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Top Ten Most Beautiful Equations in Physics
Physics can be described simply as the study of our universe and an equation as a piece of maths relating physical quantities e.g. mass, energy, temperature. The rules of our universe, technically speaking physical laws, are almost all written down in the form of equations. The concept of relating the artistic (and subjective) idea of beauty to these mathematical statements may at first seem strange and unnecessary. However, to many physicists the concept is not merely a side effect of their theories but it is intrinsic to a good theory.
Paul Dirac - "A physical law must possess mathematical beauty."
What makes an equation beautiful? This moves away from the empirical fact of whether the equation works, whether it predicts experimental data, to something more personal and subjective. In my opinion there are three criteria to consider: aesthetics, simplicity and significance. The aesthetics is simply whether it looks good when written down. Simplicity is a lack of complicated structure in the equation. The significance of the equation is more of a measure of history, both what it solved and what it lead to in future scientific advancements. Below are my top ten equations (not in any particular order).
1 - Einstein's energy-mass equivalence
A consequence of Albert Einstein's theory of special relativity and the most famous equation in physics. This equation states that mass (m) and energy (E) are equivalent. The relation is very simple, only involving multiplication of mass by a very large number (c is the speed of light). Specifically, this equation first showed that even mass not in motion has an intrinsic "rest" energy. It has since been used in nuclear and particle physics. The biggest impact of this equation and perhaps the event that secured its legacy was the development and subsequent use of atomic bombs at the end of WW2. These bombs horrifically demonstrated the extraction of a huge amount of energy from a tiny amount of mass.
2 - Newton's second law
One of the oldest physics equations, formulated by Sir Isaac Newton in his famous book Principia in 1687. It is the cornerstone of classical mechanics, which allows the motion of objects subjected to forces to be calculated. Force (F) is equivalent to mass (m) multiplied by the acceleration of the mass (a). The underline notation indicates a vector, which has both a direction and a magnitude. This equation is now the first to be learnt by every physics student due to it only requiring basic mathematical knowledge but at the same being very versatile. It has been applied to a huge number of problems from the motion of cars all the way up to the orbits of the planets around our sun. It was only usurped by the theory of quantum mechanics in the early 1900s.
3 - The Schrödinger equation(s)
Quantum mechanics was the biggest shake up in physics since Newton formulated the foundations of classical mechanics and the Schrödinger equation, formulated by Erwin Schrödinger in 1926, is the quantum analogue of Newton's 2nd law. The equation incorporates two key concepts of quantum mechanics: the wave function (ψ) and operators (anything with a hat over it) which operate on a wave function to extract information. The operator used here is the hamiltonian (H) and extracts the energy. There are two versions of this equation, depending on whether the wavefunction varies in time and space or just in space. Although quantum mechanics is a complicated topic, these equations are elegant enough to be appreciated without any knowledge. They are also a postulate of quantum mechanics, a theory which is one of the pillars of our modern electronic technology.
4 - Maxwell's laws
Maxwell's laws are a collection of four equations that were brought together and used to formulate a unified description of electricity and magnetism by scottish physicist James Clerk Maxwell in 1862. They were since refined, using calculus, into the most elegant form shown below or technically speaking in "differential form". The first equation relates the flow of electric field (E) to the charge density (ρ). The second law states that magnetic fields (B) have no monopoles. Whereas electric fields can have a source of positive or negative charge, such as an electron, magnetic fields always come with a north and south pole and hence there is no net "source". The last two equations show that a changing magnetic field creates an electric field and vice versa. Maxwell combined these equations into wave equations for electric and magnetic fields, with their propagation speed being equal to a constant value that was the same as the measured speed of light. This lead him to conclude that light is actually an electromagnetic wave. It would also inspire Einstein's theory of special relativity, which is based upon the speed of light being a constant. These consequences would be huge enough without the obvious fact that these equations led to understanding of electricity which laid the foundations for the digital revolution and the computer you're using to read this article.
