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 :
In a previous post on using the Feynman’s trick for Discrete calculus, I used a very strange operator ( ). And whose function is the following :
What is this operator? Well, to be quite frank I am not sure of the name, but I used it as an analogy to Integration. i.e
What are the properties of this operator ? Let’s use the known fact that
And applying the operator twice yields:
We can clearly see a pattern emerging from this already, applying the operator once more :
Or in general, the operator that has the characteristic prescribed in the previous post is the following:
If you guys are aware of the name of this operator, do ping me !
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 )