This page contains a brief summary of all the important introductory topics in theoretical physics. Once you’ve picked a topic, click on it and have fun learning! Note that each of these topics use various tools from math (calculus, linear algebra, differential equations, complex analysis, etc.). Don’t worry — each physics topic will have a reference to the relevant math topics on the mathematics page. $\require{physics}$

Classical Mechanics

Classical mechanics is a broad topic. It has many formulations, including Newtonian Mechanics, Lagrangian Mechanics, and Hamiltonian Mechanics. Classical mechanics develops elegant equations of motion that describe the behaviour of objects.

\begin{equation*}
\bbox[5px, border: 1px solid #800020]{\vb{F}=\frac{d\vb{p}}{dt}} \qquad \textcolor{#800020}{\text{Newton’s Second Law}}
\end{equation*}

In the language of forces, Newton’s second law tells us that the net force acting on an object is equal to the rate of change of the object’s momentum. Essentially, forces can cause objects to accelerate (change velocity). There are four fundamental forces of the universe: gravity, electromagnetism, the weak interaction, and the strong interaction. The forces that we are able to perceive in our day-to-day lives are gravity and the electromagnetic force (responsible for magnets and the repulsion we feel when we touch things). Solving Newton’s second law (a second-order ordinary differential equation) gives us the position of the particle as a function of time.

\begin{equation*}
\bbox[5px, border: 1px solid #800020]{\pdv{L}{q}=\frac{d}{dt}\pdv{L}{\dot q}} \qquad \textcolor{#800020}{\text{Euler-Lagrange Equation}}
\end{equation*}

Lagrangian mechanics is a formulation of classical mechanics beginning from a completely different viewpoint compared to Newtonian mechanics. It starts with the Principle of Stationary Action to derive the Euler-Lagrange equation (a second-order partial differential equation). This equation is analogous to Newton’s second law, but cast in a much more general and powerful form.

\begin{equation*}
\bbox[5px, border: 1px solid #800020]{
\begin{alignedat}{1}
\dot q &= \pdv{H}{p} \\
\dot p &= -\pdv{H}{q}
\end{alignedat}
}
\qquad \textcolor{#800020}{\text{Hamilton’s Equations}}
\end{equation*}

Hamiltonian mechanics is yet another formulation of classical mechanics, producing Hamilton’s equations (a set of first-order partial differential equations). It is closely related to Lagrangian mechanics (through what is known as a Legendre Transform), but each have their own strengths and weaknesses depending on the situation. For example, sometimes it is more convenient to solve a single second-order differential equation (Euler-Lagrange equation), whereas other times it is easier to solve two first-order differential equations (Hamilton’s equations).

The success of these equations is hard to overstate. They can model any classical particle (including planetary orbits), they provide us with deep insights regarding the symmetries of the universe, and they even pop up in quantum mechanics. Classical mechanics provides us with foundational tools that we will continue to use in our discussions of all the other theories of physics. It is the reason why it triumphs at the top of this page.

Quantum Mechanics

Quantum mechanics has a reputation of being difficult, unintuitive, and weird. It began in the early 20$^{\text{th}}$ century as a revolution to our classical way of thinking. The theory puzzled many scientists (and continues to puzzle us today). Yet it lays the foundation of all modern theories of physics. The degree to which it accurately predicts the results of experiments is outstanding. The universe is fundamentally quantum mechanical.

Quantum mechanics begins with a few certain axioms and evolves into an abstract yet beautiful theory. One of the most important equations we will encounter is

\begin{equation*}
\bbox[5px, border: 1px solid #01796f]{
i\hbar \frac{d}{dt} \ket{\Psi} = H\ket{\Psi}
}
\qquad \textcolor{#01796f}{\text{Schrödinger Equation}}
\end{equation*}

The Schrödinger equation is yet another analogy to Newton’s second law. Solving it gives us the state $\ket{\Psi}$. The $H$ in this equation is known as the Hamiltonian (yes, the same thing that we encountered in Hamilton’s equations). Except, in this case, the $H$ is known as an operator. Mathematical operations involving states and operators give us the probabilities of potential outcomes when we measure a quantum particle’s location, momentum, energy, angular momentum, spin, and so on. This is one of the biggest distinctions between classical and quantum mechanics; classical mechanics always predicts outcomes with full certainty, whereas quantum mechanics may predict many potential outcomes, with different probabilities. This is not due to a flaw or imprecision in the theory, but because of how the universe truly behaves.

