Photon Momentum

Even though photons have zero rest mass, they carry momentum due to their energy. This is a purely quantum mechanical effect with no classical analogue.

The Classical Problem

In classical mechanics: $$p = mv$$

If $m = 0$, then $p = 0$. This would imply massless particles have no momentum.

But photons do have momentum — proven by experiments like the Compton Effect and phenomena like solar sails.

Modern Physics Solution

From Einstein's mass-energy relation for photons: $$E = pc$$

Since photon energy is $E = hf = \frac{hc}{\lambda}$:

$$p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda}$$

Key Formula

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

Where:

  • $p$ = photon momentum (kg·m/s)
  • $h = 6.63 \times 10^{-34}$ J·s (Planck's constant)
  • $\lambda$ = wavelength (m)

Alternative Forms

Given Formula
Wavelength $\lambda$ $p = \frac{h}{\lambda}$
Frequency $f$ $p = \frac{hf}{c}$
Energy $E$ $p = \frac{E}{c}$

Physical Implications

Light Can Push Objects

  • Solar sails: Spacecraft can use photon pressure for propulsion
  • Radiation pressure: Light exerts force on surfaces

Key Relationships

  • Shorter wavelength → Higher momentum
  • Higher frequency → Higher momentum
  • Higher energy → Higher momentum

Example Calculation

Problem: Find the momentum of a photon with $\lambda = 5.0 \times 10^{-7}$ m.

Solution: $$p = \frac{h}{\lambda} = \frac{6.63 \times 10^{-34}}{5.0 \times 10^{-7}} = 1.33 \times 10^{-27} \text{ kg·m/s}$$

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