38-1 The Photon, the Quantum of Light

38-1 The Photon, the Quantum of Light#

Prompts

  • How is light quantized? What is a photon? What is the smallest amount of energy a light wave of frequency \(f\) can have?

  • Write the photon energy formula \(E = hf\) in terms of wavelength \(\lambda\). What is the Planck constant \(h\)?

  • Describe photon absorption and photon emission by an atom. What happens to the photon in each case?

  • A 100 W lamp emits visible light (average \(\lambda \approx 550\) nm). Estimate the rate at which photons are emitted. (Hint: \(P = (\text{rate}) \times (\text{energy per photon})\).)

  • Why did classical wave theory suffice in earlier chapters, but quantum physics becomes essential for single-photon experiments?

Lecture Notes#

Overview#

  • Quantum physics studies the microscopic world, where many quantities are quantized—they exist only in certain minimum (elementary) amounts or integer multiples thereof.

  • Light is quantized: electromagnetic radiation exists in elementary amounts called photons. The energy of a photon of frequency \(f\) is \(E = hf\).

  • Absorption and emission of light occur in discrete events: one photon transfers energy \(hf\) between light and matter.

  • The classical wave picture (Ch. 33) remains valid for describing propagation; the photon concept is essential when light interacts with matter.

The photon#

In 1905, Einstein proposed that light is quantized. The quantum of light is the photon. For light of frequency \(f\) and wavelength \(\lambda\) (with \(c = f\lambda\)), the energy of a single photon is

(349)#\[ E = hf = \frac{hc}{\lambda} \]

where \(h\) is the Planck constant:

(350)#\[ h = 6.626\times 10^{-34}\ \text{J}\cdot\text{s} \]

The smallest amount of energy a light wave of frequency \(f\) can have is \(hf\)—the energy of one photon. If the wave has more energy, it must be an integer multiple of \(hf\): \(E = nhf\) with \(n = 1, 2, 3, \ldots\). Light cannot have energy \(0.6hf\) or \(75.5hf\).

Wave vs particle

We have treated light as a wave (wavelength, frequency, interference). The photon is a particle-like quantum of that wave. Both descriptions are needed—light exhibits wave-like behavior (diffraction, interference) and particle-like behavior (discrete absorption and emission). This wave–particle duality is a central theme of quantum physics.

Absorption and emission#

When light interacts with matter, the interaction occurs in atoms (or molecules). Each event involves exactly one photon.

Photon absorption: When light of frequency \(f\) is absorbed by an atom, the energy \(hf\) of one photon is transferred from the light to the atom. The photon vanishes; the atom is said to absorb it.

Photon emission: When light of frequency \(f\) is emitted by an atom, an amount of energy \(hf\) is transferred from the atom to the light. A photon appears; the atom is said to emit it.

For an object with many atoms (e.g., sunglasses absorbing light, a lamp emitting light), there are many such absorption or emission events. Each event still involves the transfer of one photon’s worth of energy, \(hf\).

Photon rate, power, and intensity#

If a light source emits power \(P\) at a single frequency \(f\), and each photon has energy \(hf\), the rate at which photons are emitted is

(351)#\[ R = \frac{P}{hf} = \frac{P\lambda}{hc} \]

For intensity \(I\) (power per unit area), the rate of photons per unit area is \(I/(hf)\).

Example: lamp

A 100 W lamp emits visible light with average wavelength \(\lambda \approx 550\) nm. Energy per photon: \(E = hc/\lambda \approx (6.6\times 10^{-34})(3\times 10^8)/(550\times 10^{-9}) \approx 3.6\times 10^{-19}\) J. Photon rate: \(R = P/E \approx 100/(3.6\times 10^{-19}) \approx 3\times 10^{20}\) photons/s. Ordinary light involves enormous numbers of photons.

When is quantum physics needed?#

In previous chapters we used classical wave theory for reflection, refraction, interference, and diffraction. That approach works when many photons are involved—the discrete nature of light is averaged out. For single-photon experiments (e.g., sensitive detectors, quantum cryptography), the quantum picture is essential. Modern technology has made such experiments routine in optical engineering.

Summary#

  • Photon: quantum of light; energy \(E = hf = hc/\lambda\); Planck constant \(h = 6.626\times 10^{-34}\) J·s.

  • Quantization: light energy exists only in integer multiples of \(hf\).

  • Absorption: photon transfers \(hf\) to atom; photon vanishes. Emission: atom transfers \(hf\) to light; photon appears.

  • Photon rate: \(R = P/(hf)\) for power \(P\) at frequency \(f\).