One of the first systems that we will solve is the hydrogen atom, which cannot be solved in classical mechanics. Studying quantum mechanics will give us a new lens through which to view the universe. It will also equip us with the tools we need to tackle Quantum Field Theory, which is often stated to be one of the two pillars of modern physics (the other pillar being General Relativity).

Electromagnetism

Electromagnetic theory helps us understand a phenomenon that cannot fully be captured by classical mechanics: light. It heavily dives into the concept of a field, namely the electric and magnetic fields. These fields obey the following four equations of motion

\begin{equation*}
\bbox[5px, border: 1px solid #0047ab]{
\begin{alignedat}{3}
\grad\cdot\vb{E} &= \rho \quad &, \quad &\curl{\vb{E}} = -\pdv{\vb{B}}{t} \\
\grad\cdot\vb{B} &= 0 \quad &, \quad &\curl{\vb{B}} = \vb{J} + \pdv{\vb{E}}{t} \\
\end{alignedat}
}
\qquad \textcolor{#0047ab}{\text{Maxwell’s equations}}
\end{equation*}

The top left equation takes the form it does because of the existence of electrical charge (positive and negative), known as a monopole. The bottom left equation, with the equality to zero, suggests that there are no magnetic charges (you can’t ever split a magnet into a positive and negative magnetic charge…you’ll simply be left with two smaller magnets). However, there are modern efforts to search for magnetic monopoles (charges) due to Quantum Field Theory. Lastly, just by looking at the two equations on the right, we can observe a deep property about electric and magnetic fields. The top right equation tells us that a changing magnetic field creates an electric field. The bottom right equations tells us that a changing electric field creates a magnetic field. Imagine taking a charge and quickly moving it side to side (accelerating it). This causes rapid changes in the electric field, which then also creates rapid changes in the magnetic field, which then creates more changes in the electric field, and so forth. Combining Maxwell’s Equations in a vacuum actually results in a wave equation, telling us that these changes in the electric and magnetic fields actually propagate through space…at the speed of light. Light is nothing other than oscillations in the electromagnetic field!

Besides the enormous application these equations have to electronics and communication systems, these equations have much to tell us about the nature of the universe. They are our first encounter of classical field equations. A mysterious result is that Einstein’s theory of Special Relativity is already baked into the heart of these equations, even though these equations were published long before Einstein! The unification of electromagnetism and quantum mechanics led to the development of Quantum Electrodynamics, which was the first ever Quantum Field Theory put into practice.

Relativity

Einstein captivated the whole world with his Relativity Theory. By introducing Special Relativity in 1905, he revolutionized physics by explaining time dilation, length contraction, and the equivalence between mass and energy. Before quantum mechanics crushed our intuition regarding our ability to make certain predictions about particles, time dilation and length contraction had already crushed our intuition that time and space are absolute. Observers moving at different speeds relative to each other may measure different elapsed times between two events, as well as different lengths of the same object. However, these effects are more apparent if we view space (three dimensions) and time (one dimension) not as separate entities, but as interwoven into a four-dimensional entity known as spacetime. Spacetime can then be treated as a manifold (a geometrical object) which, like any other shape, may be curved or flat. Special relativity is the physics of flat spacetime.

\begin{equation*}
\bbox[5px, border: 1px solid #00416a]{
E^2 = \abs{\vb{p}}^2c^2 + (mc^2)^2
}
\qquad \textcolor{#00416a}{\text{Energy-Momentum Relation}}
\end{equation*}

Special relativity not only distorts our idea of time and space, but also our idea of energy and mass. The above equation demonstrates the relationship between the energy, momentum, and mass of a particle. Now imagine that the particle is at rest (zero momentum). This particle then has an energy given by $E=mc^2$, arguably the most famous equation in history.

