Einstein's Postulate of Quantization of Energy and the Photoelectric Equation

Building directly on Planck's idea that energy comes in discrete packets, Einstein proposed in 1905 that under certain conditions light itself behaves as if made of localized PARTICLES of energy, which he called PHOTONS. Each photon of light of frequency $\nu$ carries a fixed energy

where h is Planck's constant, with the now precisely known value $h = 6.626\times10^{-34}$ J s. Equation (14.1), called EINSTEIN'S RELATION, is remarkable for linking a PARTICLE-like property (the photon's energy E) to a WAVE-like property (the radiation's frequency $\nu$); since $\nu=c/\lambda$, it equally tells us that shorter-wavelength (higher-frequency) radiation carries more energetic photons -- so ultraviolet photons are more energetic than red-light photons, which in turn are far more energetic than radio-wave photons (see Example 14.2).

Einstein used this photon picture to explain every one of the photoelectric observations of section 14.2.2, point by point:

1. When a photon strikes the metal surface and collides with a free electron, it transfers ALL of its energy $h\nu$ to that one electron in a single event, and ceases to exist. The electron escapes if and only if the energy it gains is at least equal to the work function: $h\nu \geq \phi_0$. This immediately explains the THRESHOLD FREQUENCY: $\nu_0=\phi_0/h$ is the minimum frequency at which a single photon carries enough energy to free an electron at all. Below $\nu_0$ no single photon -- however many arrive -- can do the job; above $\nu_0$, even one photon suffices, however weak the overall beam.

2. Since the whole energy exchange happens in one instantaneous collision (not a gradual accumulation), there is no time lag: emission starts the instant light strikes the surface, and stops the instant it is switched off -- there being no photons left to transfer energy.

3. Increasing the intensity of light at a fixed wavelength increases the NUMBER of photons arriving per unit area per unit time, while the energy of EACH photon ($h\nu$) stays exactly the same. More photons means more electrons knocked out per second, hence a larger photocurrent -- exactly the linear intensity dependence of observation 3, correctly explained without any change in each electron's individual energy.

4. Once free, a photoelectron is accelerated towards the collector if C is at a higher potential; a larger accelerating potential draws in more of the emitted electrons, until ALL of them are being collected -- beyond that point, raising V further cannot increase the current, which explains the saturation current $I_0$ of observation 4. Since intensity fixes the total number of photons (and hence the maximum possible number of ejected electrons) per second, saturation current scales proportionately with intensity, exactly as in observation 5.

5. Because all electrons in the metal do not have identical starting energies, and some lose energy through collisions on their way out, the ejected electrons have a SPREAD of kinetic energies, not a single value. The MAXIMUM kinetic energy among them is what remains after the minimum energy $\phi_0$ needed to escape is subtracted from the absorbed photon energy $h\nu$:

This is EINSTEIN'S PHOTOELECTRIC EQUATION, arguably the single most important result of the chapter. It depends on the emitter material (through $\phi_0$) and on the incident frequency, but has NO dependence on intensity at all -- exactly matching observation 6.

6. When the collector is held at a RETARDING (negative) potential V, an electron loses kinetic energy $eV$ overcoming the opposing field; the most energetic electron (with $KE_{max}$ at the emitter surface) just fails to reach the collector when $KE_{max}=eV_0$, defining the stopping potential $V_0$. Substituting this into Eq. (14.2) gives

This is exactly the straight-line relationship of Fig. 14.6: $V_0$ varies linearly with $\nu$, with slope $h/e$ -- a UNIVERSAL constant, the same for every metal, since it depends only on h and e -- while the intercept depends on the material through its work function $\phi_0$. This single equation explains both why all the lines in Fig. 14.6 are PARALLEL (same slope h/e) and why they are shifted horizontally from one metal to another (different $\phi_0$, hence different threshold frequency $\nu_0=\phi_0/h$). …