Chapter XX : Photon Momentum, Radiation Pressure, Matter Waves

Photon Momentum

Collision of (gamma ray) photons with electrons
Applying CoM and CoE requires photon to carry momentum, even though it has no mass. even though it has no mass

\[ p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda} \]

\[ h = 6.62607015 \times 10^{-34}~J \cdot s \]

Example: Hydrogen

Recall Bohr model of the atom:
Previously found that if we have an atom with Z protons in the nucleus but just a single electron, the electron would have very specific energy levels:
\[ E_n = -( \frac{ Z^2 e^4 m }{ 8 \epsilon^2_o h^2 } ) \frac{1}{n^2} \]

Converting units:
\[ E_n = -(13.60569~eV) \frac{Z^2}{n^2} \]

Example : The lone electron in a hydrogen (Z=1) atom is presently in the n=3 orbit. It drops down to n=1, releasing a photon.

The photon will carry off energy and momentum.
The hydrogen atom will then recoil at what speed?

The moving hydrogen atom is also carrying off energy as a result.

How will this affect the wavelength of the emitted photon?

E_n = -(13.60569 eV) / n²

What energy photon will be released? (What wavelength? Visible?)
E = (1239.842 eV·nm)/λ (with wavelength in nm)
What momentum would it have? p=E / c

Conservation of momentum : what speed must the H atom have?
mass of a single hydrogen atom:
1.007825 u where 1 u = 1.6605 × 10^-27 kg

What kinetic energy does that represent?

Is that enough to affect the wavelength of the emitted photon?

Radiation Pressure

Consider EM waves (light, radio, etc) with an intensity of I watts/m² falling on some area ΔA over some time interval Δt and suppose these photons are all completely absorbed by the material:

I = (power) / (area) = (energy / time) / (area) so (energy) = (I)(Δt)(ΔA)
Each photon carries E=hc / λ
Number of photons falling on this area in the given time: N = (energy) / (energy per photon ) = (I)(Δt)(ΔA)(λ) / (hc)
Momentum each photon carries: p=h / λ
Total momentum transferred to the object in this time interval:
Δp = (momentum of each photon ) × (number of photons) = Δp = (I)(Δt)(ΔA) / c
Force is momentum/time, so F = Δp / Δt = (I)(ΔA) / c
Finally: pressure is force/area so: P = I / c

If the photons are completely reflected by the material, the momentum change is doubled, so P=2I / c

Radiation Pressure

If 100% absorbed by target, results in pressure of P = I/c
If 100% reflected by target, results in pressure of P=2I/c

Example : in a typical well-lit room, the light intensity is about 200 W/m².
What additional pressure will this exert on us?

If we're wearing dark clothes that absorb all the light, and
if we're wearing shiny tin-foil reflecting all the light.

Compare: 1 ATM = 14.7 lb/in² = 101325 N/m²

Radiation Pressure : Examples

International Space Station solar panels:

Generate a total of 240 kW of power
total area: 3500 m²
I=1350 W/m² 'up there'
(How efficient are these solar panels?)
What total pressure will the Sun's light exert on the ISS?
ISS mass: 450,000 kg
Resulting acceleration?
Anything to worry about here?

Project Breakthrough Starshot

Idea : large swarm of small probes accelerated by a (very) powerful laser, sent towards a nearby star.

Orbiting lasers with total power output of 100 GW
Probe mass: 5 grams
Solar 'sail' about 5 meters across
Each accelerated to v=0.20 c in 10 minutes
(would reach α Centauri in about 20 years)
Consistent with what we know about radiation pressure?

These 'solar sails' have been used on actual space probes in the past.

2010 : Japan Space Exploration Agency (JAXA) : IKAROS probe
2010 : NASA NANOSEL-D2
2019 : Planetary Society LightSail-2

Matter Waves : de Broglie Wavelength

Massless Photons can behave as particles carrying momentum of: p=h / λ

Particles with mass have a 'wavelength' of λ=h / p where p=mv
called the de Broglie wavelength of the particle

.. and as such will behave as waves with that wavelength, even showing diffraction effects.

Electron 'diffraction'

Actual Electron Diffraction Experiment Results

Double-slit Experiment Using Electrons instead of light

Electron Diffraction Example

Electrons are accelerated across a 100 V potential difference.
They pass through an incredibly thin crystal lattice where the separation distance between atoms is 0.2 nm
What diffraction pattern would these electrons produce?
Treat this as a diffraction grating with d=0.2 nm

Grating: constructive interference where sin(θ)=mλ/d
At what angles on the other side of this 'grating' would we see a strong signal of 'electrons'?

