Quantum Behavior: Wave-Particle Duality

Quanta, such as photons and electrons, exhibit both wave-like and particle-like behaviors.

A quantum's momentum can be calculated through its wavelength: $p = \frac{h}{\lambda}$ where:
- $p$ is the momentum.
- $h$ is Planck's constant.
- $\lambda$ is the wavelength.
If you know a quantum's momentum and Planck's constant, you can solve for its wavelength.
This relationship applies to all quanta.

Photons, despite being massless, have momentum.
Experimental evidence: photons and electrons have been observed colliding, and momentum is conserved during these collisions.
The momentum of a photon can be calculated using its wavelength:
$p = \frac{h}{\lambda}$
Massless quanta such as photons have unique equations for momentum: $p = \frac{E}{c}$ where:
- $E$ is the energy of the photon.
- $c$ is the speed of light.
The momentum of a photon can also be expressed as:
$p = \frac{hf}{c}$
where:
* $h$ is Planck's constant
* $f$ is the photon's frequency

Matter, including particles with mass like electrons, can behave like waves.
These are called matter waves.
This concept was originally proposed by Louis de Broglie and later supported by experiments.
The momentum of all quanta can be found with the equation:
$p = \frac{h}{\lambda}$
Which can be rearranged as:
$\lambda = \frac{h}{p}$

Waves can interfere with one another because different waves can exist at the same spatial coordinates and the same time coordinates.
Constructive interference: Reinforces the wave and builds it up.
Destructive interference: Cancels the resulting wave and reduces its amplitude.

Superposition describes how two or more waves at the same point combine by algebraically adding the displacements described by the amplitudes.
If a peak and trough coincide, the waves cancel each other at that point.

Thomas Young's experiment revived the wave hypothesis of light.
Light is diffracted through a single slit to make it coherent (same wavelength and phase).
Coherent light is then passed through two narrow slits, forming two spots of light on a screen.
As the size of the slits and distance between them became smaller, the two spots of light overlapped, producing bright and dark bands.
This showed that light phenomena can be described by a wave model.

Photons, known for particle-like behavior, can interfere with one another, demonstrating wave-like behavior.
Electrons also exhibit interference, indicating their wave-like nature.
Interference is a key characteristic of wave-like behavior in quanta.

Diffraction is the bending of waves through openings and/or around edges.
When quanta are sent through slits relative to their wavelengths, an interference pattern results.
This pattern is based on diffraction and constructive/destructive interference, which are wave-like behaviors.
The larger the particle, the smaller the slit size needs to be for diffraction to be noticeable.
Electron diffraction demonstrates the wave-like nature of electrons.
Light diffraction demonstrates the wave-like nature of photons.

The square of the particle's wave function is proportional to the probability of finding the particle at a given location.
This explains why electrons are confined in certain regions around the nucleus.
Electrons can escape through potential barriers in a process called tunneling.
Reflection can be explained by either wave-like or particle-like nature.

Wave functions, like light or sound waves, can be superimposed.
A wave function describes the probability of an electron's speed or position.
Many quantum states are available to the electron.
When the wave interacts with the environment (e.g., measurement), the wave function collapses to one value.

When light passes through a vertical filter, half of the intensity is removed.
Vertically polarized light will then pass through a second vertical filter.
Light is in a superposition of states, so some light will pass through both filters until the angle between them is $90^\circ$ .

Particles behave as waves, and waves behave as particles because all particles act like waves.
There is evidence that quanta such as electrons and photons behave as waves and there is evidence that they behave as particles.
Understanding this duality allows for a deeper understanding of many phenomena in the universe.