““How long will it take me to master physics?” a prospective student asks. “How long do you expect to live?” is the only respectable response.”

That is a quote from the book Mastery by George Leonard. I replaced the word aikido with physics.

Why Self-Study Physics?
Since you are here and reading this, I assume that you are, like me, a student who wants to teach themself physics. And with “student” I don’t mean “enrolled in college or university” necessarily. But rather a student of physics. Someone curious and excited and on the path of learning more and more and always expanding and deepening their understanding of physics.

Mainly for its own sake. Because learning physics or doing physics gives you joy.

And secondarily because it will serve you well in life. Learning physics means learning to think. It means learning to solve problems. It means learning to reason from first principles. Furthermore, learning physics involves learning many (mostly technical) skills.

It’s not so hard to see then how learning physics will serve you well in your career and beyond. Even if your career is not directly in physics or even a related field.

Or maybe, you simply stumbled upon this article by accident. In which case I invite you to just give it a shot. Maybe you’ve always wanted to learn some physics.

Well this is your chance to start!

Why not now?

How to Self-Study Physics?
In the end, you can only really learn something yourself. You have to do it. Whether that’s with the guidance of a mentor, a teacher, your peers, books, lectures, a school, or the internet.

And in 2021 (and beyond) it really should be possible to learn physics by yourself, at home, with nothing more than the internet (and perhaps some books).

But that doesn’t mean it’s easy. Where are you supposed to start? What about all the misinformation out there? What is the best way? What is the ideal route? You will face many more such questions (which don’t have clear answers) when deciding to learn physics just by yourself

So You Want To Teach Yourself Physics . . . ? was originally published in Physics From Scratch on Medium, where people are continuing the conversation by highlighting and responding to this story.

What is a catenary?

A catenary (derived from the Latin catenaria meaning “chain”) is an idealized curve in physics or maths that represents the shape that a chain (or rope) assumes under its own weight when being supported only at its ends.

You can see this shape everywhere in the world and in nature. From spiderwebs to overhead powerlines.

The catenary looks very much like a parabola. And it wasn’t until the 17th century that Galileo Galilei recognized that the catenary is “only an approximate parabola.” He wasn’t, however, able to derive an exact expression for the catenary.

But today we have the amazing mathematical tool of calculus that we can use to derive the catenary equation.

Derivation of the Catenary Equation
Mechanics and Statics
We can start by using basic mechanics and statics. Let’s start with a catenary and draw the forces of tension at the ends where it is supported:

These forces are obvious (I hope). The rest of the forces require a little more thinking. If you take a single chain link, for example, what are the forces acting on it? One of the forces is obvious: Gravity.

And by using Newton’s Third Law of Motion it’s easy to see that each chain link will also have a force of tension on both sides because each chain link exerts a force on every chain link next to it:

Now that we understand that, let’s draw a free body diagram:

One of the best way to solve this problem is by taking an arc of the catenary with length s and considering the forces acting on it.

The weight (distributed force of gravity) of this arc must be represented using the linear 1D density of the chain because we are not dealing with a point mass:

Where μ[kg/m] is the linear 1D density of the chain, g[m/s²] the gravitational acceleration on Earth, and s[m] the arclength.

The other forces acting on the arc (point A to P) are the tension from the chain at point A and the tension at point P.

We assume the catenary to be ideal and thus so thin and flexible that any force of tension is parallel to the chain (to the catenary curve).

At point A the force of tension T₀[N] is therefore tangent to the catenary and only horizontal. At point P the force of tension T[N] makes an angle relative to the horizontal we will call θ.

The arc (and the entire chain) is in static equilibrium and thus the net force acting on it is zero: ΣF⃗ = 0.

