Photon Momentum Calculator
Overview
Photons, the quantum particles of light, have no rest mass, but they still carry both energy and momentum. This is a cornerstone idea in quantum mechanics, electromagnetism, and modern optics. The photon momentum calculator on this page lets you convert between three closely related quantities:
- Wavelength, λ (in metres)
- Energy, E (in joules)
- Momentum, p (in kg·m/s)
Enter any two of these values, and the missing quantity can be computed using standard relations involving Planck’s constant and the speed of light in vacuum. This is useful for problems in spectroscopy, laser physics, astrophysics, and general quantum theory exercises.
Key Relations for Photon Momentum
For photons in vacuum, several simple but powerful formulas link energy, momentum, and wavelength. The calculator uses the following standard constants (CODATA 2019 values):
- Planck’s constant: h = 6.62607015 × 10−34 J·s
- Speed of light in vacuum: c = 2.99792458 × 108 m/s
Energy and frequency
The energy of a photon is proportional to its frequency ν:
E = h ν.
Since frequency and wavelength are related by c = λν for light in vacuum, we can also express energy in terms of wavelength:
E = hc / λ.
Momentum and wavelength
The momentum of a photon is inversely proportional to its wavelength:
p = h / λ.
This has the same algebraic form as the de Broglie relation for matter waves, but for photons it follows naturally from combining quantum and relativistic relations.
Energy and momentum
For a massless particle such as a photon, the relativistic energy–momentum relation simplifies to:
E = pc, so equivalently p = E / c.
For comparison, the full relation for a particle with rest mass m is:
Setting m = 0 for photons yields the simpler E = pc used by the calculator.
How the Calculator Uses These Formulas
The calculator assumes SI units for all inputs and outputs. Internally, it follows these steps:
- If wavelength λ is known, it can compute both energy and momentum using
- E = hc / λ
- p = h / λ
- If energy E is known, it can compute
- Momentum: p = E / c
- Wavelength: λ = hc / E
- If momentum p is known, it can compute
- Energy: E = pc
- Wavelength: λ = h / p
In more advanced contexts, if both E and λ are supplied, one can check consistency using E = hc / λ. Large discrepancies would indicate that the pair of values does not describe a single physical photon state in vacuum.
Interpreting the Results
The quantities involved in photon physics often span many orders of magnitude. For example, radio photons can have wavelengths of metres or more, while gamma-ray photons may have wavelengths smaller than 10−12 m. As a result, calculated energies and momenta are frequently presented in scientific notation (e.g., 2.2 × 10−26 kg·m/s). This makes the numbers easier to compare and reduces rounding issues.
When you read the output:
- Check the sign and magnitude of the exponent; a small change in exponent can represent a huge change in physical scale.
- Compare your result with typical ranges for the part of the electromagnetic spectrum you are interested in (radio, microwave, infrared, visible, ultraviolet, X-ray, gamma-ray).
- Remember that all results assume photons traveling in vacuum with speed c.
Worked Example: Green Visible Light
Suppose you want the momentum and energy of a green photon with wavelength 550 nm (a typical value in the middle of the visible spectrum). First, convert the wavelength to metres:
- 550 nm = 550 × 10−9 m = 5.50 × 10−7 m.
You would enter 5.50e−7 in the wavelength field and leave the energy and momentum fields blank. Using the formulas above:
- Energy: E = hc / λ
Numerically,
E = (6.62607015 × 10−34 J·s)(2.99792458 × 108 m/s) / (5.50 × 10−7 m).
This gives approximately
E ≈ 3.61 × 10−19 J.
- Momentum: p = h / λ
p = 6.62607015 × 10−34 J·s / (5.50 × 10−7 m).
This yields
p ≈ 1.21 × 10−27 kg·m/s.
The calculator automates these steps, but it is helpful to understand how the expressions are combined and why the results are so small in everyday units.
