# Fun with Fundamentals

# Do you know what are these called? They are Möbius strips.

Google says Möbius strip is '*a surface with one continuous side formed by joining the ends of a rectangle after twisting one end through 180°*'. For more details, read the Wikipedia article on 'Möbius strip'. But why are we talking about this? Möbius strips are good analogy for electron spin. Electrons' spins can be either -1/2 or +1/2. If an electron spin rotates 360° it is represented with a negative symbol; isn't it weird? Let me show you in the following figure why is it weird in case you did not appreciate it yet.

It is very non-intuitive to realise a physical object that gives opposite sign after a full rotation of 360° and comes back to original sign by rotating one more full 360° rotation or in other words total 720° rotation. Möbius strip is a perfect physical object which represent the spin problem of electrons. In case of Möbius strip, if you start circulating your finger from a particular staring point along its circumference after 360° rotation you come back to the same point, however, you always end up being at the opposite facet of the starting point. That means if you started from outside you end up being inside. In mathematics, one such operator is spinor, *'unlike vectors and tensors, a spinor transforms to its negative when the space is rotated through a complete turn from 0° to 360°'. *One such physical observation is true in case of quantum hall effect which experimentally explained the relativistic quantum mechanics.

Further readings:

- Zhang, Y., Tan, Y.W., Stormer, H.L. and Kim, P., 2005. Experimental observation of quantum Hall effect and Berry's phase in graphene.
*arXiv preprint cond-mat/0509355*. [cited 10768] - Bernevig, B.A., Hughes, T.L. and Zhang, S.C., 2006. Quantum spin Hall effect and topological phase transition in HgTe quantum wells.
*Science*,*314*(5806), pp.1757-1761 [cited 3673].

28th August 2017, Sid.

# Do you know what is the path of quickest descent?

I guess, this must be the one of the important questions a skier asks. I am taking a nice simulation from Wikipedia to demonstrate what do I mean!

The question puzzled Galileo, Newton, Leibniz, L'Hospital, Johan Bernoulli, and Jacob Bernoulli. I came across this problem from Ananthasuresh who works on calculus of variations. I wanted to talk about this fundamental problem. However, 3Blues1Brown did a great job, so I am sharing his interaction with Strogatz.

I am sure after understanding the problem and its connection to Snell's law (which is inherently linked with Fresnel's equations) some of you must be wondering what is the relativistic solution for Brachistochrone curves. For further reading, I am enlisting some unpopular papers:

- Goldstein, H. F., & Bender, C. M. (1986). Relativistic brachistochrone.
*Journal of mathematical physics*,*27*(2), 507-511. - Perlick, V. (1991). The brachistochrone problem in a stationary space‐time.
*Journal of mathematical physics*,*32*(11), 3148-3157. - Giannoni, F., Piccione, P., & Verderesi, J. A. (1997). An approach to the relativistic brachistochrone problem by sub-Riemannian geometry.
*Journal of Mathematical Physics*,*38*(12), 6367-6381. - Gemmer, J. A., Nolan, M., & Umble, R. (2011). Generalizations of the brachistochrone problem.
*Pi Mu Epsilon Journal*, 207-218.

18th September 2017, Sid.

# What do you think about Pilot wave theory?

An easy way to demonstrate fundamental principles of quantum mechanics needs "special thinking" but it faces criticism from many researchers!

The pilot wave theory gives a consistent interpretation for quantum mechanics that is physical and deterministic. Thus avoids troublesome notions such as wave–particle duality, instantaneous wave function collapse and the paradox of Schrödinger's cat but introducing nonlocality. !

- "Quantum mechanics writ large", Bush, J.W.M, 2010.
- "Pilot waves, Bohmian metaphysics, and the foundations of quantum mechanics", lecture course on pilot wave theory by Mike Towler, Cambridge University (2009).
- "Pilot-Wave Hydrodynamics" Bush, J. W. M.,
*Annual Review of Fluid Mechanics*, 2015

7th December 2017, **Abhilash Asok**.

## Posted by Dr Siddharth Ghosh

Do you know protein needs to be folded optimally?

What are mathematical knots?

## Posted by Dr Siddharth Ghosh

Susskind on why does mathematics work?

## Posted by Dr Siddharth Ghosh

Do you know the toy maker Arvind Gupta?

## Posted by Dr Siddharth Ghosh

'Unending education' towards the 'No School' from the 'Last School'

## Posted by Dr Siddharth Ghosh

Why is pi here?

## Posted by Dr Moumita Ghosh

3D movie of atomic arrangements

## Posted by Dr Siddharth Ghosh

What is a Quantum Fluid?

## Posted by Alok Shankar

X-rays from the electromagnetic spectrum can be used to image human anatomy. Computed Tomography (CT) can provide a 3 dimensional volume of anatomy through multiple projections (data acquisition) and backprojection (image reconstruction) as described below in the video.

## Posted by Dr Siddharth Ghosh

Sir Lawrence Bragg on atomic crystals using bubble rafts

## Posted by Dr Siddharth Ghosh

What is a Quantum Computer?

## Posted by Dr Siddharth Ghosh

Do we need academic degrees?

Header by Sid.