It’s helpful to split the net force up into its horizontal (x) and vertical (y) components:

We can rewrite sinθ as cosθtanθ to get:

Analysis with Calculus
Now it’s useful to re-write θ in terms of x and y. We can do this by realizing that the tangent of theta is equal to the slope of the curve:

We can then write:

Now let’s define a constant κ = μg/T₀ to simplify the expression to:

Now we have a first-order differential equation. If we solve it, we will have our catenary equation y(x). So the rest is “just maths.”

“Just Maths”
We have a first-order DE but it’s not so easy to solve because it’s written in terms of s instead of only y and x. The best way I know to solve this problem is by first increasing the order of the DE by differentiating both sides of eq. 3 with respect to x:

Now we have expressed the relationship between y and x with a 2nd-order DE. So it looks more complicated now. But sometimes you first have to make things more complicated to make them more simple.

Now we need the arc length equation:

As you can see this equation is nothing more than the Pythagorean Theorem applied to an infinitesimally small arc.

We can also make things more simple by defining a new variable:

Now we can re-write our arc length equation:

We can also re-write eq. 4:

Now let’s solve this DE:

The integral on the left side of the equation is a standard form and its solution is arcsinh(z) + C. Where arcsin(z) is the inverse hyperbolic sine function. So we can write:

If we take point A to have x-coodinate 0, then C=0 because the slope at that point dy/dx=z is also zero (arcsinh 0 = 0).

Thus we have:

Now let’s isolate z:

Now we finally have a first order DE in terms of y and x.

We can easily solve it,

And eq. 6 is our catenary equation at last!

Let’s Derive The Catenary Equation With Calculus was originally published in Physics From Scratch on Medium, where people are continuing the conversation by highlighting and responding to this story.

This article is related to the study guide for: Introduction to Non-Linear Dynamics

Introduction To Non-Linear Dynamics — A Study Guide

In this article we will derive an expression for the angular displacement of the simple pendulum as a function of time: φ(t).

We will not use any approximations such as the small angle approximation but instead derive an exact solution. It must be noted however that this solution cannot be expressed in terms of elementary functions.

Definition. Elementary functions are real functions of a single real variable that can be defined by applying the operations of addition, multiplication, division, nth root, and function composition to polynomial, exponential, logarithm, and trigonometric functions.

But we can write down a truly exact solution by making use of the so-called Jacobi Ellliptic Functions. These are analogous to the sine, cosine, and tangent functions but are defined relative an ellipse instead of a circle.

Note that I still call it the “simple pendulum” because we’re still dealing with an idealized model: We’re neglecting air resistance, assuming the pivot is frictionless, and neglecting the mass of the rod.

Some Basic Mechanics
This is the easiest part of the derivation.

Above you see a free body diagram of the pendulum where the force of gravity has been split up in a component tangent to displacement and a component perpendicular to it.

I have defined the angle φ to be positive in the counter-clockwise direction and negative in the clock-wise direction. As is convention.

Now our first goal should be to derive an equation of motion. We can do this by using the rotational analogue of Newton’s Second Law of Motion (time derivatives denoted with dot notation):

Where l⃗ is the ‎p‎‎‎osition vector pointing from the mass to the pivot (with magnitude l which is the length of the cord).

Note that only the force of gravity contributes to the torque (to rotation) because the force of tension is perpendicular to displacement.

Also note that the gravitational torque is negative because it rotates the mass in the direction opposite to displacement (opposite to angular displacement φ).

The moment of intertia I for a point mass rotating around a pivot (radius l) is ml², yielding our equation of motion:

This is a second-order, non-linear differential equation. Solving this DE will yield the equation we seek: φ(t).

So the rest is just (a lot of) maths.

“Just Maths”
As I said before, the solution to this DE cannot be written in terms of elementary functions.

But there does exist an exact, analytical solution which can be written in terms of Jacobi Elliptic Functions.

Jacobi Elliptic Functions
A simplified way of thinking about these functions is to see them as some kind of “generalized trigonometric functions.”

The trigonometric functions (sin, cos, tan, etc.) are defined relative to the unit circle. The Jacobi Elliptic Functions are defined relative to a more general conic section: The unit ellipse.

