## Wave

A wave can be thought of as a disturbance that travels from one location to another. Waves are generated when energy passes through a medium like air or water. Electricity creates tiny waves that move through conductors such as copper wires.

• Waves spread through a medium as atoms and molecules collide with each other, creating a domino effect.
• Waves have shared characteristics such as height (amplitude), peaks (crests), direction of movement, rate of oscillation (frequency), and distance between peaks (wavelength).
• Electromagnetic waves are generally invisible to the human eye. An exception to this is the visible spectrum, with wavelengths between approximately 400 and 700 nanometres. Beyond this range, whether the wavelengths are longer (as in radio and microwaves) or shorter (as in ultraviolet, X-rays, and gamma rays), our eyes cannot detect them.
• Although we cannot see most electromagnetic waves, we can perceive some of them in other ways. For instance, infrared waves are felt as heat, and electric current (which produces electromagnetic waves) can cause a buzzing sensation in a wire or cause electrocution.
• The wavelength of electromagnetic waves can vary greatly, from extremely long radio waves, sometimes measured in kilometres, to very short gamma rays, measured in picometres (there are a trillion picometres in a metre (1012) .
• The frequency of electromagnetic waves can also range from extremely low (1 cycle per second, known as 1 hertz) to extremely high, such as a quadrillion cycles per second, which equals 1015 hertz(1015).
• When you throw a stone into a pond, it creates a series of ripples, or waves, that propagate outward in concentric circles until they encounter obstacles.
• Out in the world, waves are generated when forces such as wind and tide disturb the water’s surface.
• The wavelength of a wave in water can be determined by measuring the distance from the crest of one wave to the crest of the next wave.
• The frequency of waves in water can be determined by counting the number of waves that reach their peak or crest at a specific point over a set period of time.
• The amplitude of a wave is often approximated by measuring half the vertical distance from the top of a wave (the crest) to the bottom of a wave (the trough). However, strictly speaking, it should be the distance from the undisturbed surface level of the water to the next crest or trough. This is an approximation because it assumes that waves are symmetrical and that the undisturbed water level is midway between the crest and the trough.
• The direction of travel of water waves can typically be easily determined by observing their movement.
• The energy carried by waves at the beach becomes evident when you experience their force while swimming, for instance, being toppled over by a wave.

## Wave function

In quantum mechanics, a wave function is a mathematical function that describes the quantum state of a physical system, such as a particle or a collection of particles.

• A wave function provides information about the probabilities of various possible states the system might be in. It depends on the coordinates of the particles in the system (for example, position or momentum). It calculates the probability of finding the system in a particular state.
• Wave functions are used to determine the probability of various outcomes in quantum experiments.
• A wave function, in the context of quantum mechanics, must encapsulate a wealth of information about a quantum system, including its possible states, probabilities, and how it evolves over time:
• Position and Momentum: The wave function must provide information about the possible positions and momenta of particles within the system. This information is crucial for predicting the outcomes of measurements.
• Superposition: It should be able to represent the idea that a quantum system can exist in a superposition of multiple possible states. This means that the system can simultaneously occupy different states with certain probabilities until observed.
• Probability Amplitudes: The wave function encodes probability amplitudes, which are complex numbers that determine the likelihood of finding the system in a particular state upon measurement.
• Time Evolution: It should be able to evolve over time, allowing for the prediction of how the system’s state will change over the course of time.
• Observable Properties: The wave function must account for the possible values of observable properties (such as energy, angular momentum, etc.) and their corresponding probabilities.
• Normalization: It must satisfy the condition of normalization, meaning that the total probability of finding the system in any possible state must equal 1.
• Boundary Conditions: For specific physical systems, the wave function must satisfy appropriate boundary conditions that reflect the constraints imposed on the system (e.g., within a finite box or in a specific potential field).
• Interference and Entanglement: It should be capable of describing interference effects between different states and, in the case of multiple particles, account for entanglement, where the states of particles become correlated.
• Wave Function Collapse: When a measurement is made, the wave function must be capable of undergoing a transition from a superposition of states to a single, definite state, in accordance with the process of wave function collapse.
• Completeness and Orthogonality: In certain mathematical formulations of quantum mechanics, wave functions must form a complete and orthogonal set to be used as a basis for representing quantum states.
##### Wave Function Collapse

Wave function collapse is a phenomenon in quantum mechanics where the act of making a measurement on a quantum system causes it to transition from a superposition of multiple possible states to a single, definite state.

