Light: the Quantum Model

In short: The quantum model says light comes in packets of energy called photons, each with energy $E=hf$. It explains black body radiation and the photoelectric effect, where light frees electrons from a metal only if each photon has enough energy, however bright the light is.

Black body radiation

Every object gives off EM radiation because of its temperature. A black body is an ideal object that absorbs all radiation falling on it, and gives off a spectrum that depends only on its temperature. Stars are close to black bodies.

As a black body gets hotter:

• it gives off more radiation at every wavelength (the curve gets taller);
• the peak moves to shorter wavelengths (red hot, then yellow, then white, then blue-white).

Wien's law

$${\lambda }_{max}=\frac{b}{T}$$

• ${\lambda }_{max}$ = wavelength of peak intensity (m)
• $T$ = temperature (kelvin)
• $b=2.898\times 10^{-3}$ m K (Wien's constant)

The ultraviolet catastrophe

Classical wave theory predicted that a black body should give off more and more energy at shorter and shorter wavelengths, without limit. The curve would shoot up to infinity in the ultraviolet. This "ultraviolet catastrophe" did not match experiments, where the curve has a peak and falls back to zero.

Planck's quantum idea

In 1900, Max Planck fixed the curve by assuming that energy is given off in small packets called quanta:

$$E=hf$$

• $h=6.626\times 10^{-34}$ J s (Planck's constant)
• $f$ = frequency (Hz)

High-frequency quanta need a lot of energy, so they are rare. That makes the curve fall back down at short wavelengths, just as experiments showed. Planck himself thought this was a maths trick. Einstein showed it was real.

Photons

Einstein said light itself travels as packets of energy called photons. Each photon has energy:

$$E=hf=\frac{hc}{\lambda }$$

• Blue light has a higher frequency than red light, so each blue photon carries more energy.
• A brighter beam of the same colour has more photons, not more energy per photon.

The photoelectric effect

When light of high enough frequency shines on a metal, electrons are knocked out. These are photoelectrons.

Einstein's explanation

• One photon gives all its energy to one electron.
• The electron needs a minimum energy, the work function $\varphi$, to escape the metal.
• Whatever energy is left over becomes the electron's kinetic energy:

$$K_{max}=hf-\varphi$$

The threshold frequency is where the photon has just enough energy: $h{f}_0=\varphi$.

Measuring the energy: stopping voltage

In a photocell, a reverse voltage can be applied to push back on the photoelectrons. The stopping voltage $V_0$ is the smallest reverse voltage that stops even the fastest electrons, so the current drops to zero. Then:

$$K_{max}=q{V}_0$$

A stopping voltage of 0.8 V means $K_{max}=0.8$ eV.

Quick summary

• A black body's spectrum depends only on its temperature. Wien's law: ${\lambda }_{max}=\frac{b}{T}$.
• Classical theory failed at short wavelengths (the ultraviolet catastrophe). Planck fixed it with energy quanta $E=hf$.
• Einstein: light is made of photons, each with energy $E=hf=\frac{hc}{\lambda }$.
• Photoelectric effect: $K_{max}=hf-\varphi$. Below the threshold frequency, no electrons are freed however bright the light.
• Brighter light means more photons and more electrons, not faster electrons.
• Stopping voltage: $K_{max}=q{V}_0$.

Quiz