Photon Theory and Related Experiments

Problems with the Photon Theory

Photon theory replaces standing waves with photons localized within boundary conditions.
Photons are drawn with wave character because energy is proportional to frequency, even though they're particles.

Questions Arising

If photons are particles, why do they have frequency?
Is this frequency local, with energy concentrated in one area?

Issues with Photon Theory

It seems to break the symmetry of states by imbuing a particle with wave characteristics.

Particle vs. Wave Character

A key problem is assigning wave character to a particle.
For matter particles:
- Energy is $E = \gamma m0 c^2$ where $\gamma$ is the Lorentz factor, $m0$ is the rest mass, and $c$ is the speed of light.
- This equation makes sense because it includes properties associated with particles, such as rest mass and velocity.
For matterless particles (photons):
- Energy isn't related to mass or velocity, but to frequency $f$ .

Questions about Wavelength

Does a photon have a wavelength?
Is wavelength found by $\lambda = \frac{c}{f}$ ?

The 84.1 State

In a cavity, a photon is localized, implying energy and momentum.
The $n = 1$ standing wave doesn't distinguish between left and right.
- Standing waves are formed by two identical waves moving in opposite directions.

Standing Waves

Standing waves result from two sinusoidal waves with the same amplitude and wavelength but moving in opposite directions, creating nodes and antinodes.

Mathematical Representation of a Standing Wave

An electric field in a standing wave is a combination of rightward and leftward disturbances:
- $E = A \cos(kx - Bt) + \cos(kx + Bt)$
- Where $A$ is the amplitude,
- $k$ is the wave number,
- $B$ is the angular frequency, and
- $t$ is the time.
- The first term is a rightward-going wave and the second term is a leftward-going wave.

Einstein's Interpretation vs. Maxwell's

In Maxwell's view, $n = 1$ standing wave is composed of two waves moving in opposite directions.
In Einstein's view, there is a single photon moving either right or left.
If the photon weren't moving, it would have zero momentum and no rest mass:
- Using $E^2 - p^2c^2 = m0^2c^4$ , if $p = 0$ and $m0 = 0$ , then $E$ would also have to be zero, but the photon has energy, so it must be moving.

Dual Nature of Light?

Question: Does a photon have a dual nature, possibly made of two crisscrossing photons?
The photoelectric effect suggests that the particle carries all its energy (not split half to the right and half to the left).
If a particle has momentum, it's a combination of two equal waves, one right and one left, combined in the middle.

Young's Two-Slit Interference Experiment

Classic experiment: light ray goes through two slits, creating an interference pattern on a screen.
Newton's idea of light as small bundles of energy (corpuscles) is inconsistent with this experiment.

Newton's Corpuscular Theory

Newton proposed light as corpuscles in the 1700s without empirical evidence.
Corpuscular theory predicts light landing directly behind the slits (ray model).
Light travels in a straight line, evidenced by shadows.

Observed Interference Pattern

Instead of direct spots, constructive and destructive interference patterns occur, similar to water or sound waves.

Equation for Constructive Interference

$\lambdam = d \sin(\thetam)$
- Where $\lambda_m$ is the wavelength in the medium
- $d$ is the separation between the slits
- $\theta_m$ is the angle to the $m$ -th maximum.

Photon Behavior in the Two-Slit Experiment

Photons exhibit both particle and wave behavior.
When particles go through two slits, light rays appear to arrive at particular angles on the screen.

Multiple Slits

Peaks in the interference pattern become sharper with more slits.
The width of the peak is inversely proportional to the number of slits.

Single Photon Experiment

If a single photon passes through one slit and is lucky enough not to hit any walls, how does it decide which peak to go to?
Photons only arrive at certain angles, whether light has a particle or wave nature.
How do 500 photons arrange themselves to maintain equal intensity at each peak?

Low-Intensity Laser Experiment

Imagine using a low-intensity laser that emits only one photon per second. Even a single photon knows to land only at certain spots. It behaves as predicted by light that went through both slits.

Even when passing through one slit, the photon somehow knows about the presence of the other slit(s) and lands only at particular spots.
This is disturbing because the whole structure is about the relation in phase of two or more waves that went through multiple slits to create constructive interference:
- If one slit is removed, the interference pattern disappears.
- The interference pattern depends on something happening at each slit interacting with each other
- So the existence of the interference pattern is dependent on something happening in each of this list.

Plane Waves

The interference pattern makes sense for plane waves because energy arrives at the slits in sheets (wave fronts).

Huygens' Principle

Radiation goes out in all directions from each slit; constructive and destructive interference occurs at certain angles.

Feynman's Proposition

Maybe the photon momentarily splits into two photons to be aware of the other slits and then regroups after going through the slits.
This idea might seem crazy.

Summary of the Photon Theory Problems

Photon theory explains certain experiments well but fails in others.
Wave picture struggles to explain phenomena like the photoelectric effect.
Advanced physics is needed to reconcile wave and particle behavior.

Compton Scattering

Examine what happens when an electromagnetic wave interacts with an electron using both classical and photon theories.

Classical Theory

An electromagnetic wave causes the electron to oscillate and accelerate, emitting radiation in all directions at the same frequency of the incident light.
The acceleration of the electron is in the vertical direction: Acceleration $= a_{\text{max}} \cos(2 \pi f t) \hat{y}$ .
The intensity of the wave diminishes as it loses energy to the electron, which then radiates energy in all directions.

Photon Theory

The photon transfers part of its energy to the electron and scatters with lower energy.
The frequency shift is associated with a particular angle.
Mathematically, the problem looks like a collision between two billiard balls.

Conservation Laws

The shift in frequency relies on applying the conservation principles of momentum and energy.

Qualitative Difference

In the classical picture, emitted radiation has the same frequency as the incident wave; in photon theory, the scattered photon has a different frequency.

Equations for Compton Scattering

To derive $f'(\theta)$ , use:
- Conservation of energy
- Conservation of momentum in the x-direction
- Conservation of momentum in the y-direction
There is no energy at the original frequency $f$ , only energy at $f'$ at different angles.

Experimental Results

The Compton experiment favors the photon theory.

Equations Used

Conservation of momentum in the x-direction:
- $p{ix} + p{fx} = \frac{hf}{c} = \frac{hf'}{c} \cos(\theta) + \gamma m0 vf \cos(\alpha)$
Conservation of momentum in the y-direction:
- $0 = -\frac{hf'}{c} \sin(\theta) + \gamma m0 vf \sin(\alpha)$
Conservation of energy:
- $hf + m0 c^2 = hf' + \gamma m0 c^2$
The unknowns are: the angle $\beta$ , the final velocity of the electron $v_f$ , and $f'$ .
So we have three unknowns.
Final Formula
- $\Delta \lambda = \lambda' - \lambda = \frac{h}{m_e c} (1 - \cos(\theta))$
- Final frequency
- $\frac{c}{f'} - \frac{c}{f} = expression$
Two interesting points to see in the final solution:
- If $\theta = 0$ :
  - A glancing blow
  - Barely interacts with the electrons
  - There is almost no change
- If $\theta = 180$ :
  - Direct collision with the electron
  - Maximum energy loss

Blue Shift

A photon falling into a planet's gravity gains kinetic energy, meaning its frequency shifts towards the blue end of the spectrum.

Energy Gain

$\Delta E = hf{final} - hf{initial}.$