How to Learn Calculus The Simple Way

So I guess it's safe to say that calculus is one of those topics, which are for some reason really hard to grasp for most of the students in high school and later in college. Maybe, it's because calculus is really a unique branch of mathematics, that is, it's different from most of the mathematics that we are taught in high school (most of which was developed by ancient Greeks, of course with some slight improvements by later mathematicians). After all, it was developed by greatly gifted mathematician and scientist Isaac Newton (and Leibniz too).

I myself remember having some problems with calculus. It's not that I didn't grasp the mathematics, it was just most of the time I didn't have the feeling what I was calculating, or what's the sense of derivatives, integrals and so on. And what really helped me, when I was revising before going to college, was some great books and video lectures.

Firstly, I really enjoyed these simple YouTube lectures by Prof. R.Delaware. I like this guy because he really doesn't presuppose that you have some past knowledge about calculus (except of course some basic mathematics from high school), he simply explains everything as it was being explained for the first time.

These lectures cover all the basic high school calculus and some of the basic college calculus. Basically the two main parts are differentiation and integration.

So here's the first lecture:

And you can find all the other lectures on calculus and other topics HERE.

If you're looking for a book specifically that would help you with calculus I highly recommend Calculus for Dummies. It's a very basic and fun to read book, which will give you all the required knowledge of high school and some college calculus. Also it contains very nice diagrams, which give you a great visualisation of calculus concepts. Also it contains some neat tricks how to do calculus problems.

Source

Thanks for reading! If you enjoy my blog feel free to subscribe and comment. Cheers!

Read more »

Basics of Quantum Mechanics Lesson 4

So last time we calculated the wave function of a free moving particle and got the following answer:

SE for a free moving particle:

Solution for this equation:  

So what do these equations tell us about the free moving particle? To find the answer to this question we need to find the probability density function. If you recall, the probability density function is basically just the square of an absolute value of the wave function. So by calculating the probability density function we would get:


|Psi |^2 = Psi x complex conjugate of Psi =
= A(cos(kx − wt) + i sin(kx − wt))A(cos(kx − wt) − i sin(kx − wt))
= A^2(cos2(kx − wt) + sin2(kx − wt))
= A^2


Now if you look at the result more carefully, you should realise that there's some strange stuff going on. The probability density function is constant, that means that the probability of finding a the particle in any point of space is equal. In more simple words, you have the same probability of finding the free moving particle in your room and somewhere in the other side of the universe. This is due to the fact that we have not taken into account the uncertainty of the momentum of a particle.



So you might be asking  - why are we studying free moving particles, while in reality most of the matter is situated in confined atoms and molecules. And you're definitely right, so let's look at another system, which might help us understand the behaviour of electrons, which are "trapped" in an orbit of an atom.


The easiest way of understanding simple atoms or similar systems is using a thing called infinite potential well. That is a system in which a particle is trapped between infinite potential "walls", which can be for instance be various electric of magnetic fields or anything similar, which confines the particle in a given space. After all, in a sense electrons orbiting a nucleus are also in a similar potential well - if they get too close to the nucleus, they are pushed back, if the get too far they are attracted back. 


Infinite potential well:

So how does particle behave in such a well? Well let's firstly imagine what would happen in a classical situation, let's say for a tennis ball trapped between two walls. It's clear that (neglecting gravity, which would eventually bring the ball down) any position between the walls has the same probability of finding the ball there.

However, in quantum mechanics, things are different. We already know that particles tend to behave as probability waves, so it's clear that the probability of finding the particle in any point of space will not be equal( a good analogy of a wave in a sort of potential well is a rope fixed at two places and oscillating).

So how can we find out at which points of space we are most likely to find the particle? By solving the Schrodinger equation of course.

And since this is a basic course we're not gonna bother ourselves with the process of solving the SE. We're just gonna skip right to the solution which is:

Wave function for a particle confined in a infinite potential well: Psi(x) = Asin kx

Now to calculate the probability density function, we need to apply the initial conditions of the system - as we know the particle can't be in points of space, which are beyond the boundaries.

