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Chapter 6: Photons and Massless Particles

Photons are massless yet carry energy and momentum. They always travel at c in all frames, have zero proper time, and obey E = pc. Their behavior illustrates the extreme limits of special relativity.

Photon Properties

$E = pc$

For massless particles, the energy-momentum relation gives $E = pc$ (since m = 0).

$E = hf$

Photon energy is quantized: $E = hf = hc/\lambda$ (Planck's relation).

p = h/λ

Photon momentum from E = pc and E = hf: p = h/λ (de Broglie relation).

No Rest Frame

Photons always travel at c—there's no frame where a photon is at rest.

Relativistic Doppler Effect

$ f_{\text{obs}} = f_{\text{source}} \sqrt{\frac{1 - \beta}{1 + \beta}} $

(for motion directly toward/away)

Redshift (receding)

Source moving away: observed frequency lower, wavelength longer (redder).

Blueshift (approaching)

Source moving toward: observed frequency higher, wavelength shorter (bluer).

Radiation Pressure

Light carries momentum, so it exerts pressure when absorbed or reflected:

$ P = \frac{I}{c} $ (absorbed) or $ P = \frac{2I}{c} $ (reflected)

Applications: solar sails, laser cooling, comet tails pointing away from Sun.

Interactive Simulations

Compton Scattering: Photon-Electron Interaction

Python

script.py52 lines

import numpy as np

print("=" * 70)
print("COMPTON SCATTERING SIMULATION")
print("Photon + electron (at rest) -> scattered photon + recoil electron")
print("=" * 70)

m_e = 0.511  # electron mass in MeV/c^2
h = 4.136e-15  # Planck constant in eV*s
c = 3e8

# Compton wavelength
lambda_C = 2.426e-12  # meters (h / m_e c)
print(f"Compton wavelength: {lambda_C*1e12:.4f} pm")

# Compton formula: lambda' - lambda = lambda_C * (1 - cos(theta))
print(f"\nCompton formula: lambda' - lambda = lambda_C * (1 - cos(theta))")

# Incident photon energies
E_photon_values = [0.1, 0.511, 1.0, 5.0, 10.0, 100.0]  # MeV

for E_gamma in E_photon_values:
    print(f"\n--- Incident photon energy: {E_gamma} MeV ---")
    print(f"{'theta (deg)':>12} {'E_scattered (MeV)':>18} {'E_electron (MeV)':>18} {'Shift/lambda_C':>16}")
    print("-" * 68)

for theta_deg in [0, 30, 60, 90, 120, 150, 180]:
        theta = np.radians(theta_deg)
        # E' = E / (1 + (E/m_e)(1 - cos(theta)))
        E_scattered = E_gamma / (1.0 + (E_gamma / m_e) * (1.0 - np.cos(theta)))
        E_electron = E_gamma - E_scattered
        shift = (E_gamma / m_e) * (1.0 - np.cos(theta))

print(f"{theta_deg:12d} {E_scattered:18.4f} {E_electron:18.4f} {shift:16.4f}")

print("\n" + "=" * 70)
print("PHOTON PROPERTIES SUMMARY")
print("=" * 70)
print("Photons have:")
print("  Mass = 0")
print("  Energy E = hf = pc")
print("  Momentum p = E/c = h/lambda")
print("  Always travel at speed c in vacuum")
print(f"\n{'E (eV)':>12} {'lambda (nm)':>14} {'Type':>16}")
print("-" * 44)
types = [(1.8, "Infrared"), (3.0, "Red"), (4.0, "Violet"),
         (100, "UV"), (1e4, "X-ray"), (1e6, "Gamma")]
for E_eV, name in types:
    lam_nm = 1240.0 / E_eV
    print(f"{E_eV:12.1f} {lam_nm:14.3f} {name:>16}")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Compton Scattering Calculator

Fortran

program.f9056 lines

program compton_scattering
  implicit none
  double precision :: m_e, E_gamma, E_scat, E_elec
  double precision :: theta, theta_deg, pi
  double precision :: lambda_C, shift
  integer :: i, j
  double precision :: energies(6)

pi = 4.0d0 * atan(1.0d0)
  m_e = 0.511d0  ! MeV/c^2
  lambda_C = 2.426d-12  ! Compton wavelength (m)

energies = (/0.1d0, 0.511d0, 1.0d0, 5.0d0, 10.0d0, 100.0d0/)

print '(A)', repeat('=', 65)
  print '(A)', 'COMPTON SCATTERING: E'' = E/(1 + (E/mc^2)(1-cos(theta)))'
  print '(A)', repeat('=', 65)
  print '(A,F10.4,A)', 'Electron mass: ', m_e, ' MeV/c^2'
  print '(A,ES10.4,A)', 'Compton wavelength: ', lambda_C, ' m'

do i = 1, 6
    E_gamma = energies(i)
    print '(/,A,F8.3,A)', '--- E_photon = ', E_gamma, ' MeV ---'
    print '(A12, A16, A16, A14)', &
      'theta(deg)', 'E_scat(MeV)', 'E_elec(MeV)', 'KE_frac'
    print '(A)', repeat('-', 60)

do j = 0, 6
      theta_deg = dble(j) * 30.0d0
      theta = theta_deg * pi / 180.0d0

E_scat = E_gamma / (1.0d0 + (E_gamma/m_e)*(1.0d0 - cos(theta)))
      E_elec = E_gamma - E_scat

print '(F12.1, F16.4, F16.4, F14.4)', &
        theta_deg, E_scat, E_elec, E_elec/E_gamma
    end do
  end do

! Maximum energy transfer (backscatter)
  print '(/,A)', repeat('=', 65)
  print '(A)', 'MAXIMUM ENERGY TRANSFER (theta = 180 deg)'
  print '(A)', repeat('=', 65)
  print '(A12, A16, A14)', 'E_in(MeV)', 'E_max_elec', 'Fraction'
  print '(A)', repeat('-', 45)

do i = 1, 6
    E_gamma = energies(i)
    E_scat = E_gamma / (1.0d0 + 2.0d0*E_gamma/m_e)
    E_elec = E_gamma - E_scat
    print '(F12.3, F16.4, F14.4)', E_gamma, E_elec, E_elec/E_gamma
  end do

end program compton_scattering

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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