• Prior to measurement, a quantum system can exist in a superposition of states, meaning it simultaneously occupies multiple possible states with different probabilities – these are described by the wave. However, when a measurement is made, the wave function collapses, and the system assumes one of the possible states with certainty.
• Wave function collapse illustrates the profound influence that observation has on the behaviour of quantum systems.
• In the context of quantum mechanics, “observation” refers to the act of making a measurement or carrying out an experiment to gain information about a quantum system. When we observe a quantum system, we are attempting to determine one of its properties, such as position, momentum, energy, etc.
• The interpretation of wave function collapse is a subject of ongoing debate among physicists, with various interpretations positing different explanations for the phenomenon.

Summary

## Wave-cycle

A wave-cycle refers to the path of a wave measured from any point through the course of a single oscillation to the same point on the next oscillation.

• Imagine a wave-cycle as a series of points marked on the path of the wave between one crest and the next.
• All electromagnetic waves share features such as crests, troughs, oscillations, wavelength, frequency, amplitude, direction of travel.
• Whilst a wave-cycle is the path from one point on a wave during a single oscillation to the same point on completion of that oscillation, wavelength is a measurement of the same phenomenon along the axis of the wave.

## Wave-fronts, diffraction & interference

###### Wavefronts

Parallel electromagnetic waves with a common point of origin, the same frequency and phase, and propagating through the same medium, produce an advancing wavefront perpendicular to their direction of travel.

• Lasers that form a pencil of light made of parallel rays produce waves with flat wavefronts.
• An electromagnetic wave with a  flat wavefront is called a plane wave.

Point sources emitting electromagnetic waves in all directions, at same frequency and phase, and propagating through the same medium, produce spherical wavefronts tangental to their origin.

• Diffraction describes the way light waves bend around the edges of an obstacle into regions that would otherwise be in shadow.
• An object or aperture that causes diffraction is treated as being the location of a secondary source of wave propagation.
• Diffraction causes a propagating wave to produce a distinctive pattern when it subsequently strikes a surface.
• Diffraction produces a circular pattern of concentric bands when a narrow beam of light passes through a small circular aperture.
• In classical physics, the diffraction of electromagnetic waves is described by treating each point in a propagating wavefront as an individual spherical wavelet.
• As each wavelet encounters the edge of an obstacle it bends independently of every other. However, interference between wavelets alters the angle to which they bend and the distance they must travel before striking a surface.
• The explanations that best describe the process of diffraction belong to Wave Theory and are the result of two centuries of study in the field of optics.
• In modern quantum mechanics, diffusion is explained by referring to the wave function and probability distribution of each photon of light when it encounters the corner of an obstacle or the edge of an aperture.
• A wave function is a mathematical description concerning the probable distribution of outcomes of every possible measurement of a photon’s behaviour.

## Wave-particle duality

Wave-particle duality is a fundamental concept in quantum mechanics that describes the dual nature of particles, which can exhibit both wave-like and particle-like behaviour.

• Wave-particle duality refers to the phenomenon where entities like light can exhibit characteristics of both waves and particles.
•  Electromagnetic radiation, including light, is often described using wave properties. However, when it interacts with matter, it behaves like particles.
•  A photon is a quantum of electromagnetic radiation and represents the smallest discrete amount of light energy.
• When a photon is absorbed by matter, the energy becomes localized at specific points. This phenomenon is termed ‘wave function collapse.’ It describes the transition of a quantum system from a superposition of states to a definite state upon measurement.
• Wave-particle duality is a fundamental aspect of quantum mechanics and applies to all particles, not just light. Particles like electrons also exhibit wave-like and particle-like behaviour.
• The double-slit experiment is an experiment in quantum physics that demonstrates the wave-like behaviour of particles, including photons and electrons, and is a key illustration of wave-particle duality.