In an infinite potential well the boundary conditions imply:
 Psi (0) = Psi  (L) = 0.
Resulting in the limitation that k must have one of the discrete set of values
kn = (pi . n)/L , where
n = 1, 2, 3 . . .


Now when we know k, we can find allowed energies.
Using  E= (p^2)/m and De Broglies wavelength p= h/ lambda = hk/ 2pi gives the following:

En = (h^2 . n^2)/(8mL^2)
 n = 1, 2, 3 . . .


The allowed energies for the trapped particle are quantised according to the value of
n which is known as a quantum number.


Note that we can calculate the quantum number using simple calculations, which we'll look at later. It's important to notice that for macroscopic objects n is very larger, which gives rise to the so called correspondence principle - the idea that if n is very large (which is definately the case for macroscopic objects), quantum mechanics prediction become similar to classical physics prediction.

And we can see this by finding the probability function of the infinite potential well, and graphing it with different values of n:

 

As we can see as n grows, the wavelenght of the wave in the graph becomes shorter. In n was very very big, it would look as every point in the confined space has the same probability (like in the classical physics case).


So that's all for now, thanks for reading!


LINK to lesson 3
LINK to lesson 2
LINK to lesson 1

Read more »

A Great Book to Learn General Relativity

When it comes to GR, it's easy to find good books, which explain you the basics and the non-mathematical side of the subject, however, it's really hard to find a book, which would explain the basics of mathematics of GR in a easy to understand way. Thus I would like to share my experience of reading a book called "Firs Course in General Relativity" by Bernard Schultz.

As the books half title says - "Clarity, readability, and rigor combine in the second edition of this widely used textbook to provide the first step into general relativity for undergraduate students with a minimal
background in mathematics." This is truely the case, because the book is written in a easy to understand way.

The book contais theory, including special relativity in algebraic form and geometric form and general relativity basics. Furthermore, it has some exercises with solved examples.

From my own experience I can say that it's one of the best books to get the basics of both SR and GR. The difficulty level is not over the top like in the most of similar books (I'm a fresher in Physics, and my knowledge in maths and physics is basically enough to read and understand this book). Also, it's a great book to refresh your knowledge of SR and GR.

Thanks for reading!


The book can be found here:

Read more »

So You Want to Learn Physics (The Fun Way)

These days when internet is easily accessible, it's easy to learn anything you like from home. Internet, being the biggest virtual library, gives you an opportunity to learn various science fields as well. Today I will show you some nice books that you can easily find on the internet. These books helped me to learn various subjects an get better at high school.

If you want to learn physics you should start from simple high school level physics, however there are some very nice books, which can teach you the basics of quantum mechanics, string theory, general and special theories of relativity and other.

There's a very nice book about time travel, which covers many aspects of general relativity, special relativity, quantum mechanics and some history of physics. This book is called Black Holes Wormholes and Time Machines by Jim Al-Khalili. It's a simple to read book that contains basically no maths, so it can be read by anyone.

Another great book for those, who want to learn the basics of modern physics is Elegant Universe by Brian Greene. This book also covers the basics of Einstein's theories, quantum mechanics and some history of physics. The main focus of the book is string theory.

After reading these books, you could find Stephen Hawking's books useful. The most popular of his books are The Brief History of Time and The Universe in a Nutshell.  Both of the books cover a wide range of physics subjects, are easy to read and contain some valuable knowledge.

The series of books called For Dummies contain a lot of useful books, basically about anything. You can find many useful books like The String Theory for Dummies, Physics for Dummies and so on.  I highly recommend these books for starters, because they're easy to use and contain all that a starter needs.

Well that's it for now. Thanks for reading. I hope these books, will be useful for you. Please subscribe or follow my blog if you like it.

Read more »