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!
You might have seen animations like this that show an electron undergoing a transition from a lower energy to a higher energy state and vice versa like so:
There is something really important about this image that one must understand clearly.
The diagram represents the transition in energy of an electron BUT this does not mean that the electron
is magically jumping from one position and respawning at another
The electron’s position is NOT doing this i
If you want to know about the probability of finding an electron around the nucleus at a certain energy level, you look at its wavefunction and not at the energy diagram.
Here is the wavefunction of a hydrogen atom and each stationary state defines a specific energy
level of the atom.
The electron makes a transition between these wavefunctions by the absorption/emission of photons. *
This might not sound like a big deal but one might be surprised to know that there are a lot of people who think that the electron is magically transported from energy level to another which, most certainly is not true.
Thank you so much! Well.. There is an amazing playlist on YouTube compiled by Ludwig Maximilian University of Munich featuring a set of 27 frequently asked questions about Bohmian Mechanics. It is definitely a rich source of information on the topic.
* NOTE: You guys have been asking us a ton of interesting questions over the past month or two and we have been trying our very best to get to all of your questions ASAP. It would help if you did not ask Anonymously since it’s hard to address some personal questions on a global platform. Thanks!
TBH cleaning your car is a rather mundane task. But when you fill your head with some interesting physics the task actually gets rather pretty interesting. Here’s some good for thought on such an occasion :
The dust on your windshield might actually be from the Sahara desert
To understand how, lets start with some simple physics.
The stacked ball drop
You basically take couple of balls, align them up and drop them to the ground. The ball at the top reaches the most highest due to the subsequent transfer of energy from the other balls.
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 )