illuminated by light of photon energy \(hf=3.5\text{ eV}\), find the maximum kinetic energy \(K_{\max}\) and the stopping potential \(V_s\).
Use \(K_{\max}=hf-\Phi\) and \(V_s=K_{\max}/e\).
Subtracting the work function gives \(K_{\max}=3.5\text{ eV}-2.0\text{ eV}=1.5\text{ eV}\), which corresponds to a stopping potential of \(V_s=1.5\text{ V}\).
In the unsaturated single-photon regime, does doubling the incident light intensity at fixed frequency \(f\) double the stopping voltage \(V_s\)?
Distinguish the energy per absorbed quantum \(hf\) from the number of quanta arriving per second.
No. Doubling the intensity doubles the photon flux and therefore doubles the saturated photocurrent, but each single-photon absorption still transfers energy \(hf\), leaving \(K_{\max}=hf-\Phi\) and \(V_s\) unchanged.
Why does measuring a set of optical transition frequencies \(\{f_{ij}\}\) not fix the absolute energies \(\{E_i\}\) without an additional convention?
Consider shifting every energy level by \(E_i\to E_i+C\).
Because transition frequencies depend only on differences \(hf_{ij}=|E_i-E_j|\), adding a uniform constant \(C\) to every level cancels out of every observable line frequency.
Photoelectric stopping potentials and atomic spectra measure quantized energy differences \(hf\). Always separate the per-event energy transfer \(hf-\Phi\) from the flux-dependent count rate and from higher-order photon-correlation tests.
These readings provide historical context, experimental implementations, or deeper derivations. The interactive display above is an educational model, not a reproduction of their measured data.