What is the wavelength of these electrons? λ=h / p with p=mv so how fast are these electrons travelling?
m_electron = 9.11 × 10^-31 kg K = ½mv² = 100 eV

Baseball Diffraction

We design an apparatus that perfectly drops a baseball so that it always hits exactly the same spot on a floor.
Now we put a barrier with a 10 cm circular aperture in the way so that the ball passes through this hole.
What is the de Broglie wavelength of these baseball if it has a mass of m=0.145 kg and a speed of 90 miles/hour (v=40.225 m/s)?
A wave passing through a circular aperture 'spreads out' by an angle of θ=1.22λ / D

Photon Momentum
Collision of (gamma ray) photons with electrons Applying CoM and CoE requires photon to carry momentum, even though it has no mass. even though it has no mass \[ p = \frac{E}{c} = \frac{hf}{c} = \frac{h}{\lambda} \] \[ h = 6.62607015 \times 10^{-34}~J \cdot s \]
Example: Hydrogen
Recall Bohr model of the atom: Previously found that if we have an atom with Z protons in the nucleus but just a single electron, the electron would have very specific energy levels: \[ E_n = -( \frac{ Z^2 e^4 m }{ 8 \epsilon^2_o h^2 } ) \frac{1}{n^2} \] Converting units: \[ E_n = -(13.60569~eV) \frac{Z^2}{n^2} \] Example : The lone electron in a hydrogen (Z=1) atom is presently in the n=3 orbit. It drops down to n=1, releasing a photon. The photon will carry off energy and momentum. The hydrogen atom will then recoil at what speed? The moving hydrogen atom is also carrying off energy as a result. How will this affect the wavelength of the emitted photon?
E_n = -(13.60569 eV) / n² What energy photon will be released? (What wavelength? Visible?) E = (1239.842 eV·nm)/λ (with wavelength in nm) What momentum would it have? p=E / c Conservation of momentum : what speed must the H atom have? mass of a single hydrogen atom: 1.007825 u where 1 u = 1.6605 × 10^-27 kg What kinetic energy does that represent? Is that enough to affect the wavelength of the emitted photon?

Radiation Pressure : Examples
International Space Station solar panels: Generate a total of 240 kW of power total area: 3500 m² I=1350 W/m² 'up there' (How efficient are these solar panels?) What total pressure will the Sun's light exert on the ISS? ISS mass: 450,000 kg Resulting acceleration? Anything to worry about here?

Project Breakthrough Starshot
Idea : large swarm of small probes accelerated by a (very) powerful laser, sent towards a nearby star. Orbiting lasers with total power output of 100 GW Probe mass: 5 grams Solar 'sail' about 5 meters across Each accelerated to v=0.20 c in 10 minutes (would reach α Centauri in about 20 years) Consistent with what we know about radiation pressure?
These 'solar sails' have been used on actual space probes in the past. 2010 : Japan Space Exploration Agency (JAXA) : IKAROS probe 2010 : NASA NANOSEL-D2 2019 : Planetary Society LightSail-2

Matter Waves : de Broglie Wavelength
Massless Photons can behave as particles carrying momentum of: p=h / λ Particles with mass have a 'wavelength' of λ=h / p where p=mv called the de Broglie wavelength of the particle .. and as such will behave as waves with that wavelength, even showing diffraction effects.	Electron 'diffraction'
Actual Electron Diffraction Experiment Results

Double-slit Experiment Using Electrons instead of light

Electron Diffraction Example
Electrons are accelerated across a 100 V potential difference. They pass through an incredibly thin crystal lattice where the separation distance between atoms is 0.2 nm What diffraction pattern would these electrons produce? Treat this as a diffraction grating with d=0.2 nm Grating: constructive interference where sin(θ)=mλ/d At what angles on the other side of this 'grating' would we see a strong signal of 'electrons'? What is the wavelength of these electrons? λ=h / p with p=mv so how fast are these electrons travelling? m_electron = 9.11 × 10^-31 kg K = ½mv² = 100 eV
Baseball Diffraction
We design an apparatus that perfectly drops a baseball so that it always hits exactly the same spot on a floor. Now we put a barrier with a 10 cm circular aperture in the way so that the ball passes through this hole. What is the de Broglie wavelength of these baseball if it has a mass of m=0.145 kg and a speed of 90 miles/hour (v=40.225 m/s)? A wave passing through a circular aperture 'spreads out' by an angle of θ=1.22λ / D