E.2.1 Wave–particle duality (HL only)

The Photoelectric Effect and Wave-Particle Duality

1. Imagine you’re holding a metal plate in a brightly lit room.
2. As light shines on the plate, something extraordinary happens: electrons are ejected from its surface.
3. But here’s the twist—this only occurs if the light’s frequency is above a certain threshold, no matter how intense the light is.

Why doesn’t brighter light always eject electrons?

Evidence for Light’s Particle Nature: The Photoelectric Effect

When light strikes the metal, electrons are ejected and travel to the collecting plate, completing an electric circuit and generating a measurable current.

Key Observations of the Photoelectric Effect:

• Threshold Frequency:
  • Electrons are emitted only if the light’s frequency exceeds a minimum value, called the threshold frequency ($f_{\text{c}}$).

• Frequency Determines Kinetic Energy:
  • The kinetic energy ($E_{\text{max}}$) of the emitted electrons increases with the frequency of the light but is unaffected by its intensity.
• Instantaneous Emission:
  • Electrons are emitted immediately after light strikes the surface, with no delay.
• Intensity Affects Quantity, Not Energy:
  • Increasing the light’s intensity increases the number of emitted electrons (current) but does not affect their kinetic energy.

Why Light as a Wave Fails to Explain These Observations

If light were purely a wave:

• Increasing intensity (wave amplitude) would increase the energy of emitted electrons, but this is not observed.
• Electrons would eventually absorb enough energy to escape, even at low frequencies, contradicting the threshold frequency requirement.
• A time delay would occur as electrons accumulate energy from low-intensity light, but emission is instantaneous.

Einstein’s Explanation: Light as Photons

1. Albert Einstein resolved this dilemma by proposing that light consists of discrete packets of energy called photons.
2. The energy ($E$) of a photon is proportional to the frequency ($f$) of the light:

$$E=hf$$

where $h$ is Planck’s constant ($6.63\times {10}^{-34}\text{Js}$).

• When a photon strikes the metal surface, its energy is transferred to a single electron.
• The electron uses some of this energy to overcome the attractive forces holding it in the metal.

This minimum energy is called the work function ($\varphi$) of the metal.

• Any remaining energy becomes the electron’s kinetic energy ($E_{\text{max}}$):

$$E_{\text{max}}=hf-\varphi$$

Stopping Voltage and Maximum Kinetic Energy

The maximum kinetic energy of the emitted electrons can be measured using the stopping voltage ($V_s$).

The relationship is:

$$e{V}_s=E_{\text{max}}$$

Threshold Frequency and Work Function

At this frequency, the photon’s energy equals the work function ($h{f}_{\text{c}}=\varphi$), and the electrons have zero kinetic energy:

$$f_{\text{c}}=\frac{\varphi }{h}$$

Wave-Like Behavior of Matter

In 1924, Louis de Broglie proposed that all particles exhibit wave-like properties, with a wavelength ($\lambda$) given by:

$$\lambda =\frac{h}{p}$$

where $p$ is the particle’s momentum ($p=mv$):

$$\lambda =\frac{h}{mv}$$

Reflection

The photoelectric effect and de Broglie’s hypothesis reveal a profound truth: light and matter exhibit both wave-like and particle-like properties. This duality is a cornerstone of quantum mechanics, driving advancements in solar cells, electron microscopes, and quantum computing.