Grade 12

Advanced

6 min

Lesson 6 of 12

Wave-Particle Duality and De Broglie

Not Started

Exploring the dual nature of matter and radiation, where particles exhibit wave-like properties.

If light—the ultimate wave—can behave like a stream of particles, is it possible that you, your phone, and every atom in the universe are actually vibrating waves?

Learning Objectives

1.Calculate the de Broglie wavelength for particles ranging from electrons to everyday objects
2.Explain why wave-like behavior is observable in subatomic particles but invisible in the macroscopic world
3.Analyze the Davisson-Germer experiment as the definitive evidence for electron diffraction

The Radical Hypothesis

In 1924, Louis de Broglie made a bold suggestion: if radiation has a dual nature, matter must too. He proposed that any moving particle with momentum

p

has an associated wavelength

\lambda

. This is defined by the de Broglie relation:

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

where

h

is Planck’s constant (

6.626 \times 10^{-34} \text{ J}\cdot\text{s}

). This equation bridges the gap between the particle world (defined by mass

m

and velocity

v

) and the wave world (defined by wavelength

\lambda

). It suggests that 'matter waves' are not just metaphors, but physical realities. However, because

h

is so incredibly small, the wavelength only becomes significant when the momentum

p

is also extremely small—typical of subatomic particles.

Calculate the de Broglie wavelength of an electron ( $m = 9.11 \times 10^{-31} \text{ kg}$ ) moving at $1.0 \times 10^6 \text{ m/s}$ .

1. Identify knowns: $m = 9.11 \times 10^{-31} \text{ kg}$ , $v = 1.0 \times 10^6 \text{ m/s}$ , $h = 6.626 \times 10^{-34} \text{ J}\cdot\text{s}$ . 2. Use the formula: $\lambda = \frac{h}{mv}$ . 3. Substitute values: $\lambda = \frac{6.626 \times 10^{-34}}{(9.11 \times 10^{-31})(1.0 \times 10^6)}$ . 4. Calculate: $\lambda \approx 7.27 \times 10^{-10} \text{ m}$ (or $0.727 \text{ nm}$ ).

Quick Check

If a particle's velocity doubles while its mass stays the same, what happens to its de Broglie wavelength?

Look at the denominator of the de Broglie equation.

Answer

The wavelength is halved, because wavelength is inversely proportional to velocity ( $v$ ).

The Macro-Micro Divide

Why don't we see a car diffracting as it passes through a tunnel? The answer lies in the scale of $h$ . For a $1500 \text{ kg}$ car moving at $20 \text{ m/s}$ , the momentum is $30,000 \text{ kg}\cdot\text{m/s}$ . Plugging this into $\lambda = h/p$ gives a wavelength of roughly $10^{-38} \text{ m}$ . This is billions of times smaller than a single atomic nucleus! For a wave to show noticeable effects like diffraction or interference, it must encounter an obstacle or opening roughly the same size as its wavelength. Since no physical 'slits' exist at the $10^{-38} \text{ m}$ scale, the wave nature of macroscopic objects remains completely hidden from our senses.

Compare the wavelength of a $0.145 \text{ kg}$ baseball thrown at $40 \text{ m/s}$ to the size of an atom (approx. $10^{-10} \text{ m}$ ).

1. Calculate momentum: $p = mv = (0.145)(40) = 5.8 \text{ kg}\cdot\text{m/s}$ . 2. Calculate wavelength: $\lambda = \frac{6.626 \times 10^{-34}}{5.8} \approx 1.14 \times 10^{-34} \text{ m}$ . 3. Conclusion: The baseball's wavelength is $10^{24}$ times smaller than an atom, making its wave properties undetectable by any current technology.

Quick Check

Why is a crystal lattice used to observe electron diffraction instead of a standard laboratory slit?

Think about the scale required for diffraction to occur.

Answer

Because the spacing between atoms in a crystal is small enough (on the order of $10^{-10} \text{ m}$ ) to match the tiny de Broglie wavelength of electrons.

The Proof: Davisson-Germer Experiment

In 1927, Davisson and Germer provided the 'smoking gun' for de Broglie’s theory. They fired a beam of slow-moving electrons at a crystalline nickel target. Crystals act as natural diffraction gratings. They observed that electrons didn't just bounce off like tiny marbles; they formed a specific interference pattern—a phenomenon exclusive to waves. By measuring the angles of the diffracted beam, they calculated a wavelength that matched de Broglie’s prediction perfectly. This proved that matter waves are not just theoretical constructs but a fundamental part of the universe's architecture.

An electron is accelerated from rest through a potential difference of $V = 54 \text{ V}$ . Find its de Broglie wavelength.

1. Kinetic Energy $K = eV = (1.6 \times 10^{-19} \text{ C})(54 \text{ V}) = 8.64 \times 10^{-18} \text{ J}$ . 2. Relate $K$ to momentum $p$ : $K = \frac{p^2}{2m} \implies p = \sqrt{2mK}$ . 3. Calculate $p$ : $p = \sqrt{2(9.11 \times 10^{-31})(8.64 \times 10^{-18})} \approx 3.97 \times 10^{-24} \text{ kg}\cdot\text{m/s}$ . 4. Calculate $\lambda$ : $\lambda = \frac{h}{p} = \frac{6.626 \times 10^{-34}}{3.97 \times 10^{-24}} \approx 1.67 \times 10^{-10} \text{ m}$ .

Key Takeaways

1Wave-particle duality applies to all matter, not just light.
2The de Broglie wavelength is calculated using $\lambda = \frac{h}{p}$ .
3Macroscopic objects have wavelengths too small to produce observable diffraction.
4The Davisson-Germer experiment confirmed the wave nature of electrons using nickel crystals.

Quick Check

Multiple Choice

Which of the following would have the longest de Broglie wavelength if all are moving at the same speed?

Multiple Choice

What was the primary result of the Davisson-Germer experiment?

True/False

True or False: A stationary particle has a de Broglie wavelength of zero.

What's Next?

Review Tomorrow

In 24 hours, try to write down the de Broglie formula and explain why you can't see your own wave nature while walking.

Practice Activity

Calculate the wavelength of a 60kg human running at 5 m/s and compare it to the width of a typical doorway.

Browse

Browse

Wave-Particle Duality and De Broglie

Learning Objectives

The Radical Hypothesis

The Macro-Micro Divide

The Proof: Davisson-Germer Experiment

Key Takeaways

Quick Check

What's Next?

Wave-Particle Duality and De Broglie

Learning Objectives

The Radical Hypothesis

The Macro-Micro Divide

The Proof: Davisson-Germer Experiment

Key Takeaways

Quick Check

What's Next?