Wave-particle duality - Physics (9702) - Cambridge A-Level

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Welcome to one of the most exciting and mind-bending topics in A-Level Physics: Wave-Particle Duality! This chapter marks the transition from classical physics (Newton, Maxwell) into the mysterious world of quantum mechanics.

Classical physics told us that things are either waves (like light and sound) or particles (like electrons and dust). Quantum physics says: Why not both?

You will learn that light sometimes acts like a particle, and particles (like electrons) sometimes act like a wave. This revolutionary idea explains phenomena that classical physics simply couldn't touch. Don't worry if it feels counter-intuitive—it felt the same way to the physicists who discovered it!

Section 1: The Particle Nature of Electromagnetic Radiation (Photons)

1.1 Understanding the Photon (Quantum of Energy)

In the early 20th century, physicists were struggling to explain how energy was emitted and absorbed by matter. Max Planck suggested that energy wasn't continuous (like pouring water) but came in discrete packets, or "quanta."

The photon is the quantum (packet) of electromagnetic energy. It is the particle equivalent of light.
Photons travel at the speed of light, $c$.
Photons have zero rest mass.

1.2 Energy of a Photon

The energy $E$ carried by a single photon is directly proportional to the frequency $f$ of the electromagnetic radiation.

Key Formula: Photon Energy
$$E = hf$$ Where:

$E$ is the energy of the photon (J)
$h$ is the Planck constant ($6.63 \times 10^{-34}\text{ J s}$)
$f$ is the frequency of the radiation (Hz)

Since $c = f\lambda$, we can also express the photon energy in terms of wavelength ($\lambda$):
$$E = \frac{hc}{\lambda}$$

Analogy: Imagine a vending machine for light. You can’t put in 1.5 units of currency; you must use whole coins (quanta). Higher frequency light means each coin is worth more energy.

Did you know?

Because the Planck constant $h$ is so tiny ($10^{-34}$), we only notice the quantization of energy when dealing with microscopic phenomena, like individual electrons or atoms. In everyday life, energy transfer appears continuous.

1.3 Using the Electronvolt (eV)

Energy calculations involving single photons and electrons often result in extremely small values when measured in Joules (J). Therefore, we often use a more convenient unit: the electronvolt (eV).

Definition: One electronvolt (1 eV) is the kinetic energy gained by a single electron when it is accelerated through a potential difference of 1 volt.
Conversion: Since $E = QV$, and the charge $Q$ of an electron is $e = 1.60 \times 10^{-19}\text{ C}$:
$$1\text{ eV} = 1.60 \times 10^{-19}\text{ J}$$

Memory Aid: If you need to convert from eV to J, multiply by the charge of the electron, $e$.

1.4 Momentum of a Photon

Particles have momentum ($p = mv$). If light is made of particles (photons), they must also have momentum, even though their rest mass is zero.

Key Formula: Photon Momentum
$$p = \frac{E}{c}$$ Where:

$p$ is the momentum of the photon ($\text{kg m s}^{-1}$ or $\text{N s}$)
$E$ is the photon energy (J)
$c$ is the speed of light ($3.00 \times 10^8\text{ m s}^{-1}$)

By substituting $E = hf$ and using $c = f\lambda$, we get an alternative, useful formula:
$$p = \frac{h}{\lambda}$$

Key Takeaway for Section 1: Photons are energy packets. Their energy is linked to frequency ($E=hf$) and they carry momentum ($p=E/c$).

Section 2: Evidence for the Particle Nature of Light (Photoelectric Effect)

The photoelectric effect is the definitive experiment that proves light behaves like a particle.

2.1 What is the Photoelectric Effect?

Definition: The emission of electrons (called photoelectrons) from a metal surface when it is illuminated by electromagnetic radiation (typically UV or visible light).

