How Light Can Behave Like Both a Wave and a Particle

Light forms interference fringes like a wave, yet in the photoelectric effect it acts like a stream of separate packets of energy. Electrons — clearly particles — can behave like waves too. This dual nature is one of the central ideas of modern physics.

This part covers why the wave theory fails, Einstein's photoelectric equation, the properties of photons, and the de Broglie wavelength.

The wave theory spreads light energy evenly over the wavefront, so it predicts that brighter light should eject faster electrons, that any frequency should work if the light is intense enough, and that dim light should take time to eject electrons — and experiment contradicts all three.

Wave theory against experiment:

- Maximum kinetic energy — predicted to rise with intensity; actually depends only on frequency
- Threshold frequency — predicted not to exist; actually no emission below ${\nu }_0$
- Time lag — predicted for dim light; actually emission is almost instant

Worked example — the missing time lag. Suppose light of intensity $10^{-5}$ W m${}^{-2}$ falls on a metal, and one electron absorbs the energy landing on an atom-sized area of $10^{-19}$ m${}^2$:

$$P=10^{-5}\times 10^{-19}=10^{-24}\text{ W}$$

Gathering a work function of $2$ eV $=3.2\times 10^{-19}$ J would take

$$t=\frac{3.2\times 10^{-19}}{10^{-24}}=3.2\times 10^5\text{ s}\approx 89\text{ hours}$$

In practice, electrons appear the moment the light is switched on.

An everyday example. A very bright red torch cannot free electrons from zinc, while a faint ultraviolet lamp can — brightness cannot make up for low frequency.

The substance. The wave theory still explains interference and diffraction perfectly — it fails when light exchanges energy with matter.

**Light behaves as a stream of photons, each carrying energy $h\nu$ and giving all of it to a single electron, so $h\nu ={\varphi }_0+K_{max}$, or $K_{max}=e{V}_0=h(\nu -{\nu }_0)$.

How the equation explains each observation:

- Threshold** — emission needs $h\nu \ge {\varphi }_0$, so no electrons below ${\nu }_0$
- Frequency dependence — $K_{max}$ rises linearly with $\nu$
- Intensity — more photons eject more electrons, raising the current but not $K_{max}$
- No time lag — each photon delivers its energy in one go

Stopping potential graph:

$$V_0=\frac{h}{e}\nu -\frac{{\varphi }_0}{e}$$

The slope $\frac{h}{e}\approx 4.14\times 10^{-15}$ V s is the same for every metal.

Worked example. Light of wavelength $400$ nm falls on cesium (${\varphi }_0=2.14$ eV). Using $hc\approx 1240$ eV nm,

$$E=\frac{1240}{400}=3.10\text{ eV},K_{max}=3.10-2.14=0.96\text{ eV},V_0=0.96\text{ V}$$

An everyday example. Rooftop solar panels rely on a closely related idea: each photon must bring enough energy to free an electron, so light below a certain frequency produces no current.

The substance. **$K_{max}$ is only a maximum** — electrons from below the surface lose some energy on the way out and emerge slower.

**A photon is a packet of electromagnetic energy with $E=h\nu$ and momentum $p=\frac{h}{\lambda }$, travelling at $c$ in vacuum, with no charge and zero rest mass.

Properties:

- Energy** $E=h\nu =\frac{hc}{\lambda }$, depending only on frequency
- Momentum $p=\frac{h\nu }{c}=\frac{h}{\lambda }$
- Speed $c$ in vacuum, with zero rest mass
- Electrically neutral, so not deflected by electric or magnetic fields
- Intensity depends on the number of photons crossing unit area per second
- Photons can be created or absorbed, but energy and momentum are conserved in every collision

Worked example. A red laser pointer emits $1$ mW at $650$ nm:

$$E=\frac{hc}{\lambda }=\frac{6.63\times 10^{-34}\times 3\times 10^8}{650\times 10^{-9}}\approx 3.06\times 10^{-19}\text{ J}$$

$$n=\frac{10^{-3}}{3.06\times 10^{-19}}\approx 3.3\times 10^{15}{\text{ s}}^{-1},p=\frac{h}{\lambda }\approx 1.0\times 10^{-27}{\text{ kg m s}}^{-1}$$

An everyday example. Ultraviolet light can cause sunburn, but the warm glow of a room heater cannot — each ultraviolet photon carries enough energy to damage skin molecules, while infrared photons do not, however many arrive.

The substance. Brighter light means more photons, not more energetic ones.

**Every moving particle has a wave associated with it, of wavelength $\lambda =\frac{h}{p}=\frac{h}{mv}$, and for an electron accelerated from rest through a potential difference $V$ this becomes $\lambda =\frac{h}{\sqrt{2meV}}\approx \frac{1.227}{\sqrt{V}}$ nm.

Derivation for an electron.** Gaining kinetic energy $K=eV$ gives momentum $p=\sqrt{2mK}=\sqrt{2meV}$, so

$$\lambda =\frac{h}{\sqrt{2meV}}$$

Worked example 1 — an electron accelerated through $100$ V:

$$\lambda =\frac{1.227}{\sqrt{100}}\approx 0.123\text{ nm}$$

This is comparable to the spacing of atoms in a crystal, so electron beams show diffraction.

Worked example 2 — a cricket ball of mass $0.15$ kg moving at $30$ m s${}^{-1}$:

$$\lambda =\frac{6.63\times 10^{-34}}{0.15\times 30}\approx 1.5\times 10^{-34}\text{ m}$$

Far too small to detect.

An everyday example. Electron microscopes in research laboratories use electron waves far shorter than visible light, which is why they can show the detail of a virus.

The substance. **Wave behaviour shows up only when $\lambda$ is comparable to the size of an obstacle or gap** — which is why a cricket ball never diffracts through a doorway.