Photon Momentum Across the Electromagnetic Spectrum
The momentum of a photon increases as its wavelength decreases. The table below shows representative values for different parts of the spectrum, assuming photons in vacuum.
| Type of light | Example wavelength (m) | Approx. momentum (kg·m/s) |
|---|---|---|
| Radio (100 MHz) | 3.0 | 2.21 × 10−34 |
| Microwave (10 GHz) | 3.0 × 10−2 | 2.21 × 10−32 |
| Green visible | 5.5 × 10−7 | 1.20 × 10−27 |
| Ultraviolet | 1.0 × 10−7 | 6.63 × 10−27 |
| X-ray | 1.0 × 10−10 | 6.63 × 10−24 |
Each row illustrates how dramatically photon momentum grows as the wavelength becomes shorter. While a single photon has a tiny momentum, intense X-ray or ultraviolet beams can deliver a noticeable transfer of momentum to matter.
Applications: Radiation Pressure, Solar Sails, and Laser Cooling
Photon momentum has concrete, measurable effects:
- Radiation pressure and solar sails: When light reflects or is absorbed by a surface, the change in photon momentum exerts a force. Solar sail concepts use huge, lightweight reflective sheets in space. Continuous bombardment by solar photons transfers enough momentum over time to accelerate spacecraft without consuming conventional propellant.
- Optical tweezers and trapping: Highly focused laser beams can trap and move microscopic particles. The gradient in photon momentum and intensity creates forces that confine small beads, cells, or even neutral atoms near the beam focus.
- Laser cooling: In laser cooling and trapping experiments, photons are tuned so that atoms preferentially absorb light when moving toward a laser beam. Repeated absorption and re-emission events transfer momentum to the atoms, gradually reducing their average kinetic energy and cooling them to microkelvin or even nanokelvin temperatures.
- Atomic recoil: When an atom spontaneously emits a photon, conservation of momentum requires that the atom recoil in the opposite direction. This recoil sets a fundamental limit on how precisely certain transitions can be measured and influences the lowest temperatures achievable with specific cooling schemes.
Assumptions and Limitations
The photon momentum calculator is designed for clarity and educational use. It makes several simplifying assumptions:
- Vacuum propagation: All formulas assume photons travel in vacuum, where the speed of light is exactly c = 2.99792458 × 108 m/s. In media (glass, water, optical fibers), the effective propagation speed and relationships between wavelength and frequency can differ.
- Idealized, monochromatic photons: The tool treats each input as referring to a single, well-defined photon energy or wavelength. Real light sources may emit a range (spectrum) of wavelengths and energies.
- Neglect of gravitational and relativistic effects beyond E = pc: Curved spacetime, gravitational redshift, and detailed field-theoretic corrections are not considered. For most laboratory and classroom situations, these corrections are negligible.
- Constant physical constants: Planck’s constant and the speed of light are taken as exact CODATA 2019 values. Any future refinements to recommended constants are not dynamically included.
- SI units only: Inputs and outputs are in metres, joules, and kg·m/s. If you work in electronvolts (eV) or nanometres, you must convert to SI before using the calculator and convert back if needed.
- Educational, not safety-critical: The calculator is intended for learning, demonstrations, and approximate design work. It should not be used as the sole basis for safety-critical engineering, high-power laser system design, or medical dosimetry.
Within these assumptions, the underlying physics is straightforward and robust. Being aware of the scope of the formulas helps you judge when a more detailed model or specialist tool is required.
Further Exploration
Photon momentum is just one aspect of light’s quantum behavior. Closely related quantities include photon frequency, de Broglie wavelength for matter particles, and the relationship between intensity, photon flux, and delivered power. Exploring how energy, wavelength, and momentum transform between different frames, or how they behave in materials with high refractive index, can deepen your understanding of both classical and quantum optics.
Use the calculator as a starting point: vary the wavelength from radio to gamma rays, observe how the momentum changes, and relate this to real-world phenomena such as communication technologies, medical imaging, and astrophysical observations.