Concepts used to describe light as waves and particles can be paired to illustrate the wave-particle character of light

##### Explanation
FrequencyEnergyFrequency is related to the energy of a photon. Higher frequency light corresponds to higher energy photons. This is described by Planck's equation E = hf, where E is energy, h is Planck's constant, and f is frequency.
WavelengthMomentumThe wavelength of a wave is inversely proportional to its momentum. This is described by the de Broglie wavelength formula λ = h/p, where λ is wavelength, h is Planck's constant, and p is momentum.
Wave speedMomentumThe speed of a wave is related to its momentum through its frequency and wavelength. For light, the speed of propagation (c, the speed of light) is equal to its frequency times wavelength (c = λf), which is also related to its momentum as described above.
AmplitudeIntensityThe amplitude of a wave determines its intensity, which is related to the brightness or energy density of light. Higher amplitude corresponds to higher intensity.
PeriodLifetimeThe period of a wave is the time it takes for one complete cycle. In the context of light particles (photons), their "lifetime" refers to the duration of their existence before they are absorbed or undergo a different process.
PhaseQuantum statePhase refers to the position of a point in a wave cycle, and it can be related to the quantum state of a particle. In quantum mechanics, a particle's state is described by a complex-valued wavefunction, and the phase of this wavefunction is significant in determining probabilities and interference patterns.

These pairings are all based on the wave-particle duality of matter, which states that all objects have both wave-like and particle-like properties.

The specific pairings in the table can be derived from the following equations:

• E = hf (Planck’s equation describes the relationship between the energy of a photon and its frequency.)
• p = h/λ (de Broglie wavelength equation describes the relationship between the momentum of a particle and its wavelength.)
• p = mv (momentum equation describes the relationship between the momentum of a particle and its mass and velocity.)
• I = 1/2 cε_0 E^2 (intensity equation describes the relationship between the intensity of an electromagnetic wave and its electric field.)

where:

• E is energy
• f is frequency
• p is momentum
• m is mass
• v is velocity
• λ is wavelength
• I is intensity
• c is the speed of light
• ε_0 is the permittivity of free space. The permittivity of free space, also known as the electric constant, is a fundamental constant of nature. It is a measure of how easily an electric field can penetrate a vacuum.

The equation E = hf tells us that the energy of a particle is proportional to its frequency. This means that the higher the frequency of a particle, the more energy it has. This is why high-frequency radiation, such as gamma rays and X-rays, is so dangerous.

The equation p = h/λ tells us that the momentum of a particle is inversely proportional to its wavelength. This means that the shorter the wavelength of a particle, the more momentum it has. This is why high-energy particles, such as protons and electrons, can be used to accelerate other particles to high speeds.

## Wavefront

Electromagnetic waves that are parallel, share a common starting point, have the same frequency and phase, and move through the same medium, form an advancing wavefront at right angles to their direction of travel

• Sources that emit light in all directions, known as point sources, generate spherical wavefronts.
• Lasers, which produce a narrow beam of parallel rays, create waves with flat wavefronts.
• An electromagnetic wave with a flat wavefront is known as a plane wave.
• In addition to plane waves and spherical waves, there are also cylindrical waves which are produced when a point source is extended along a straight line.
• The shape of the wavefront can be influenced by interaction with different mediums or obstacles, such as when:
• Refraction causes the path of light to bend as it crosses the boundary between two transparent media.
• Dispersion causes light to separate into its constituent wavelengths, each of which bends to a different degree as it crosses the boundary between two transparent media.
• Diffraction causes light to bend around the edges of obstacles into regions that would otherwise be in shadow.
• The concept of wavefronts applies not only to light but also to other types of waves such as sound waves or water waves.

## Wavelength

Wavelength is a measurement from any point on the path of a wave to the same point on its next oscillation. The measurement is made parallel to the centre-line of the wave.

• The wavelength of an electromagnetic wave is measured in metres.
• Each type of electromagnetic radiation, such as radio waves, visible light and gamma waves,  forms a band of wavelengths on the electromagnetic spectrum.
• The visible part of the electromagnetic spectrum is composed of the range of wavelengths that correspond with all the different colours we see in the world.
• Human beings don’t see wavelengths of visible light, but they do see the spectral colours that correspond with each wavelength and the other colours produced when different wavelengths are combined.
• The wavelength of visible light is measured in nanometres. There are 1,000,000,000 nanometres to a metre.