2.2 Classical Wave Theory vs. Experimental Observations

If light were purely a wave (as classical theory suggested), its energy would be spread out continuously across the wave front. This leads to predictions that contradict what is observed:

Observation	Classical Wave Prediction	Actual Observation
Effect of Intensity	Higher intensity should give electrons more energy, leading to higher kinetic energy.	Higher intensity only increases the number (current) of photoelectrons, but not their maximum kinetic energy.
Effect of Frequency	Any frequency should eventually cause emission if the intensity is high enough (energy builds up over time).	Emission only occurs if the light frequency is above a certain minimum value (the threshold frequency, $f_0$).
Time Delay	Low intensity light should require a time delay for enough wave energy to accumulate to eject an electron.	Emission is instantaneous, even at very low intensities, provided $f \ge f_0$.

2.3 Einstein's Photon Explanation

In 1905, Albert Einstein explained the effect by using Planck’s idea of quantization (photons).

Light consists of discrete photons, each carrying energy $E = hf$.
One photon interacts with one electron. It is an "all-or-nothing" collision.
The energy of the photon, $hf$, is used for two purposes:

A. Work Function ($\Phi$)

Definition: The minimum energy required to remove an electron from the surface of the metal.
This is the "entrance fee" the electron must pay to escape the metal's attractive forces.

B. Kinetic Energy of the Electron ($KE_{\text{max}}$)

Any remaining energy from the photon is converted into the electron's maximum kinetic energy.

2.4 The Photoelectric Equation

The conservation of energy in this process gives us the photoelectric equation (Einstein's formula):

Key Formula: Photoelectric Effect
$$hf = \Phi + \frac{1}{2}mv_{\text{max}}^2$$ Where:

$hf$ is the incident photon energy.
$\Phi$ (Phi) is the Work Function of the metal (J or eV).
$\frac{1}{2}mv_{\text{max}}^2$ is the maximum kinetic energy ($KE_{\text{max}}$) of the emitted photoelectron (J or eV).

Threshold Frequency ($f_0$):

If the photoelectron just escapes (meaning its kinetic energy is zero, $KE_{\text{max}} = 0$), the photon energy equals the work function:
$$hf_0 = \Phi$$ Therefore, the threshold frequency $f_0 = \Phi/h$ is the minimum frequency of light required for emission to occur.

Similarly, the threshold wavelength ($\lambda_0$) is the maximum wavelength that will cause emission:

$$\lambda_0 = \frac{c}{f_0} = \frac{hc}{\Phi}$$

2.5 Explaining the Observations (Critical Thinking)

Frequency vs. KE: If the frequency $f$ is increased, the photon energy $hf$ increases. Since $\Phi$ is constant for a given metal, the excess energy goes directly into $KE_{\text{max}}$. Thus, maximum kinetic energy depends only on frequency.
Intensity vs. Current: Increasing the intensity of light means sending more photons per second. More photons mean more collisions with electrons, resulting in a higher rate of electron emission (i.e., a larger photoelectric current). Intensity does NOT affect the energy of individual photons.
Instantaneous Emission: Because the energy is concentrated in a particle (photon), the electron gets all the required energy ($hf$) instantly in a single collision, meaning there is no time delay.

Quick Review Box

Light behaves as WAVES when: Interference, Diffraction, Polarization.
Light behaves as PARTICLES (photons) when: Photoelectric effect.

Common Mistake: Students often think higher intensity means higher KE. Remember: Intensity = Quantity of photons. Frequency = Quality (Energy) of photons.

Section 3: Wave-Particle Duality of Matter (De Broglie Wavelength)

3.1 Matter Waves

We established that light (which we classically thought of as a wave) can act as a particle. In 1924, Louis de Broglie hypothesized the opposite: if waves can act as particles, then particles (like electrons, protons, and baseballs) should also be able to act as waves.

This revolutionary concept states that every moving particle has a wave associated with it—a matter wave.

3.2 The De Broglie Wavelength ($\lambda$)

De Broglie connected the wave properties (wavelength $\lambda$) to the particle properties (momentum $p$) using the Planck constant, $h$. He used the photon momentum equation and applied it to matter.