The figure above geometrically defines the Jacobi Elliptic Functions relative to the unit ellipse:

This is still quite abstract, but we can define these elliptic functions as inverses of the incomplete elliptic integral of the first kind (and so arrive at a more useful expression).

An “Elliptic Integral” is defined in the following way:

Where R is any rational function and P a polynomial with degree 3 or 4 with no repeated roots. c is a constant.

Historically, these integrals were used to solve problems related to ellipses (such as finding the arc length of an ellipse).

The specific Elliptic Integral of our interest is the special case known as the “Incomplete Elliptic Integral of the First Kind.”

But let us first write down the definition for the “Complete Elliptic Integral of the First Kind”, which will also prove useful in our derivation:

The integral is considered “complete” because the amplide φ is equal to π/2 (t = 1). And here k is the so-called “Elliptic Modulus.” We will use the definition where 0 ≤ k² ≤ 1.

Note: In some definitions the parameter m will be used instead where m=k²

Now let’s write down the definition for the “Incomplete Elliptic Integral of the First Kind”:

This integral is considered “incomplete” because the amplitude φ is not fixed at π/2 but variable (note that it differs from the complete integral only in the integration limit but as a result has two arguments).

In our derivation it will be useful to write our incomplete elliptic integral as an indefinite integral. We can do this by making use of the following identity that arises from the Fundamental Theorem of Calculus:

Yielding,

Defining the Jacobi Elliptic Functions
Now let’s go back to the Jacobi Elliptic Functions. Because we still need to define them properly.

One of the most important ones is the Jacobi Amplitude am, which is defined in terms of the Incomplete Elliptic Integral in the following way:

Other Jacobi Elliptic Functions that will be relevant for us and their definition:

And an important addition identity:

Deriving that final elliptic function as the ratio of two others and the addition identity would be an entire article in itself requiring to explain the Jacobi Theta Function.

Instead I’ll cite a source:

Jacobi Elliptic Functions

And I challenge you to derive these two facts for yourself :)

Solving the Equation of Motion
Now we finally have all the tools to start solving the equation of motion we derived:

Let’s start by re-writing the DE into a form that’s easier to solve.

First we will re-write our DE to this:

Using the following two identities will help us re-write it again. They can be derived by applying the chain rule:

Using above two identities we can re-write our eq. of motion to get:

Now we can bring everything into the time-derivative:

The above equation implies that the expression being differentiated with respect to time is constant (because its rate of change is zero). So we can write:

We can find an expression for our constant C by considering the initial conditions of our problem: At the initial angle φ(t=0)=φ₀, the speed of the mass is zero, and thus the angular speed must also be zero. Yielding:

So we are assuming that the system starts in rest.

Now we can write:

Let’s re-write it again to find an expression for the angular speed:

You might be able to see now that the goal is to re-write our equation of motion such that we can express it in terms of Jacobi Elliptic Functions. The next step is to re-write it again by making use of the following trig identity:

Substitution of this identity yields:

Now let us define a few constants to simplify our expression:

Where ω₀ is known as the “natural frequency.” It’s acually the frequency of the pendulum’s motion in the small angle approximation where you can speak of simple harmonic motion.

Let’s also define our Elliptic Modulus:

It is thus defined by and dependent on our initial angle. Also note that it obeys the rule that 0 ≤ k² ≤1.

Using these constants we can re-write our eq. of motion again:

The equation is still quite complex. How do we get an expression for φ?

We can do this through integration. But in order to do that, defining yet another variable for substitution will prove useful:

Note that:

So we can write:

Now we can use this substitution to make the integration easier.

Let’s begin by re-writing our eq. of motion into:

And let’s integrate the entire expression with respect to time:

We can also write:

(as dφ/dt · dt = dφ).

