Einstein framed three postulates about light quanta and their interaction with matter:

1. Light of frequency $f$ consists of discrete quanta, each of energy $E=hf$. Each photon travels at the speed of light $c$.

2. Light quanta are emitted or absorbed on an all-or-nothing basis. A substance can emit 1 or 2 or 3 quanta, but not 1.5. Similarly, an electron in a metal must absorb an entire quantum of light at once; it cannot absorb half a quantum.

3. A light quantum, when absorbed by a metal, delivers its entire energy to one electron.

Let’s look at how Einstein’s postulates apply to the photoelectric effect. If Einstein is correct, the light of frequency $f$ shining on the metal is a flow of light quanta, each of energy $hf$. Each quantum is absorbed by one electron, giving that electron an energy $E_{\text{elec}}=hf$. This leads us to several interesting conclusions:

1. An electron that has just absorbed a quantum of light energy has $E_{\text{elec}}=hf$. (The electron’s thermal energy at room temperature is so much less than $hf$ that we can neglect it.) FIGURE 38.7 shows that this electron can escape from the metal, becoming a photoelectron, if

   $$E_{\text{elec}}=hf={\displaystyle \frac{hc}{\lambda }}\ge E_0$$ (38.5)

   where, you will recall, the work function $E_0$ is the minimum energy needed to free an electron from the metal. As a result, there is a threshold frequency

   $$f_0={\displaystyle \frac{E_0}{h}}$$ (38.6)

   for the ejection of photoelectrons. If $f$ is less than $f_0$, even by just a small amount, none of the electrons will have sufficient energy to escape no matter how intense the light. But even very weak light with $f\ge f_0$ will give a few electrons sufficient energy to escape because each light quantum delivers all of its energy to one electron. This threshold behavior is exactly what Lenard observed.

   FIGURE 38.7 The creation of a photoelectron.

   

   FIGURE 38.7 Full Alternative Text

2. A more intense light means more quanta of the same energy, not more energetic quanta. These quanta eject a larger number of photoelectrons and cause a larger current, exactly as observed.

3. There is a distribution of kinetic energies, because different photoelectrons require different amounts of energy to escape, but the maximum kinetic energy is

   $$K_{\text{max}}=E_{\text{elec}}-E_0=hf-E_0$$ (38.7)

   As we noted in Equation 38.3, the stopping potential $V_{\text{stop}}$ is directly proportional to $K_{\text{max}}$. Einstein’s theory predicts that the stopping potential is related to the light frequency by

   $$V_{\text{stop}}={\displaystyle \frac{K_{\text{max}}}{e}}={\displaystyle \frac{hf-E_0}{e}}$$ (38.8)

   The stopping potential does not depend on the intensity of the light. Both weak light and intense light will have the same stopping potential, which Lenard had observed but which could not previously be explained.

4. If each light quantum transfers its energy $hf$ to just one electron, that electron immediately has enough energy to escape. The current should begin instantly, with no delay, exactly as Lenard had observed.

Using the swimming pool analogy again, FIGURE 38.8 shows a pebble being thrown into the pool. The pebble increases the energy of the water, but the increase is shared among all the molecules in the pool. The increase in the water’s energy is barely enough to make ripples, not nearly enough to splash water out of the pool. But suppose all the pebble’s energy could go to one drop of water that didn’t have to share it. That one drop of water could easily have enough energy to leap out of the pool. Einstein’s hypothesis that a light quantum transfers all its energy to one electron is equivalent to the pebble transferring all its energy to one drop of water.

FIGURE 38.8 A pebble transfers energy to the water.