Planck Units, a.k.a. Natural Units are a set of Units of Measurement defined solely in terms of four Universal Natural Constants. They are established by setting the Speed of Light c, the Gravitational constant G, Planck's constant ℏ, and the Boltzmann constant k_B equal to ONE UNIT. This is the ultimate Natural World, non-arbitrary, measurement system. For those who study the quantum world it greatly simplifies calculating solutions to problems.

The Planck Length (L_p) is the Smallest Measurement of Length that has any real meaning in our universe. It is about 10^-20 the size of a proton. It derives directly from Planck's Constant and tells us about the actual 'fabric' of our universe. Imagine our universe as an enormous TV screen with the Planck Length representing the size of each pixel. What is happening BETWEEN the pixels has no meaning.

Planck Time (t_p) is the Smallest Measurement of Time. Planck Time is defined as the time required for a Photon moving at the Speed of Light (c) to cross a distance of one Planck Length.

Since this is the Smallest Division of Time that has ANY meaning, we can only say that our universe came into existence when its age was One Planck Duration (10^-43 seconds) after the Big Bang.

The Planck Temperature (T_p) is the Highest Temperature theoretically possible for matter, most likely occurring at One Planck Duration (10^-43 seconds) after the Big Bang singularity.

1900 - Planck's Constant
Karl Ernst Ludwig Max Planck
Planck proposed that energy (e), could only exist in discrete packets known as Quanta (plural of Quantum, from the Latin, Quantus, meaning 'how much'). These Quanta were proportional to the frequency (v) of light by a universal constant he called h, so…

This solved a serious nagging problem known as the Ultraviolet Catastrophe. By using his mathematical solution, now known as Planck's Law of Blackbody Radiation, he solved the problem completely, but he did not understand why it should work. His clever mathematical solution however turned out to be the Cornerstone of all of Quantum Mechanics. By using Planck's Constant & the concept of Quanta, everything began to fall into place.

When frequency is expressed in radians per second (angular frequency) instead of cycles per second, it is useful to incorporate 2π into the Planck Constant. The result is called the Reduced Planck Constant or H-Bar, written as ℏ. It is a very, very, small number equal to:

1905 - Special Relativity
Albert Einstein
Einstein proposed two basic principles that apply for all Inertial (Non-accelerating) systems.

1. The speed of light (c) is the same for all observers.
2. The Laws of Physics remain constant in all such systems.

From these two basic principles it directly follows that all events in the universe (spacetime) must be considered relative to each other. Events are related to each other by a factor known as The Lorentz Transform Factor γ = 1/√(1- v²⁄c²). Special Relativity did away with any universal or absolute reference system. Also from this Principle can be derived the famous equivalence of matter to energy:

Speed of Light in a Vacuum:     c = 299,792,458 m ⁄ s
The Lorentz Transform Factor:     γ = 1 ⁄ √(1- v²⁄c²)

1913 - Atomic Structure
Niels Henrik David Bohr
Bohr proposed the theory that describes the postion of electrons in an atom. Since the nucleus of an atom is massive, its position is well defined, however since the mass of an electron is so small, its position must be defined by an 'Electron Cloud' of probability. These Probabilistic Regions or 'Clouds' exist in different energy levels defined by the first Quantum Number N and also take on different shapes as defined by their 2nd Quantum Number L. In addition, each Orbital State has Sub-Orbital states s, p, d, f (named by Bohr as Sharp, Principal, Diffuse, and Fundamental). Some examples are:

1915 - Equivalence Principle → General Theory of Relativity
Albert Einstein
Einstein postulated that the effects of gravity are indistinguishable from the effects of acceleration. His 'Equivalency Thought Experiment' explained his reasoning. The forces acting upon someone who is in a Space Ship in deep space that is accelerated constantly is Equivalent to the forces acting upon someone in a similar stationary box on Earth, which is only being acting upon by gravity. Therefore: Gravity ≡ Acceleration.

In the Space-Ship light always bends down towards the accelerating force. This leads to the conclusion that light must also bend in any gravitational field. BRILLIANT ! This in turn leads to the conclusion that space must be curved in any gravitational field since light always takes the shortest path. These brilliant conclusions lead to Einstein developing his General Theory of Relativity. As a test, the theory correctly predicted the apparent shift of the planet Mercury during the Total Solar Eclipse of 1919.

1924 - Wave ⁄ Particle Duality of Matter
Louis de Broglie
In his 1924 PhD thesis Louis de Broglie hypothesized that all matter has wave properties. He postulated that electrons zipping around a nucleus should be thought of as waves instead of particles. The electrons as waves certainly cannot destructively interfere with themselves or they would quickly self-annihilate. Therefore, each electron's frequency (λ = h ⁄ mv) could only be allowed at unique quantized orbits. This reinforced electron shell theory and for this he won the Nobel Prize for Physics in 1929.