## Wavelength

Wavelength is the distance from any point on a wave to the corresponding point on the next wave. This measurement is taken along the middle line of the wave.

• While wavelength can be measured from any point on a wave, it is often simplest to measure from the peak of one wave to the peak of the next, or from the bottom of one trough to the bottom of the next, ensuring the measurement covers a whole wave cycle.
• The wavelength of an electromagnetic wave is usually given in metres.
• The wavelength of visible light is typically measured in nanometres, with 1,000,000,000 nanometres making up a metre.
• Each type of electromagnetic radiation – such as radio waves, visible light, and gamma waves – corresponds to a specific range of wavelengths on the electromagnetic spectrum.
• As energy increases, frequency also increases and wavelength decreases. Therefore, shorter wavelengths correspond to higher energy levels, and longer wavelengths correspond to lower energy levels.
##### Wavelength & the Visible Spectrum
• The visible part of the electromagnetic spectrum consists of a range of wavelengths that correspond with all the different colours we see in the world.
• The visible spectrum, which spans wavelengths from approximately 400 to 700 nanometres, is commonly described in terms of bands of colour—red, orange, yellow, green, blue, indigo, and violet. However, these divisions are somewhat arbitrary since the spectrum is continuous, and the exact wavelength boundaries between colours is subjective.
• Humans don’t see the wavelengths of visible light directly but do see the specific colour associated with each wavelength and the colour produced when different wavelengths mix.
• The visible spectrum encompasses all the colours from red to violet, with each colour corresponding to a single light wavelength.
• The concept of colour is a perceptual property, not a property of light itself. It’s how our brain interprets the wavelengths of light that reach our eyes.
• The perceived colour (hue) of a light stimulus is dependent on its wavelength.
• A colour produced by a single wavelength is known as a pure spectral colour.
##### Wavelength & Colour
• Light that we perceive as white when it is reflected towards an observer contains a mix of different wavelengths of light.
• Light that contains wavelengths corresponding with red, green and blue appear white when it is reflected off a neutral-coloured surface.
• Light is seldom made up of a single wavelength. It is typically a mix of many different wavelengths that together determine the colour an observer sees as they reflect off a surface.
• The greater the number of different wavelengths incident light contains, the lower the saturation of the reflected colour. So a light comprising mixed wavelengths tends to produce lighter (whiter) colours.
• Wavelengths matching the colours of the visible spectrum are typically measured in nanometres. Hence, there are 300 distinct colours between 400 nanometres (violet) and 700 nanometres (red). If picometres are the unit used to measure wavelength then there are 300,000 different wavelengths in the visible spectrum, each corresponding to a unique colour.
###### References

Wavelength is a measurement from any point on the path of a wave to the same point on its next oscillation. The measurement is made parallel to the centre-line of the wave.

• The wavelength of an electromagnetic wave is measured in metres.
• Each type of electromagnetic radiation, such as radio waves, visible light and gamma waves,  forms a band of wavelengths on the electromagnetic spectrum.
• The visible part of the electromagnetic spectrum is composed of the range of wavelengths that correspond with all the different colours we see in the world.
• Human beings don’t see wavelengths of visible light, but they do see the spectral colours that correspond with each wavelength and the other colours produced when different wavelengths are combined.
• The wavelength of visible light is measured in nanometres. There are 1,000,000,000 nanometres to a metre.

## Waves in water

• When you throw a stone into a pond, it creates a series of ripples, or waves, that propagate outward in concentric circles until they encounter obstacles.
• Out in the world, waves are generated when forces such as wind and tide disturb the water’s surface.
• The wavelength of a wave in water can be determined by measuring the distance from the crest of one wave to the crest of the next wave.
• The frequency of waves in water can be determined by counting the number of waves that reach their peak or crest at a specific point over a set period of time.
• The amplitude of a wave is often approximated by measuring half the vertical distance from the top of a wave (the crest) to the bottom of a wave (the trough). However, strictly speaking, it should be the distance from the undisturbed surface level of the water to the next crest or trough. This is an approximation because it assumes that waves are symmetrical and that the undisturbed water level is midway between the crest and the trough.
• The direction of travel of water waves can typically be easily determined by observing their movement.
• The energy carried by waves at the beach becomes evident when you experience their force while swimming, for instance, being toppled over by a wave.

