Wave–particle duality - Physics - HKDSE

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Wave-Particle Duality: The Weird and Wonderful World of Quantum Physics

Hello! Welcome to one of the most mind-bending but fascinating topics in Physics. For centuries, scientists thought of the world in two simple categories: particles (like tiny billiard balls) and waves (like ripples in water). Particles have mass and a specific location. Waves spread out, carry energy, and do things like diffract and interfere. Simple, right? Well, when we zoom into the atomic world, things get weird. It turns out that tiny things like light and electrons can't be put into just one box. They can act like a particle one moment and a wave the next! This strange but true idea is called wave-particle duality.

Don't worry if this sounds confusing at first! We'll break it down step-by-step. By the end of these notes, you'll understand the key evidence that turned physics upside down and how to handle the calculations.

Part 1: Light Acting as a Particle (The Photon)

We've already learned that light behaves like a wave. It shows diffraction (bending around corners) and interference (creating bright and dark fringes), which are classic wave behaviours. For a long time, everyone was happy with the wave model of light. But then, an experiment called the photoelectric effect came along and broke all the rules.

What is the Photoelectric Effect?

The setup is simple: you shine light onto a clean metal surface, and if the light is right, electrons get knocked out of the metal. These ejected electrons are called photoelectrons.

Here's what scientists observed, which the wave theory couldn't explain:

Observation 1: Instant Emission
Electrons are ejected the very instant the light hits the metal. There is no time delay, even for very dim light.
Wave Theory's Problem: The wave theory predicted that for dim light, it should take time for the electrons to absorb enough energy to escape, like a bucket slowly filling with water. This wasn't happening!

Observation 2: The Threshold Frequency
For each metal, there is a certain minimum frequency of light, called the threshold frequency (f₀), needed to eject electrons. If the light's frequency is below f₀, no electrons are ejected, no matter how bright (intense) the light is!
Wave Theory's Problem: The wave theory said that any frequency of light should work, as long as it's bright enough. A very bright, low-frequency wave should carry lots of energy. But it didn't eject any electrons.

Observation 3: Kinetic Energy depends on Frequency
The maximum kinetic energy (speed) of the ejected electrons depends only on the frequency of the light, not its intensity (brightness). Higher frequency light gives faster electrons. Brighter light just gives MORE electrons, but not faster ones.
Wave Theory's Problem: Wave theory linked energy to amplitude (intensity), so it predicted that brighter light should produce faster electrons. This was wrong.

Einstein's Genius Solution: The Photon

In 1905, Albert Einstein proposed a revolutionary idea: Light isn't a continuous wave, but is made of tiny, discrete packets of energy called photons. Think of it like the difference between a ramp (continuous) and stairs (discrete packets).

The energy of a single photon is directly proportional to its frequency:

$$ E = hf $$

Where:
• E is the energy of one photon (in Joules, J).
• h is Planck's constant, a fundamental constant of nature ($$6.63 \times 10^{-34} \text{ J s}$$).
• f is the frequency of the light (in Hertz, Hz).

Explaining the Photoelectric Effect with Photons

Imagine the metal surface is like a vending machine, and an electron is a snack trapped inside. To get the snack out, you need to pay a certain price. This "price" is the energy needed for the electron to escape the metal, called the work function (Φ).

• An incoming photon is like a single coin. It gives ALL its energy ($$E = hf$$) to ONE electron in a single hit.

• This explains instant emission: The energy transfer is one-on-one and immediate. No waiting around!

• This explains threshold frequency: If the photon's energy ($$hf$$) is less than the work function ($$\Phi$$), the electron can't "pay the price" to escape. So, you need a minimum frequency, the threshold frequency ($$f_0$$), where the photon has just enough energy: $$hf_0 = \Phi$$.

• This explains the kinetic energy: If the photon's energy is more than the work function, the extra energy becomes the electron's kinetic energy ($$K_{max}$$). This gives us Einstein's Photoelectric Equation:

$$ hf = \Phi + K_{max} $$

Incoming Photon Energy = Energy to Escape + Max Kinetic Energy

Analogy Time!
Think of the equation like this: You use your money ($$hf$$) to buy a snack from a vending machine. The price of the snack is ($$\Phi$$). The leftover money you get back as change is your kinetic energy ($$K_{max}$$). If you don't have enough money to pay the price, you get no snack (no electron emission)! More money ($$hf$$) means more change ($$K_{max}$$).

Key Takeaway for Part 1

The photoelectric effect provides strong evidence that light, which we know can behave like a wave, can also behave like a stream of particles called photons. This was the first piece of the wave-particle duality puzzle.

Part 2: Particles Acting as Waves (Matter Waves)

So, if waves can act like particles, could particles act like waves? In 1924, a young physicist named Louis de Broglie made exactly this bold suggestion. He proposed that all matter has wave-like properties.

The de Broglie Wavelength

De Broglie came up with an equation to calculate the wavelength of any particle, now called the de Broglie wavelength (λ):

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

Where:
• λ is the de Broglie wavelength (in metres, m).
• h is Planck's constant ($$6.63 \times 10^{-34} \text{ J s}$$).
• p is the momentum of the particle (in kg m/s).