Recall the photon momentum: $p = h/\lambda$. Rearranging this gives the de Broglie wavelength:

Key Formula: De Broglie Wavelength
$$\lambda = \frac{h}{p}$$ Where:

$\lambda$ is the de Broglie wavelength (m)
$h$ is the Planck constant ($6.63 \times 10^{-34}\text{ J s}$)
$p$ is the momentum of the particle (p = mv) ($\text{kg m s}^{-1}$)

This means any moving particle with momentum $p$ has an associated wavelength $\lambda$.

Why don't we see the wave nature of everyday objects?

Let's consider two examples:

A baseball (m = 0.15 kg, v = 30 m/s):
$p = 4.5\text{ kg m s}^{-1}$
$\lambda = (6.63 \times 10^{-34}) / 4.5 \approx 10^{-34}\text{ m}$
This wavelength is impossibly small to measure or observe.

An electron (m = $9.11 \times 10^{-31}$ kg, v = $1.0 \times 10^7$ m/s):
$p = 9.11 \times 10^{-24}\text{ kg m s}^{-1}$
$\lambda = (6.63 \times 10^{-34}) / (9.11 \times 10^{-24}) \approx 7.3 \times 10^{-11}\text{ m}$
This wavelength is in the size range of X-rays and atom spacing, which is detectable!

Because $h$ is so small, only particles with very small mass or very small velocity (and thus very small momentum) have a detectable wavelength. This explains why we only observe matter duality in the quantum realm.

3.3 Evidence for the Wave Nature of Particles (Electron Diffraction)

How do we prove that electrons, which we usually treat as particles, can act as waves?

We use the signature experiment for waves: Diffraction.

A beam of electrons is accelerated through a potential difference to give them a specific kinetic energy (and thus a specific momentum $p$ and de Broglie wavelength $\lambda$).
This beam is directed at a thin crystalline material, often graphite. The spacing between the atoms in the crystal lattice acts like a natural diffraction grating.
If the electrons were purely particles, they would simply pass through and create a central spot.
Observation: A pattern of concentric bright and dark rings is observed on a screen placed behind the crystal.

Interpretation:

The appearance of an interference/diffraction pattern is conclusive evidence that the electrons were behaving as waves.
The observed diffraction pattern matches the pattern predicted using the de Broglie wavelength $\lambda = h/p$ and the known spacing of the crystal lattice.

Did you know? This experiment confirmed de Broglie’s hypothesis and fundamentally changed our understanding of reality, paving the way for technologies like the Electron Microscope.

3.4 Summary of Wave-Particle Duality (Syllabus 22.3)

The concept of duality is a central pillar of quantum physics:

Electromagnetic Radiation (Light):
- Exhibits Wave Nature via: Interference, Diffraction, Refraction, Polarization.
- Exhibits Particle Nature via: Photoelectric effect.
Matter (Electrons/Protons):
- Exhibits Particle Nature via: Collisions, momentum/kinetic energy, electric/magnetic deflection.
- Exhibits Wave Nature via: Electron diffraction (de Broglie waves).

The "true" nature of an entity is neither just wave nor just particle; it is something more fundamental that expresses itself as one or the other depending on the type of experiment being performed.

Key Takeaway for Section 3: All matter has an associated wave nature, quantified by the de Broglie wavelength ($\lambda = h/p$). This is experimentally proven by observing electron diffraction.

Final Quick Review: Key Concepts and Formulas

Photon Energy: $E = hf$ or $E = hc/\lambda$
Photon Momentum: $p = E/c$ or $p = h/\lambda$
Photoelectric Effect: $hf = \Phi + \frac{1}{2}mv_{\text{max}}^2$
De Broglie Wavelength: $\lambda = h/p$

The concepts in this chapter require you to be comfortable switching between wave language (frequency, wavelength) and particle language (momentum, kinetic energy) using the Planck constant, $h$, as the bridge.