Photons

1. Quantum Nature of Radiation

Classical wave theory could not explain phenomena such as black body radiation and the photoelectric effect. To resolve them, radiation is treated as being made of discrete packets of energy rather than a continuous wave.

Planck's quantum hypothesis (1900). A body emits or absorbs radiant energy not continuously but in separate packets called quanta. Each quantum of radiation of frequency $f$ carries a definite energy:

$$E = h f$$

where $h = 6.63 \times 10^{-34}\ \text{J s}$ is Planck's constant. A quantum of electromagnetic radiation is called a photon.

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2. Photons and Their Properties

A photon is the smallest discrete amount (quantum) of electromagnetic energy. Its main properties are:

1. A photon is a chargeless, massless packet of energy that travels through vacuum at the speed of light $c = 3 \times 10^8\ \text{m s}^{-1}$.
2. The energy of a photon of frequency $f$ (wavelength $\lambda$) is entirely radiant energy:

$$ \begin{aligned} E &= h f \ &= \frac{h c}{\lambda} \end{aligned} $$

3. A photon carries momentum even though its rest mass is zero. From the mass energy relation $E = mc^2$ and $E = hf$, the equivalent (dynamic) mass and momentum are:

$$ \begin{aligned} m &= \frac{E}{c^2} \ &= \frac{h f}{c^2} \ \qquad p &= m c \ &= \frac{E}{c} \ &= \frac{h f}{c} \ &= \frac{h}{\lambda} \end{aligned} $$

4. Photons are electrically neutral, so they are not deflected by electric or magnetic fields.
5. In a beam, the intensity of radiation depends on the number of photons crossing unit area per unit time; the energy of each photon depends only on its frequency.

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3. Photoelectric Effect

When electromagnetic radiation of a suitable frequency falls on a metal surface, electrons are ejected from the surface. This phenomenon is called the photoelectric effect, and the ejected electrons are called photoelectrons. It is a direct consequence of the conservation of energy for the photon-electron interaction.

Key terms:

• Threshold frequency ($f_0$): the minimum frequency of incident radiation that can just eject photoelectrons from a given metal. Below $f_0$ no emission occurs, however intense the light.
• Threshold wavelength ($\lambda_0$): the wavelength corresponding to the threshold frequency, $\lambda_0 = \dfrac{c}{f_0}$. It is the maximum wavelength that can cause emission.
• Work function ($\phi$ or $W_0$): the minimum energy required to just liberate an electron from the metal surface:

$$ \begin{aligned} \phi &= h f_0 \ &= \frac{h c}{\lambda_0} \end{aligned} $$

Laws of the photoelectric effect:

1. Emission occurs only when $f \ge f_0$ (there is a threshold frequency for each metal).
2. The number of photoelectrons emitted per second is proportional to the intensity of the incident light.
3. The maximum kinetic energy of the photoelectrons depends only on the frequency, not the intensity.
4. The emission is instantaneous (no measurable time lag).

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