And by making use of our definition of θ (I’m not going through all the steps), we get:

Now we managed to get an expression we can write in terms of Jacobi Elliptic Functions. Remember the definition of the Incomplete Elliptic Integral of The First Kind? We can use it to write:

Or, in terms of φ,

Now let’s find our constant C.

At t=0 we have φ=φ₀,

Turns out our constant is the negative of the Complete Elliptic Integral of the First Kind!

Now we can finally write our solution in terms of Jacobi Elliptic Functions:

And using the definition of the Jacobi Amplitude Function am, we get,

And then using the definition for the Jacobi Ellipitic Function sn=sin(am):

That’s already a solution. But we can make it more compact.

By using the identity introduced above, sn(u+K(k), k) = cd(u, k), we finally arrive at:

That’s an expression of φ(t) in terms of Jacobi Elliptic Functions.

Before we analyze this function, is this solution truly exact?

The answer is yes, of course!

The solution is just not expressed in terms of “elementary functions.” But just like trignometric functions, Jacobi Elliptic Functions can be computed and are therefore useful. See:

DLMF: 22.20 Methods of Computation

Analyzing The Solution
Small Angle Approximation
Let’s compare the solution to the Small Angle Approximation, which we can derive from our solution as well:

(cd(x, k=0)=cos x)

Which is the solution if the Small Angle Approximation is applied.

Comparison

Let’s compare the exact solution against the small angle approximation. For the following graphs the following initial conditions were used: g=9.81 m/s², l=0.1 m.

As you can see in the figure below, if the initial angle is sufficiently small (like π/16 rad), and the time interval too, the small angle approximation is almost indistinguishable from the exact solution:

If we make the initial angle a bit larger (like π/3 rad), you can see that the small angle approximation begins to deviate from the exact solution as time passes (see figure below).

A Complete Solution To The Non-Linear Pendulum was originally published in Physics From Scratch on Medium, where people are continuing the conversation by highlighting and responding to this story.

“A physicist is just an atom’s way of looking at itself.” — Niels Bohr

“If you think you understand quantum mechanics, you don’t understand quantum mechanics.” — Richard Feynman

“If quantum mechanics hasn’t profoundly shocked you, you haven’t understood it yet.” — Niels Bohr

“Anyone who claims to understand quantum theory is either lying or crazy” — Richard Feynman

Opening

What is light?

Is light a particle or a wave?

After studying electrodynamics and optics and even just observing nature you might conclude light is a wave. And historically it became clear that “light must be a wave” when Maxwell showed that light really is an electromagnetic wave.

Physics seemed finished. The foundations were laid. Classical Physics, Thermodynamics, Statistical Physics, Optics, and Electrodynamics. These theories explained everything in the natural world. And light was understood.

“There is nothing new to be discovered in physics now. All that remains is more and more precise measurement.” — Lord Kelvin

There were however some problems like the “Ultraviolet Catastrophe” and the “Photoelectric Effect.” Which the theories at the time could not explain.

• Photoelectric effect - Wikipedia
• Ultraviolet catastrophe - Wikipedia

Then came Max Planck. He solved the ultraviolet catastrophe by assuming the existence of “light quanta,” a smallest discrete packet of light with particle-like properties, which would later be called a photon.

Albert Einstein then used this concept to explain the photoelectric effect (which later earned him a Nobel Prize).

“According to the assumption to be contemplated here, when a light ray is spreading from a point, the energy is not distributed continuously over ever-increasing spaces, but consists of a finite number of ‘energy quanta’ that are localized in points in space, move without dividing, and can be absorbed or generated only as a whole.” — Albert Einstein

So light is a particle, then?

No. And yes, sort of. It’s complicated.

So it seems Kelvin was wrong. But he wasn’t just wrong. He was very wrong. Until now we have only dicussed what is considered “The Old Quantum Theory”

Old quantum theory - Wikipedia

Things were about to get a whole lot weirder.

The Double Slit Experiment showed that particles such as electrons produced interference patterns you would expect from waves.