For about ten years, Einstein worked on generalizing his works by studying what happens when spacetime is curved. This became his theory of General Relativity. As soon as you introduce curvature to spacetime, particles begin to behave in very peculiar ways. In fact, they behave exactly as if they were under the influence of gravity in Newton’s picture. Gravity manifests itself as curvature in spacetime! That is the brilliance and beauty of general relativity.

\begin{equation*}
\bbox[5px, border: 1px solid #00416a]{
R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R + \Lambda g_{\mu \nu} = \frac{8\pi G}{c^4} T_{\mu \nu}
}
\qquad \textcolor{#00416a}{\text{Einstein Field Equations}}
\end{equation*}

Although it looks like a single equation, the Einstein Field Equations actually encode ten partial differential equations. This is because the indices $\mu$ and $\nu$ (attached to objects known as tensors) encode multiple equations within one expression (these equations also exhibit certain symmetries, which reduce the total number of independent equations). The above expression teaches us the dynamics between the curvature of spacetime (the left-hand side) and the distribution of matter (the right-hand side). The curvature of spacetime tells matter how to move, and matter tells spacetime how to curve.

Statistical Mechanics

Thermodynamics is the study of heat, energy, and work. By understanding and applying the laws of thermodynamics, we can create steam engines and kickstart the industrial revolution. This can all be done without ever thinking about the individual motion of the countless particles that are constantly colliding with each other and the walls of their container.

\begin{equation*}
\bbox[5px, border: 1px solid #967117]{dE = TdS-PdV+\mu dN}
\qquad \textcolor{#967117}{\text{First Law of Thermodynamics}}
\end{equation*}

The First Law of Thermodynamics is essentially a statement of energy conservation. A small change in energy (left-hand side) is due to heat flow ($TdS$ term), mechanical work ($PdV$) term, and chemical work ($\mu dN$ term). There is no mention or care of what the particles are doing at the microscopic level.

Classical mechanics, on the other hand, teaches us how to study and understand the individual motion of particles at the microscopic level. This raises the question of whether we can apply the tools of classical mechanics to systems involving many particles to derive the laws of thermodynamics. The answer is yes, and it lies within the study of Classical Statistical Mechanics.

\begin{equation*}
\bbox[5px, border: 1px solid #967117]{S=k\log{\Omega}}
\qquad \textcolor{#967117}{\text{Boltzmann’s Equation}}
\end{equation*}

Boltzmann’s equation is one of the key triumphs of statistical mechanics. It establishes a bridge between the entropy of a system $S$, which can be understood at the macroscopic level in thermodynamics, and the number of microstates of a system $\Omega$, which captures the microscopic behaviour of a system.

Is it possible to study the thermodynamics of systems involving quantum particles using the tools of quantum mechanics? Yes! The tools we use for such a task arise from (you guessed it) Quantum Statistical Mechanics.

\begin{equation*}
\bbox[5px, border: 1px solid #967117]{
\expval{n_j}_\pm = \frac{1}{e^{\frac{1}{kT}({\varepsilon}_j-\mu)}\mp 1}
}
\qquad \textcolor{#967117}{\text{Bose-Einstein & Fermi-Dirac Distributions}}
\end{equation*}

$\expval{n_j}$ is the average number of quantum particles occupying a state with energy $\varepsilon _j$, temperature $T$, and chemical potential $\mu$. These two distributions ($\expval{n_j}_+$ and $\expval{n_j}_-$) point to the behaviour of systems consisting of two types of quantum particles. Bosons (photons, hydrogen atoms, Higgs boson, etc.) are quantum particles that are able to occupy the same quantum state, and thus follow Bose-Einstein statistics ($\expval{n_j}_+$). Fermions (electrons, neutrons, protons, quarks, etc.) are quantum particles that are unable to occupy the same quantum state, and thus follow Fermi-Dirac statistics ($\expval{n_j}_-$). Whether a particle is a boson or a fermion depends on an intrinsic quantum mechanical property called spin.

These quantum systems have vastly different properties, especially when we reach very low temperatures (near absolute zero). For bosons at very low temperatures, they can form a Bose-Einstein Condensate, a state where all of the bosons occupy the same quantum state. Such systems are at the frontier of research in condensed matter physics and can be used to make superconductors.