When one is solving problems on the two dimensional plane and you are using polar coordinates, it is always a challenge to remember what the velocity/acceleration in the radial and angular directions () are. Here’s one failsafe way using complex numbers that made things really easy :
From the above expression, we can obtain and
From this we can obtain and with absolute ease.
Something that I realized only after a mechanics course in college was done and dusted but nevertheless a really cool and interesting place where complex numbers come in handy!
What ?!! There exists such an elegant decimal representation of the Fibonacci sequence? Well yes! and the only thing that you need to know to prove this is that if the Fibonacci numbers were the coefficients to a power series expansion, then the Fibonacci generating function is given as follows:
Subsituting the value of , we get :
So, I read this post on the the area of the sine curve some time ago and in the bottom was this equally amazing comment :
Diameter of the circle/ The distance covered along the x axis starting from and ending up at .
And therefore by the same logic, it is extremely intuitive to see why:
Because if a dude starts at and ends at , the effective distance that he covers is 0.
If you still have trouble understanding, follow the blue point in the above gif and hopefully things become clearer.
When one is dealing with complex numbers, it is many a times useful to think of them as transformations. The problem at hand is to find the nth roots of unity. i.e
Multiplication as a Transformation
Multiplication in the complex plane is mere rotation and scaling. i.e
Now what does finding the n roots of unity mean?
If you start at 1 and perform n equal rotations( because multiplication is nothing but rotation + scaling ), you should again end up at 1.
We just need to find the complex numbers that do this.i.e
This implies that :
And therefore :
Take a circle, slice it into n equal parts and voila you have your n roots of unity.
Okay, but what does this imply ?
Multiplication by 1 is a rotation.
When you say that you are multiplying a positive real number(say 1) with 1 , we get a number(1) that is on the same positive real axis.
Multiplication by (-1) is a rotation.
When you multiply a positive real number (say 1) with -1, then we get a number (-1) that is on the negative real axis
The act of multiplying 1 by (-1) has resulted in a 180o transformation. And doing it again gets us back to 1.
Multiplication by is a rotation.
Similarly multiplying by i takes 1 from real axis to the imaginary axis, which is a 90o rotation.
This applies to -i as well.
That’s about it! – That’s what the nth roots of unity mean geometrically. Have a good one!
Now flip this over by 90 degree counter clockwise :
Now flip this over again by 90 degree clockwise :
We now understand that Matrix multiplication is not commutative (Why?). What has this have to do anything with Quantum Mechanics ?
Behold the commutator operator:
where are operators that are acting on the wavefunction . This is equal to 0 if they commute and something else if they don’t.
One of the most important formulations in Quantum mechanics is the Heisenberg’s Uncertainty principle and it can be written as the commutation of the momentum operator (p) and the position operator (x):
If you think of p and x as some Linear transformations. (just for the sake of simplicity).
This means that measuring distance and then momentum is not the same thing as measuring momentum and then distance. Those two operators do not commute! You can sort of visualize them in the same way as in the post.
But in Quantum Mechanics, the matrices that are associated with and are infinite dimensional. ( The harmonic oscillator being the simple example to this )
Say what? This one blew my mind when I first encountered it. But it turns out Euler was the one who came up with it and it’s proof is just beautiful!
Say you have a quadratic equation whose roots are , then you can write as follows:
As for as this proof is concerned we are only worried about the coefficient of , which you can prove that for a n-degree polynomial is:
where are the n-roots of the polynomial.
Now begins the proof
It was known to Euler that
But this could also be written in terms of the roots of the equation as:
Now what are the roots of ?. Well, when i.e *
The roots of the equation are
Comparing the coefficient of y on both sides of the equation we get that:
* n=0 is not a root since
at y = 0
** Now if all that made sense but you are still thinking : Why on earth did Euler use this particular form of the polynomial for this problem, read the first three pages of this article. (It has to do with convergence)
One is commonly asked to prove in college as part of a linear algebra problem set that matrix multiplication is not commutative. i.e If A and B are two matrices then :
But without getting into the Algebra part of it, why should this even be true ? Let’s use linear transformations to get a feel for it.
If A and B are two Linear Transformations namely Rotation and Shear. Then it means that.
Is that true? Well, lets perform these linear operations on a unit square and find out:
You can clearly see that the resultant shape is not the same upon the two transformations. This means that the order of matrix multiplication matters a lot ! ( or matrix multiplication is not commutative.)
Basis vectors are best thought of in the context of roads.
Imagine you are in a city – X which has only roads that are perpendicular to one another.
You can reach any part of the city but the only constraint is that you need to move along these perpendicular roads to get there.
Now lets say you go to another city-Y which has a different structure of roads.
In this case as well you can get from one part of the city to any other, but you have to travel these ‘Sheared cubic’ pathways to get there.
Just like these roads determine how you move about in the city, Basis Vectors encode information on how you move about on a plane. What do I mean by that ?
The basis vector of City-X is given as:
This to be read as – ” If you would like to move in City-X you can only do so by taking 1 step in the x-direction or 1 step in the y-direction ”
The basis vector of City-Y is given as:
This to be read as – ” If you would like to move in City-Y you can only do so by taking 1 step in the x-direction or 1 step along the diagonal OB ”
By having the knowledge about the Basis Vectors of any city, you can travel to any destination by merely scaling these basis vectors.
As an example, lets say need to get to the point (3,2), then in City-X, you would take 2 steps in the x-direction and 3 steps in the y-direction
And similarly in City-Y, you would take 1 step along the x -direction and 2 steps along the diagonal OB.
Destination Arrived 😀
The aim of this post is to understand the travelling wave solution. It is sometimes not explained in textbook as to why the solution “travels”.
We all know about our friend – ‘The sinusoid’.
y becomes 0 whenever sin(x) = 0 i.e
Now the form of the travelling sine wave is as follows:
When does the value for y become 0 ? Well, it is when
As you can see this value of x is dependent on the value of time ‘t’, which means as time ticks, the value of x is pushed forward/backward by a .
When the value of , the wave moves forward and when , the wave moves backward.
Here is a slowly moving forward sine wave.