Strange. But physicists got clever and decided to shoot the electrons through the slits one at a time. This way they could not interact with eachother and produce “wave-like behavior” by some strange mechanism.

The result: The same pattern. The same exact wave interference pattern.

When learning this for the first time the only proper response is to fall out of your chair.

How could these individual electrons possibly interfere with eachother when shot through the slits one by one?

But still physicists tried to be clever and decided to use a detector to observe which slit each electron went through. They placed a detector at one of the slits.

The result: The detector ‘beeped’ or detected an electron 50% of the time (as expected).

But there was no wave inteference pattern.

It turns out the act of measurement interfered with the outcome of the experiment. The act of observation or measurement resulted in the electron behaving like a particle again and not like a wave.

Again, if you were to learn this for the first time and did not fall out of your chair yet, surely now you must be on the floor.

Or else you haven’t grasped the significance of what is being told yet.

I highly recommend you watch the folllowing video on the double slit experiment:

This was a long introduction. But the point was to explain a bit about the history of Quantum Mechanics, and, more importantly, expose you to the fact that this theory is quite radical, strange, and counter-intuitive.

This theory has huge physical as well as metaphysical implications. It’s no mystery as to why it took so long for this theory to be accepted as real physics.

But today Quantum Mechanics is one of the most succesful theories in physics. And our modern world: nanotechnology, semiconductors, transistors, modern computers, is all and only made possible because of this theory.

Introductory Quantum Mechanics

A subject such as “Introductory Quantum Mechanics” or “Quantum 1” will typically be given in the 1st or 2nd year of undergrad physics.

This subject will mostly just introduce you to the concepts in Quantum Mechanics and some of the formalisms used.

Scope of Subject

Quantum Mechanics is a physical theory that describes the “microscopic world.” This microscopic world is nature at the scale of atoms and subatomic particles (systems with dimensions smaller than ~ 0.1 Å).

The major ways in which Quantum Mechanics differ from Classical Mechanics are that physical quantities (energy, momentum, etc.) are quantized (restricted to discrete values), Wave-Particle Duality, and the Uncertainty Principle.

In modern QM, a fundamental mathematical function called “The Wave Function” contains all the information of a system in the form of probabiltiy amplitudes of what a measurement of physical quantities will yield.

The theory is very succesful. But it fails at speeds approaching the speed of light (~ 3×10⁸ m/s)

See the image below:

Important topics for this subject (I have divided it into two parts):

Part 1:

• The Wave Function
• The Schrödinger Equation
• The Hydrogen Atom
• Wave-Particle Duality, De Broglie Relation, Quantum Numbers
• Heisenberg Uncertainty Principle

Part 2:

• Operators
• Dirac Notation
• Generalized Uncertainty Principle
• Perturbation Theory
• Varitional Methods

Important maths:

• (Partial) Differential Equations
• Complex Numbers
• Linear Algebra

Where to begin?

I recommend you start by reading the following Wikipedia article:

Introduction to quantum mechanics - Wikipedia

Next it’s important to get really familiar with two new concepts:

• The Wave Function. The following is a must-watch:

• Schrödingers Wave Equation: Another must-watch (1-min long):

Make sure you also read about these concepts and think about what they really are.

It’s also a good idea to learn about The Infinite Potential Well Problem and The Finite Potential Well Problem.

Particle in a box - Wikipedia

Another great video:

Step Plan

• If you have an introductory physics books (like Giancoli): Read the chapters on QM and do the exercises
• The following lecture notes are a great resource when first learning QM: https://oer.physics.manchester.ac.uk/QM/Notes/Notes.pdf
• I also recommend an introductory QM book (like Griffith’s Introduction to Quantum Mechanics)
• Basically every chapter in Griffith’s book is important

Thanks for reading!

Introductory Quantum Mechanics — A Study Guide was originally published in Physics From Scratch on Medium, where people are continuing the conversation by highlighting and responding to this story.