Quick Review: Momentum

Remember that momentum (p) is a measure of an object's motion. For an object with mass (m) moving at velocity (v), its momentum is:
$$ p = mv $$

Why Don't We See Everyday Objects Behaving Like Waves?

Let's calculate the de Broglie wavelength of a soccer ball (mass = 0.45 kg) kicked at 20 m/s.
First, find its momentum: $$ p = mv = (0.45 \text{ kg}) \times (20 \text{ m/s}) = 9 \text{ kg m/s} $$
Now, find its wavelength: $$ \lambda = \frac{h}{p} = \frac{6.63 \times 10^{-34}}{9} \approx 7.4 \times 10^{-35} \text{ m} $$
This wavelength is incredibly tiny! It's trillions of trillions of times smaller than an atom. To see wave effects like diffraction, the wave must pass through an opening similar in size to its wavelength. Since there are no openings this small, we never see soccer balls diffracting!

The Proof: Electron Diffraction

But what about a tiny particle, like an electron? Electrons have a very small mass ($$9.11 \times 10^{-31} \text{ kg}$$). If you accelerate one through a voltage, you can get it to a high speed. Its de Broglie wavelength turns out to be about the same size as the spacing between atoms in a crystal.

This is the key! Scientists fired a beam of electrons at a very thin piece of graphite (which has a regular, crystalline atomic structure). The gaps between the atoms acted like a diffraction grating.

The result? The electrons produced a pattern of concentric rings on a screen behind the graphite. This is a classic diffraction pattern, exactly what you'd expect from a wave! This experiment was the stunning proof that particles like electrons do indeed have a wave nature.

Did you know?

The wave nature of electrons is not just a weird theory—it's the basis for powerful electron microscopes! Because electrons can have much shorter wavelengths than visible light, electron microscopes can see things in much greater detail than regular light microscopes.

Key Takeaway for Part 2

The electron diffraction experiment provides strong evidence that particles (like electrons) can behave like waves. The wavelength of these "matter waves" is given by the de Broglie equation, $$\lambda = h/p$$.

Part 3: The Big Picture - Wave-Particle Duality

So, is light a wave or a particle? Is an electron a wave or a particle? The amazing answer is: It's both!

Wave-particle duality is the central concept of quantum mechanics. It states that every quantum object (like a photon or an electron) exhibits the properties of both particles and waves. Which property you see depends on the experiment you perform.

• If you do an experiment to measure interference (like the double-slit experiment), light and electrons will behave like waves.
• If you do an experiment to measure collisions or specific locations (like the photoelectric effect or detecting where an electron hits a screen), they will behave like particles.

It's not that they "switch" between being a wave and a particle. They just have both natures at the same time. It's one of the deepest and strangest ideas in all of science!

Summary of Evidence

Light
• Evidence as a Wave: Interference, Diffraction.
• Evidence as a Particle: Photoelectric effect.

Electrons
• Evidence as a Wave: Electron diffraction.
• Evidence as a Particle: They have mass and charge; they hit a specific point on a screen.

Part 4: Solving Problems with the de Broglie Equation

You will need to be comfortable using the de Broglie equation to solve problems. Let's walk through an example.

Example Problem

An electron (mass = $$9.11 \times 10^{-31} \text{ kg}$$) is moving at a speed of $$2.0 \times 10^6 \text{ m/s}$$. Calculate its de Broglie wavelength.

Step-by-step Solution:

1. Identify your goal.
We need to find the de Broglie wavelength (λ).

2. Write down the formula.
The formula is $$ \lambda = \frac{h}{p} $$. We also know that $$ p = mv $$.

3. Calculate the momentum (p) first.
$$ p = mv = (9.11 \times 10^{-31} \text{ kg}) \times (2.0 \times 10^6 \text{ m/s}) $$
$$ p = 1.822 \times 10^{-24} \text{ kg m/s} $$

4. Substitute the momentum into the de Broglie equation.
$$ \lambda = \frac{h}{p} = \frac{6.63 \times 10^{-34} \text{ J s}}{1.822 \times 10^{-24} \text{ kg m/s}} $$
$$ \lambda \approx 3.64 \times 10^{-10} \text{ m} $$

5. State the final answer with units.
The de Broglie wavelength of the electron is $$3.64 \times 10^{-10} \text{ m}$$.

Common Mistakes to Avoid

• Forgetting to calculate momentum: Don't just plug velocity into the equation! The formula uses momentum (p), not velocity (v). Always calculate $$p = mv$$ first.
• Unit errors: Make sure mass is in kg and velocity is in m/s to get momentum in kg m/s. The value of Planck's constant $$h$$ is in J s, which is consistent with these SI units.
• Calculator errors: Be careful when entering scientific notation into your calculator. Use the EXP or EE button correctly.

Final Summary: Key Takeaways

1. Duality is Real: At the quantum level, things like light and electrons have both wave and particle properties.
2. Light as a Particle: The photoelectric effect shows light exists in energy packets called photons ($$E = hf$$).
3. Particles as Waves: Electron diffraction shows that particles have a wave nature, with a de Broglie wavelength given by the equation below.
4. The Connecting Equation: The de Broglie equation, $$ \lambda = h/p $$, is the mathematical link between a particle's momentum (a particle property) and its wavelength (a wave property).