Louis de Broglie's theory can also be applied to the 'Particle In A Box' thought experiment. Consider a single sub-atomic particle bouncing about inside a box. Ever so slowly we reduce the size of the box. As the box becomes smaller and smaller we know the position of the particle with ever more precision. Classically, if we reduce the size of the box down to the size of the particle itself, then we would know its exact position. This should work, right? BUT NO! Since the Particle has BOTH Particle AND Wave Descriptions, as the box gets smaller the Particle's Wave component becomes more and more important. When the box reaches a certain very small limit the Particle's wave description actually extends 'beyond the box' and the particle can no longer be considered as being 'inside the box'. It appears to have tunneled through the barriers and is suddenly outside the box. 1925 - Pauli Exclusion Principle
Wolfgang Ernst Pauli
For any of the electrons in an atom their quantum numbers must be different. This prevents any two electrons from simultaneously having the same quantum numbers N, L, & M and helps explain how electrons occupy certain probabilistic regions around the nucleus.
1925 - Probability Wave Equation
Erwin Rudolf Josef Alexander Schrödinger
Schrödinger derived a partial differential equation using the variables of space, momentum, particle spin, and time that describes how any system changes with time. It accurately predicts the Probability Amplitude of the position and momentum for sub-atomic as well as macro systems. This was a major leap forward towards understanding the effect known as Quantum Tunneling. This suggested that particles could sometimes be thought of as Wave Packets, spread out over SpaceTime. The Schrödinger Equation (general case):

1927 - Heisenberg Uncertainty Principle and The Ideal Oscillator
Werner Karl Heisenberg
Heisenberg proposed that the POSITION and MOMENTUM of any particle are COMPLEMENTARY properties. The more you pinpoint the POSITION, the less you know about the MOMENTUM, and vice-versa. Their product can only be more than a very small number (Planck's Constant). This is a Basic Law of Physics and is independent of the ability of ANY aparatus to measure. This Fundamental Principle is used to accurately explain many quantum effects and forces us to use Laws of Probabilities at sub-atomic levels. Among the possibilities introduced was Quantum Tunneling ⬇.

Heisenberg also brilliantly suggested discarding the electron orbital description of the atom, and instead treating the entire Atom as an Ideal Oscillator - without visualizing the exact internal mechanics and just focusing upon measureable quantities. As such it's Total Energy H would oscillate as a function of the Observables A with respect to time t, according to the following formula:

Where H is the Hamiltonian (an operator representing the total energy of a system), and A are Observables (physical properties that can be measured).

1928 - The Dirac Equation
Paul Adrien Maurice Dirac
Arguably the Most Important Physicist of the 20th Century and probably the least known even though he was easily on a par with Sir Isaac Newton. Dirac derived an equation that was both Mathematically beautiful and powerful beyond even his dreams. He worked on deriving the equation for 3 full years, virtually in seclusion, finally presenting it in 1928. It seamlessly combined all the aspects of Quantum Physics with Einstein's Theory of General Relativity and among other things Predicted the Existence of Anti-Matter a full 4 years before it was discovered. Each term in the Dirac Equation is just the tip of a Mathematical Iceberg.

1935 - Schrödinger's Cat
Erwin Rudolf Josef Alexander Schrödinger
Schrödinger's Cat is a thought experiment, first presented in 1935, to help scientists understand in our everyday world what was happening in the odd quantum world. In this experiment a live cat is enclosed in a sealed box with no windows for one hour. Inside the box is a vial of poison and an apparatus to break the vial, triggered by the decay of a small amount of radioactive material with a half-life of one hour. The vial will be broken at some random time killing the cat. Schrödinger suggested that until we open the box and actually LOOK the Copenhagen Interpretation of quantum mechanics implies that the cat is BOTH alive and dead. It's fate is ONLY known when we look, and the quantum superposition collapses into only one possibility.

1948 - Feynman Diagrams
Richard Phillips Feynman
Feynman was a brilliant theoretical physicist who developed a theory of sub-atomic particle interactions that came to be known as Quantum ElectroDynamics (QED). In 1948 he presented his theory to the world's top physicists through the use of his special graphics known as Feynman Diagrams that boiled down the complex mathematics into simple drawings.

Time is shown along the bottom axis. Antiparticles are shown as moving backwards through time. i.e. a Positron p⁺ is an Electron e^- moving backwards in time.

Shown above is an Electron e^- meeting it's antiparticle the Positron p⁺. Their mutual annihilation generates a Electron e^-Photon (gamma wave 𝛾 ), which soon decays into a Quark (q) and an Anti-Quark (q̄) pair. The Anti-Quark (q̄) then expels a Gluon (g) which is a Force carrier.