## Weak Nuclear force

The weak nuclear force is one of the four fundamental forces in nature. The other forces are the electromagnetic force, the strong nuclear force and gravity.

• The weak nuclear force was responsible for the creation of elements in the early universe including hydrogen, helium, and lithium. Today it plays a key role in the nuclear fusion reactions that power the sun and other stars.
• The weak nuclear force is responsible for the decay of radioactive isotopes, as well as for other nuclear reactions such as beta decay and neutrino interactions.
• When unstable radioactive isotopes decay, they emit radiation and transform into a more stable elements.
• In beta decay, a neutron in the nucleus of an atom decays into a proton, an electron, and an antineutrino.
• neutrino interactions occur in nuclear reactors. Neutrinos are very light particles that rarely interact with matter, but they can interact with the nuclei of atoms through the weak nuclear force.
• The weak nuclear force is a very weak, but it has a long range. It is also very selective, interacting only with certain types of subatomic particles.
• The weak nuclear force is mediated by a family of particles called W and Z bosons.

Summary

## White light

White light is the name given to visible light that contains all wavelengths of the visible spectrum at equal intensities.

• As light travels through a vacuum or a medium it is described as white light if it contains all the wavelengths of visible light.
• As light travels through the air it is invisible to our eyes.
• When we look around we see through the air because it is very transparent and light passes through it.
• The term white light doesn’t mean light is white as it travels through the air.
• One situation in which light becomes visible is when it reflects off the surface of an object.
• When white light strikes a neutral coloured object and all wavelengths are reflected then it appears white to an observer.

## White light

White light is the term for visible light that contains all wavelengths of the visible spectrum at equal intensities.

• The sun emits white light because sunlight contains all the wavelengths of the visible spectrum in equal proportions.
• Light travelling through a vacuum or a medium is termed white light if it includes all wavelengths of visible light.
• Light travelling through a vacuum or air isn’t visible to our eyes unless it interacts with something.
• The term white light can have two meanings:
• It can refer to a combination of all wavelengths of visible light travelling through space, regardless of observation.
• What a person sees when all colours of the visible spectrum hit a white or neutral-coloured surface.
• The human eye sees white when the wavelengths of light associated with the three primary colours – red, green, and blue (RGB) – are projected onto an achromatic surface.
• White light appears as different colours to an observer when some wavelengths of light are reflected by an object’s surface while others are absorbed.
• Artificial light sources usually emit light with a varied distribution of wavelengths or intensities, so they don’t typically emit white light.
• While there isn’t a single, distinct definition for white light, it’s fair to say that in any given situation:
• Colour constancy, the ability to perceive colours as relatively constant, even under changing lighting conditions, enables human observers to see very different whites as appearing the same.
##### Why do Light Bulbs Glow if Light is Invisible?
• Incandescent light bulbs pass an electrical current through a fine tungsten filament with high electrical resistance.
• The resistance causes the filament’s electrons to heat up, giving off a bright yellowish-white colour. This colour is produced by:
• Besides these observed properties, a filament, along with other types of light sources, may emit wavelengths of light:
• That are outside the visible region of the electromagnetic spectrum.
• Travel from source to the retina without encountering surfaces or objects (however large or small), so remain completely invisible to an observer.
• The light emitted by a tungsten bulb spans wavelengths between 200 and 3000 nanometres.
###### References

White light is the name given to visible light that contains all wavelengths of the visible spectrum at equal intensities.

• As light travels through a vacuum or a medium it is described as white light if it contains all the wavelengths of visible light.
• As light travels through the air it is invisible to our eyes.
• When we look around we see through the air because it is very transparent and light passes through it.
• The term white light doesn’t mean light is white as it travels through the air.
• One situation in which light becomes visible is when it reflects off the surface of an object.
• When white light strikes a neutral coloured object and all wavelengths are reflected then it appears white to an observer.