5 - Second law of thermodynamics
Not an equality but an inequality, stating that the entropy (S) of our universe always increases. Entropy can be interpreted as a measure of disorder, hence the law can be stated as the disorder of the universe increasing. An alternative view of the law is heat only flows from hot to cold objects. As well as practical uses during the industrial revolution, when designing heat and steam engines, this law also has profound consequences for our universe. It allows the definition of an arrow of time. Imagine being shown a video clip of a mug being dropped and breaking. The initial state is a mug (ordered) and the final state is a collection of pieces (disordered). You would clearly be able to tell whether the video was being played forward of backward from the flow of entropy. This would also lead on to the big bang theory, with the universe getting hotter as you go into the past but also more ordered, leading towards the most ordered state at zeroth time; a singular point.
6 - The Wave equation
The wave equation is a 2nd order partial differentiation equation that describes the propagation of waves. It relates the change of propagation of the wave in time to the change of propagation in space and a factor of the wave speed (v) squared. This equation isn't as groundbreaking as others on this list but it is elegant and has been applied to things such as sound waves (instruments etc.), waves in fluids, light waves, quantum mechanics and general relativity.
7 - The Einstein field equations
Only fitting that that the greatest physicist has a second equation in this list and one arguably more important than his first. It gives the fundamental reason for gravity, mass curving spacetime (a four dimensional combination of 3D space and time). The equation actually hides 10 partial differential equations by using tensor notation (everything with indices is a tensor). The left hand side contains the Einstein tensor (G) which tells you the curvature of spacetime and this is related to the stress-energy tensor (T) which tells you the distribution of energy in the universe on the right hand side. A cosmological constant term (Λ) can be included in the equation to attribute for our expanding universe, although physicists are unsure of what is actually causing this expansion. This theory completely changed our understanding of the universe and has since been experimentally validated, a beautiful example being the bending of light around stars or planets.
8 - Heisenberg's uncertainty principle
Introduced by Werner Heisenberg in 1927, the uncertainty principle is a limit on quantum mechanics. It states that the more certain you are about a particle's momentum (P) the less certain you are about the particle's position (x) ie. momentum and position can never both be known exactly. A common misconception is that this effect is due to a problem with the measuring procedure. This is incorrect, it is a limit on accuracy fundamental to quantum mechanics. The right hand side involves Plank's constant (h) which is equal to a tiny value (a decimal with 33 zeros), which is why this effect isn't observed in our everyday, "classical", experience.
9 - Quantisation of radiation
A law initially introduced by Max Plank to solve a problem with black body radiation (specifically to do with efficient lightbulbs) that led to quantum theory. This law states that electromagnetic energy can only be emitted/absorbed in specific (quantised) amounts. This is now known to be due to electromagnetic radiation not being a continuous wave but actually many photons, "packets of light". The energy of a photon (E) is proportional to the frequency (f). At the time it was only a mathematical trick used by Plank to solve a frustrating problem and he both considered it unphysical and struggled with the implications. However, Einstein would link this concept to photons and this equation is now remembered as the birth of quantum theory.
10 - Boltzmann entropy
A key equation for statistical mechanics formulated by Ludwig Boltzmann. It relates the entropy of a macrostate (S) to the number of microstates corresponding to that macrostate (W). A microstate describes a system by specifying the properties of each particle, this involves microscopic properties such as particle momentum and particle position. A macrostate specifies collective properties of a group of particles, such as temperature, volume and pressure. The key thing here is that multiple different microstates can correspond to the same macrostate. Therefore, a simpler statement would be that the entropy is related to the arrangement of particles within the system (or the 'probability of the macrostate'). This equation can then be used to derive thermodynamic equations such as the ideal gas law.
Bonus - Feynman diagrams
Feynman diagrams are very simple pictorial representations of particle interactions. They can be appreciated superficially as a pretty picture of particle physics but do not underestimate them. Theoretical physicists use these diagrams as a key tool in complex calculations. There are rules to drawing a feynman diagram, a particular one to note is that any particle travelling backwards in time is an antiparticle (corresponding to a standard particle but with the opposite of its electrical charge). Feynman did win a noble prize for quantum electrodynamics and did lots of great work but perhaps his most well known legacy are his diagrams that every physics students learns to draw and study. Feynman even painted these diagrams all over his van.