1955 - Quantum Fluctuations (Virtual Particles)
John Archibald Wheeler
The Energy (E) and also the Time (t) accuracy of any system is limited by Plank's constant.
Wheeler postulated that spacetime was very turbulent on small scales below the Planck Length. At these extremely small scales he suggested that Energy could be 'borrowed' from the universe and then almost instantly 'repaid'. Doing so would briefly generate particles and anti-particles, that would almost instantly self-annihilate. They have also been given the moniker, "Virtual Particles". The time (t) duration would be extremely short. The shorter the time scale the more energetic can be the particle pairs. Since Einstein's General Theory of Relativity states that energy can curve spacetime, on these extremely small scales, below the Planck Length, space makes a significant departure from its smooth macro appearance resulting in what he called a 'Quantum Foam' seething with activity. This process is impossible to observe directly, but its effects can linger long enough to be detected. Please see The Casimir effect

1958 - Black Holes
David Ritz Finkelstein
Einstein's General Releativity predicts that mass curves spacetime. Taken to the extreme, there are sufficiently compact masses whose gravity is so strong that no particle or light ray entering the region can ever escape from it. The boundary of this region from which no escape is possible is called the Event Horizon. The Event Horizon's Radius was worked out by Karl Schwarzschild in 1916 purely from General Relativity, but it's interpretation was not suggested until 1958 by Finkelstein. The Schwarzschild Radius (R_s) is porportional to the Black Hole's total mass (M), the Universal Gravitational constant (G), and inversely by the speed of light (c) squared so:

Crossing the event horizon seals the fate of any object crossing it, but has no locally detectable features other than tidal gravitational effects. An object approaching an event horizon would appear to an outsider however, to slow down and gradually fade out to infra-red as it comes under the effects of Relativity. 1974 - Hawking Radiation
Stephen William Hawking
In 1974 Stephen Hawking provided a theoretical argument combining Einstein's General Theory of Relativity, the Schwarzschild Radius prediction, and Wheeler's Quantum Foam Theory. Hawking reasoned that at the event horizon of a Black Hole it might be possible for the randum fluctuations of spacetime to create pairs of objects, from which, one of the pair might proceed towards the Black Hole and the other one might proceed away from the Black Hole with enough energy to escape from the Event Horizon. These pairs would not be able to re-unite and Black Holes should therefore appear to radiate the extra particles and associated photons. This effect is known as Hawking Radiation.

Counter to intuition, the Black hole loses mass equivalent to the Energy of the Radiation, since e = mc². Over vast stretches of time the Black Hole would eventually radiate away all of it's mass and evaporate. I.E. if we stand back and view an individual Black Hole in deep space, then as it radiates energy its mass must decrease over time.

Newton's Second Law of Motion
The Force on an object is equal to its Mass (m) times its Acceleration (a)
F = ma

Kinetic Energy
Kinetic Energy (E_k) is equal to one-half the Mass (m) times the velocity squared (v²). It is a SCALAR quantity, i.e. it is a Quantity without a direction.
E_k = ½mv²

Potential Energy
Potential Energy (E_p) is the stored up energy of an object in a system with some kind of restoring force.
Formulas for E_p can take different forms depending upon the system(s) involved. Here are a few…

• Mechanical E_p (e.g. a drawn arrow in a bow, a coiled metal spring)
• Gravitational E_p (e.g. an apple in a tree)
• Magnetic E_p (e.g. a compass needle in earth's magnetic field)
• Nuclear E_p (e.g. the energy associated with sub-atomic particles in an atom)
• Electrostatic E_p (e.g. the force between two charged particles)
• Chemical E_p (e.g. a firecracker, gasoline)

Entropy
The Second Law of Thermodynamics states that in any CLOSED system the total Entropy will either remain constant or increase. Any CLOSED system will tend towards disorder and randomness and NOT towards orderliness. Entropy is a measure of a system's DISORDER. This has profound ramifications:

• Disorder can only increase or remain constant → Things decay.
• Heat Flows only from a hot object to a colder object → Things lose heat.
• Available Work from any system is always less than 100% efficient → No free energy.

Einstein's Mass-Energy Relationship
e = mc²

Heisenberg Uncertainty Principle
Δx Δp ≥ ℏ ⁄ 2
ΔE Δt ≥ ℏ ⁄ 2

Lorentz Transform Factor
1 ⁄ √1 - (v²/c²) 

Wheeler's Quantum Foam Prediction
from ΔE Δt ≥ ℏ , as Δt → 0 , ΔE → ∞   ∴     E ⇒ e^- + p⁺ ⇒ E

Planck's Constant
h = 6.626070150 x 10^-34 kg m² s^-1

Reduced Planck's Constant
ℏ = 1.054571596 x 10^-34 kg m² s^-1

Boltzmann Constant
k = 1.38964852 x 10^-23 kg m² s^-2 K^-1

Imaginary Number ( i )
i = √-1 
i² = -1

Euler's Formula
e ^{π i} = -1

Ratio of Diameter of a Circle to its Circumference ( π )
π = 3.14159265358…

Natural Logarithm - ( log_e )
ln (e) = -1
e = 2.718281828459045…

The Golden Ratio
φ - 1 = ¹⁄_φ
φ  = 1.6180339887… (Golden Ratio)
¹⁄_φ = .6180339887… (Inverse Golden Ratio)