## Why the sky is blue

Perhaps the most common of atmospheric effects, the blueness of the sky, is caused by the way sunlight is scattered by tiny particles of gas and dust as it travels through the atmosphere.

The sky is blue because more photons corresponding with blue reach an observer than any other colour.

In outer space, the Sun forms a blinding disk of white light set against a completely black sky. The only other light is produced by stars and planets (etc.) that appear as precise white dots against a black background. The sharpness of each of these distant objects results from the fact that photons travel through the vacuum of space in straight lines from their source to an observer’s eyes. In the absence of gas and dust, there is nothing to scatter or diffuse light into different colours and no surfaces for it to mirror or reflect off.

All of this changes when sunlight enters the atmosphere. Here, the majority of photons do not travel in straight lines because the air is formed of gases, vapours and dust and each and every particle represents a tiny obstacle that refracts and reflects light. Each time a photon encounters an obstacle both its speed and direction of travel change resulting in dispersion and scattering. The outcome is that, from horizon to horizon, the sky is full of light travelling in every possible direction and it reaches an observer from every corner.

The following factors help to account for why blue photons reach an observer in the greatest numbers:

• The sky around the Sun is intensely white in colour because vast numbers of photons of all wavelengths make the journey from Sun to an observer in an almost straight line.
• In every other area of the sky, light has to bend towards an observer if they are to see colour. It is this scattering of light that fills the sky with diffuse light throughout the day.
• Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water in the atmosphere so scatter the least so few are redirected towards an observer.
• Shorter wavelengths (blue and violet) are just the right size to interact with obstacles in the atmosphere. These collisions scatter light in every possible direction including towards an observer.
• Because blue is relative intense compared with violet in normal conditions and in the absence of the longer wavelengths the sky appears blue.
• However, there is a whole band of wavelengths corresponding with what we simply call blue. As a result, different atmospheric conditions fill the sky with an enormous variety of distinctly different blues during the course of the day.

## Why the sky is sometimes red

If we understand why the sky is usually blue it’s easier to understand why it can be filled with reds and pinks at sunrise and sunset.

###### Let’s review why the sky is blue
• In most weather conditions, the Sun and the area around it appear intensely white to an observer because vast numbers of photons of every wavelength make the journey from Sun to their eyes in an almost straight line.
• The Sun, and the area around it, appears white because it contains a mixture of all wavelengths of light (white light).
• In every other area of the sky, sunlight is striking billions of particles that make up the atmosphere and scattering in every possible direction.
• If it were not for this scattering (deflection of light in all directions), the sky would be as black as night. In reality, an observer is bathed in light arriving from every direction and the sky, as a result, appears to be full of diffuse light.
• Not all wavelengths of light behave in the same way when scattered by the small particles that make up the atmosphere.
• Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water so scatter the least.
• Shorter wavelengths (blue and violet) are just the right size and are affected by reflection, refraction and scattering as they strike successions of particles. It is these collisions that direct light in every possible direction including towards an observer.
• Because human eyes are more sensitive to blue than violet, in most atmospheric conditions, and in the absence of the longer wavelengths, the sky appears blue.
• A wide band of wavelengths corresponds with what we often describe as blue. As a result, the sky is filled with an enormous variety of distinctly different blues during the course of every day.
###### Why the sky is sometimes red
• A red sky suggests an atmosphere loaded with dust or moisture and that the Sun is near the horizon.
• In the morning and evening, photons must travel much further through the atmosphere than at mid-day.
• Assuming the air above our heads is around 20 km, the total distance light travels increases fivefold to around 500 km when the Sun is on the horizon.
• Remember that:
• Longer wavelengths of light (red, yellow, orange and green) are too big to be affected by tiny molecules of dust and water so scatter the least.
• Shorter wavelengths (blue and violet) are just the right size and are affected by reflection, refraction and scattering as they strike successions of particles.
• In the right weather conditions, light travelling horizontally through the atmosphere undergoes so much scattering that no yellow, green, blue or violet wavelengths remain.
• In these conditions, the light that reaches us, illuminating the sky and clouds and reflecting off every surface around us, is composed of wavelengths that bath the world in red and orange.