Using Complex numbers in Classical Mechanics

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 (v_r , v_{\theta}, a_r, a_{\theta} ) are. Here’s one failsafe way using complex numbers that made things really easy :

z = re^{i \theta}

\dot{z} = \dot{r}e^{i \theta} + ir\dot{\theta}e^{i \theta} = (\dot{r} + ir\dot{\theta} ) e^{i \theta}

From the above expression, we can obtain v_r = \dot{r} and v_{\theta} = r\dot{\theta}

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta} ) e^{i \theta}   + (\dot{r} + ir\dot{\theta} )i \dot{\theta} e^{i \theta} 

\ddot{z} =  (\ddot{r} + ir\ddot{\theta} + i\dot{r}\dot{\theta}  + i  \dot{r} \dot{\theta} - r\dot{\theta}\dot{\theta} )e^{i \theta} 

\ddot{z} =  (\ddot{r} - r(\dot{\theta})^2+ i(r\ddot{\theta} + 2\dot{r}\dot{\theta} ) )e^{i \theta} 

From this we can obtain a_r = \ddot{r} - r(\dot{\theta})^2 and a_{\theta} = (r\ddot{\theta} + 2\dot{r}\dot{\theta}) 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!

 

 

Note on Electron level transitions

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:

image

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
position.

image

                        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.

image

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.

Have a good one!

– A2A

EDIT : * suggested by @imsureiforgotsomething.

Your series on Pilot wave was fascinating.. Any good resources for learing about Bohmian Mechanics?

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.

Thanks for asking! *

PDF Version: Click here

Home page: Department of Mathematics, Ludwig Maximilian University of Munich

* 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!

How does sand from Sahara end up in your windshield ?

image

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.

image

                                    Source Video : Physics Girl

Here is an exaggerated but amazing slow motion of the same energy transfer with a water balloon. Notice how the transfer of energy takes place between the water balloon and the tennis ball.

image

                                     Source Video : Slow Mo Lab


Sandstorms in the desert

image

Sandstorms/ Dust storms as you might be aware, are pretty common in the desert. . Dust storms arise when a gust front or other strong wind blows loose sand and dirt from a dry surface.

image

And this can cause something phenomenal to happen:

If the wind speed is sufficient then larger sand particles can propel finer ones high into the atmosphere. ( just like the stacked ball )

Then these fine particles are caught in the global wind pattern and are transported across the globe until they fall down to the earth as rain.

How cool is that ! Have a great day!

* Tracking saharan dust in 3D – NASA video

** All the World’s a Stage … for Dust – NASA article

** Wiki on Saltation

Matrix Multiplication and Heisenberg Uncertainty Principle

We now understand that Matrix multiplication is not commutative (Why?). What has this have to do anything with Quantum Mechanics ?

Behold the commutator operator:
[\hat{A}, \hat{B}] = \hat{A}\hat{B} - \hat{B}\hat{A}

where \hat{A},\hat{B} are operators that are acting on the wavefunction \psi . 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):

[\hat{p}, \hat{x}] = \hat{p}\hat{x} - \hat{x}\hat{p} = i\hbar

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 \hat{p} and \hat{x} are infinite dimensional. ( The harmonic oscillator being the simple example to this )

\hat{x} = \sqrt{\frac{\hbar}{2m \omega}} \begin{bmatrix} 0 & \sqrt{1} & 0 & 0 & \hdots \\ \sqrt{1} &  0 &\sqrt{2} & 0 & \hdots \\ 0 & \sqrt{2} &  0 &\sqrt{3}  & \hdots \\  0 & 0 & \sqrt{3} &  0  & \hdots \\  \vdots & \vdots & \vdots & \vdots \end{bmatrix}

\hat{p} = \sqrt{\frac{\hbar m \omega}{2}} \begin{bmatrix} 0 & -i & 0 & 0 & \hdots \\ i &  0 & -i \sqrt{2} & 0 & \hdots \\ 0 & i\sqrt{2} &  0 &\-i \sqrt{3}  & \hdots \\  0 & 0 & i\sqrt{3} &  0  & \hdots \\  \vdots & \vdots & \vdots & \vdots \end{bmatrix}

 

 

 

fuckyeahphysica:

So, this is called Blobbing!

Potential Energy of jumping dude — > Kinetic energy upon jumping — > hits the blob( some energy lost here ) — > Remaining energy propels the launching dude.

What i am trying to say is :

image
image

m1 = combined mass of the jumping dudes,

m2 = mass of the launched dude

Projectile is not exactly a parabola

image

If you look at the projectile, you would notice that it is not exactly a true parabola. This is because the projectile resembles a parabola without the effect of air resistance.

But under its influence, the shape of the projectile is a bit different.

image

Simple, but aesthetically pleasing!

Have a good day.

More:

Complete physics of the world record blob jump

So, this is called Blobbing!

Potential Energy of jumping dude — > Kinetic energy upon jumping — > hits the blob( some energy lost here ) — > Remaining energy propels the launching dude.

What i am trying to say is :

image
image

m1 = combined mass of the jumping dudes,

m2 = mass of the launched dude

Projectile is not exactly a parabola

image

If you look at the projectile, you would notice that it is not exactly a true parabola. This is because the projectile resembles a parabola without the effect of air resistance.

But under its influence, the shape of the projectile is a bit different.

image

Simple, but aesthetically pleasing!

Have a good day.

More:

Complete physics of the world record blob jump

fuckyeahphysica:

That’s how a revolving fan works

This gif has been circulating throughout the internet for many years, and I am not denying it It’s pretty darn cool!

Although it gets the mechanism spot on, but it misses out on one of the most amazing part of the mechanism : the oscillation itself.

How does it oscillate?

image

                       Source

The oscillation is simple but ingenious. So, you see that pink rod? Yeah that is attached to v – type wedge, like so:

image

89 is a pin. And that pin is constrained to oscillate between the two wedges. The angle of oscillation is primarily between 40-100 degrees.

It is this wedge that controls the oscillation.

Why does it oscillate?

The oscillation is due to an eccentric round cam present under the yellow gear. ( seen in animation too )

image

An eccentric cam

The rotation of this cam induces a torque which causes the linkage ( the pink rod ) to sweep across the plane.

Important Note

I understand that this explanation is not exactly the finest. But this mechanism has not been explored anywhere else on the Internet .This is what i could reckon of the Patent.

If you have a better explanation/sources/anything at all then please reblog it with this post, that way everyone wins. It is our humble request.

Have a good day!

That’s how a revolving fan works

This gif has been circulating throughout the internet for many years, and I am not denying it It’s pretty darn cool!

Although it gets the mechanism spot on, but it misses out on one of the most amazing part of the mechanism : the oscillation itself.

How does it oscillate?

image

                       Source

The oscillation is simple but ingenious. So, you see that pink rod? Yeah that is attached to v – type wedge, like so:

image

89 is a pin. And that pin is constrained to oscillate between the two wedges. The angle of oscillation is primarily between 40-100 degrees.

It is this wedge that controls the oscillation.

Why does it oscillate?

The oscillation is due to an eccentric round cam present under the yellow gear. ( seen in animation too )

image

An eccentric cam

The rotation of this cam induces a torque which causes the linkage ( the pink rod ) to sweep across the plane.

Important Note

I understand that this explanation is not exactly the finest. But this mechanism has not been explored anywhere else on the Internet .This is what i could reckon of the Patent.

If you have a better explanation/sources/anything at all then please reblog it with this post, that way everyone wins. It is our humble request.

Have a good day!