Momentum of Photons and Matter

Introduction

This section explores the momentum of photons and matter, focusing on their wave-particle duality and how momentum relates to wavelength.

KEY TAKEAWAY: Both photons and matter exhibit wave-particle duality, meaning they possess both wave-like and particle-like properties.

Momentum of Photons

Photon Momentum Formula

Photons, despite having no mass, possess momentum. The momentum of a photon is related to its energy and wavelength by the following formulas:

• $E_{ph} = pc$
• $p = \frac{E_{ph}}{c}$
• $p = \frac{h}{\lambda}$

Where:

• $E_{ph}$ = Photon energy (J)
• $p$ = Momentum (kg m s$^{-1}$)
• $c$ = Speed of light in a vacuum (3.0 × 10$^8$ m s$^{-1}$)
• $h$ = Planck’s constant (6.63 × 10$^{-34}$ J s)
• $\lambda$ = Wavelength (m)

REMEMBER: Photons have momentum even though they have no mass.

Derivation of Photon Momentum

The relationship $E_{ph} = pc$ can be derived from Einstein’s mass-energy equivalence ($E=mc^2$) and relativistic momentum. However, at the VCE level, understanding the formula and its application is more important than the derivation.

STUDY HINT: Focus on understanding how to use the formula $p = \frac{h}{\lambda}$ in different scenarios.

Momentum of Matter

Matter Momentum Formula

The momentum of matter (particles with mass) is given by:

• $p = mv$
• $p = \sqrt{2m \times KE}$

Where:

• $m$ = mass (kg)
• $v$ = velocity (m s$^{-1}$)
• $KE$ = Kinetic Energy (J)

EXAM TIP: Be careful to use consistent units when performing calculations.

de Broglie Wavelength

Louis de Broglie proposed that matter also exhibits wave-like properties. The de Broglie wavelength of a particle is given by:

• $\lambda = \frac{h}{p}$
• $\lambda = \frac{h}{mv}$
• $\lambda = \frac{h}{\sqrt{2m \times KE}}$

APPLICATION: Electron microscopes use the wave-like properties of electrons to achieve higher resolution than optical microscopes.

Comparing Photon and Matter Momentum

Same Wavelength

If a photon and matter particle have the same wavelength ($\lambda$), they have the same momentum ($p$).

Same Momentum

If a photon and matter particle have the same momentum ($p$), they have the same wavelength ($\lambda$).

COMMON MISTAKE: Forgetting that although the formulas linking momentum and wavelength are the same for photons and matter, the reason they have momentum differs (photons due to energy, matter due to mass and velocity).

Diffraction

When photons and matter with the same wavelength diffract through a gap of the same width, they produce identical diffraction patterns. The amount of diffraction is proportional to the ratio of wavelength to gap width ($\text{diffraction} \propto \frac{\lambda}{w}$).

• Same spacing of bright bands in diffraction patterns implies same wavelengths and momentums.
• Different spacing of bright bands in diffraction patterns implies different wavelengths and momentums.

VCAA FOCUS: Expect questions that involve comparing the wavelengths and momentums of photons and matter based on diffraction patterns.

Table: Comparing Photon and Matter

KEY TAKEAWAY: The formula $p = \frac{h}{\lambda}$ is universally applicable to both photons and matter, highlighting their wave-particle duality.