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Chapter 3: Energy-Momentum Relation

The energy-momentum relation is the fundamental equation connecting energy, momentum, and mass. It's the relativistic version of the kinetic energy formula and reveals the deep connection between these quantities.

The Energy-Momentum Relation

$ E^2 = (pc)^2 + (mc^2)^2 $

The "mass shell" condition—true for all free particles

At Rest (p = 0)

$E = mc^2$ (rest energy)

Massless (m = 0)

$E = pc$ (photons)

General

$ E = \gamma mc^2 $

Relativistic Kinetic Energy

$ K = E - mc^2 = (\gamma - 1)mc^2 $

Total energy minus rest energy

At low speeds (v ≪ c), expanding γ - 1 ≈ v²/2c² gives:

$ K \approx \frac{1}{2}mv^2 $ (classical result)

The Dispersion Relation

In quantum mechanics, E = ℏω and p = ℏk. The energy-momentum relation becomes:

$ \omega^2 = c^2k^2 + \frac{m^2c^4}{\hbar^2} $

This is the Klein-Gordon dispersion relation, fundamental to relativistic quantum field theory.

Interactive Simulations

Energy-Momentum Relation: E^2 = (pc)^2 + (mc^2)^2

Python

script.py44 lines

import numpy as np

print("=" * 70)
print("ENERGY-MOMENTUM RELATION")
print("E^2 = (pc)^2 + (mc^2)^2    [c = 1]")
print("=" * 70)

m = 1.0  # rest mass

print(f"\nRest mass m = {m:.1f}")
print(f"{'v/c':>8} {'gamma':>8} {'E':>10} {'p':>10} {'KE':>10} {'E^2-p^2':>10} {'= m^2?':>8}")
print("-" * 68)

for v in [0.0, 0.1, 0.3, 0.5, 0.7, 0.8, 0.9, 0.95, 0.99, 0.999]:
    gamma = 1.0 / np.sqrt(1.0 - v**2) if v > 0 else 1.0
    E = gamma * m
    p = gamma * m * v
    KE = E - m
    check = E**2 - p**2
    print(f"{v:8.3f} {gamma:8.4f} {E:10.4f} {p:10.4f} {KE:10.4f} {check:10.6f} {'YES' if abs(check - m**2) < 1e-8 else 'NO':>8}")

# Ultra-relativistic and non-relativistic limits
print("\n" + "=" * 70)
print("LIMITING CASES")
print("=" * 70)

print("\nNon-relativistic limit (v << c): E ~ mc^2 + p^2/(2m)")
for v in [0.001, 0.01, 0.05, 0.1]:
    gamma = 1.0 / np.sqrt(1.0 - v**2)
    E_exact = gamma * m
    p = gamma * m * v
    E_NR = m + p**2 / (2*m)  # non-relativistic approximation
    print(f"  v = {v:.3f}c: E_exact = {E_exact:.8f}, E_NR = {E_NR:.8f}, error = {abs(E_exact-E_NR)/E_exact*100:.6f}%")

print("\nUltra-relativistic limit (v -> c): E ~ pc")
for v in [0.9, 0.99, 0.999, 0.9999]:
    gamma = 1.0 / np.sqrt(1.0 - v**2)
    E = gamma * m
    p = gamma * m * v
    print(f"  v = {v:.4f}c: E = {E:.4f}, pc = {p:.4f}, E/pc = {E/p:.6f}")

print("\nPhoton (m=0): E = pc exactly, for any energy.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Energy-Momentum Dispersion Relation

Fortran

program.f9082 lines

program energy_momentum
  implicit none
  double precision :: m, v, gamma_val, E, p, KE, check
  double precision :: speeds(10)
  integer :: i

m = 0.938d0  ! proton mass in GeV/c^2

speeds = (/0.0d0, 0.1d0, 0.3d0, 0.5d0, 0.7d0, &
             0.8d0, 0.9d0, 0.95d0, 0.99d0, 0.999d0/)

print '(A)', repeat('=', 70)
  print '(A)', 'ENERGY-MOMENTUM RELATION FOR A PROTON'
  print '(A,F6.3,A)', 'Rest mass m = ', m, ' GeV/c^2'
  print '(A)', 'E^2 = p^2 + m^2  (c = 1)'
  print '(A)', repeat('=', 70)

print '(A8, A10, A12, A12, A12, A12)', &
    'v/c', 'gamma', 'E (GeV)', 'p (GeV/c)', 'KE (GeV)', 'E^2-p^2'
  print '(A)', repeat('-', 68)

do i = 1, 10
    v = speeds(i)
    if (v < 1.0d-15) then
      gamma_val = 1.0d0
    else
      gamma_val = 1.0d0 / sqrt(1.0d0 - v**2)
    end if

E = gamma_val * m
    p = gamma_val * m * v
    KE = E - m
    check = E**2 - p**2

print '(F8.3, F10.4, F12.4, F12.4, F12.4, F12.6)', &
      v, gamma_val, E, p, KE, check
  end do

print '(/,A,F10.6)', 'All E^2 - p^2 = m^2 = ', m**2
  print '(A)', 'The mass-shell condition is always satisfied!'

! Particle identification via E-p measurement
  print '(/,A)', repeat('=', 70)
  print '(A)', 'PARTICLE IDENTIFICATION FROM E AND p'
  print '(A)', repeat('=', 70)

block
    double precision :: E_meas, p_meas, m_calc
    double precision :: E_list(4), p_list(4)
    character(len=12) :: particle

E_list = (/1.5d0, 0.55d0, 10.0d0, 5.0d0/)
    p_list = (/1.2d0, 0.13d0, 9.99d0, 4.95d0/)

print '(A12, A12, A12, A16)', 'E (GeV)', 'p (GeV/c)', 'm (GeV/c^2)', 'Particle?'
    print '(A)', repeat('-', 55)

do i = 1, 4
      E_meas = E_list(i)
      p_meas = p_list(i)
      m_calc = sqrt(abs(E_meas**2 - p_meas**2))

if (abs(m_calc - 0.938d0) < 0.05d0) then
        particle = 'Proton'
      else if (abs(m_calc - 0.511d-3) < 0.001d0) then
        particle = 'Electron'
      else if (abs(m_calc - 0.140d0) < 0.02d0) then
        particle = 'Pion'
      else if (m_calc < 0.01d0) then
        particle = 'Near photon'
      else
        particle = 'Unknown'
      end if

print '(F12.4, F12.4, F12.4, A16)', &
        E_meas, p_meas, m_calc, particle
    end do
  end block

end program energy_momentum

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

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