Matter Waves

De Broglie proposed that electrons travel through space as a wave
- This would explain why they can exhibit behaviour such as diffraction

He therefore suggested that electrons must also hold wave properties, such as wavelength
- This came to be known as the de Broglie wavelength
However, he realised all particles can show wave-like properties, not just electrons
- He hypothesised that all moving particles have a matter wave associated with them

This is known as the de Broglie wavelength, and can be defined as:

The wavelength associated with a moving particle

The majority of the time, and for everyday objects travelling at normal speeds, the de Broglie wavelength is far too small for any quantum effects to be observed
- A typical electron in a metal has a de Broglie wavelength of about 10 nm
Therefore, quantum mechanical effects will only be observable when the width of the sample is around that value

$Electron Diffraction Experiment$

The electron diffraction tube can be used to investigate how the wavelength of electrons depends on their speed
- The smaller the radius of the rings, the smaller the de Broglie wavelength of the electrons
As the voltage is increased:
- The energy of the electrons increases
- The radius of the diffraction pattern decreases
This shows as the speed of the electrons increases, the de Broglie wavelength of the electrons decreases

Using ideas based upon the quantum theory and Einstein’s theory of relativity, de Broglie suggested that the momentum (p) of a particle and its associated wavelength (λ) are related by the equation:

$λ = \frac{h}{p}$

Since momentum p = mv, the de Broglie wavelength can be related to the speed of a moving particle (v) by the equation:

$λ = \frac{h}{m v}$

Kinetic Energy

Since kinetic energy E = ½ mv²
Momentum and kinetic energy can be related by:

$E = \frac{p^{2}}{2 m} \to p = \sqrt{2 m E}$

Combining this with the de Broglie equation gives a form which relates the de Broglie wavelength of a particle to its kinetic energy:

$λ = \frac{h}{\sqrt{2 m E}}$

Where:
- λ = the de Broglie wavelength (m)
- h = Planck’s constant (J s)
- p = momentum of the particle (kg m s^-1)
- E = kinetic energy of the particle (J)
- m = mass of the particle (kg)
- v = speed of the particle (m s^-1)

Worked example

A proton and an electron are each accelerated from rest through the same potential difference.

Determine the ratio: $\frac{de Broglie wavelength of the proton}{de Broglie wavelength of the electron}$

Mass of a proton = 1.67 × 10^–27 kg
Mass of an electron = 9.11 × 10^–31 kg

2.5.4 De Broglie Wavelength Worked Example

Matter Waves (DP IB Physics: HL)

Revision Note

Kinetic Energy

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1. Measurement & Uncertainties

1.1 Measurements in Physics

1.1.1 SI Units

1.1.2 Using Scientific Notation

1.1.3 Estimating Physical Quantities

1.2 Uncertainties & Errors

1.2.1 Random & Systematic Errors

1.2.2 Calculating Uncertainties

1.2.3 Determining Uncertainties from Graphs

1.3 Vectors & Scalars

1.3.1 Vector & Scalar Quantities

1.3.2 Combining & Resolving Vectors

1.3.3 Solving Vector Problems

2. Mechanics

2.1 Motion

2.1.1 Distance & Displacement

2.1.2 Speed & Velocity

2.1.3 Acceleration

2.1.4 Graphs Describing Motion

2.1.5 Uniform Acceleration

2.1.6 Acceleration of Free Fall Experiment

2.1.7 Projectile Motion

2.1.8 Terminal Speed

2.2 Forces

2.2.1 Free-Body Diagrams

2.2.2 Newton’s First Law

2.2.3 Newton’s Second Law

2.2.4 Newton’s Third Law

2.2.5 Applying Newton’s Laws of Motion

2.2.6 Friction

2.3 Work, Energy & Power

2.3.1 Kinetic Energy

2.3.2 Gravitational Potential Energy

2.3.3 Elastic Potential Energy

2.3.4 Work Done

2.3.5 Power

2.3.6 Principle of Conservation of Energy

2.3.7 Efficiency

2.4 Momentum & Impulse

2.4.1 Force & Momentum

2.4.2 Impulse

2.4.3 Conservation of Linear Momentum

2.4.4 Collisions & Explosions

3. Thermal Physics

3.1 Thermal Concepts

3.1.1 Solids, Liquids & Gases

3.1.2 Temperature

3.1.3 Internal Energy

3.1.4 Specific Heat Capacity

3.1.5 Specific Latent Heat

3.1.6 Phase Change

3.1.7 Investigating Thermal Energy

3.2 Modelling a Gas

3.2.1 Ideal Gas Laws

3.2.2 Ideal Gas Equation

3.2.3 Kinetic Model of an Ideal Gas

3.2.4 Mole Calculations

3.2.5 Investigating Gas Laws

4. Waves

4.1 Oscillations

4.1.1 Properties of Oscillations

4.1.2 Simple Harmonic Oscillations

4.1.3 SHM Graphs

4.1.4 Energy in SHM

4.2 Travelling Waves

4.2.1 Properties of Waves

4.2.2 Transverse & Longitudinal Waves

4.2.3 Electromagnetic Waves

4.2.4 Sound Waves

4.3 Wave Characteristics

4.3.1 Wavefronts

4.3.2 Amplitude & Intensity

4.3.3 Superposition

4.3.4 Polarisation