First, install Julia from its website. Use the link for the latest version compatible with Apple M1 processor. The current version I installed is Julia 1.7. It’s as simple as extracting the app ins copying it to the application folder.

You can run Julia like any other app from from the application folder or the spotlight search. It opens Julia in the terminal app. If you will need to run it directly from the terminal, create an alias to run the executable in your bash or zsh profile. I use zsh and I simply appended my .zshrc file with the following line:

alias julia="/Applications/Julia-1.7.app/Contents/Resources/julia/bin/julia"

Next, you need to install package IJulia using Pkg. Follow the instructions at the official IJulia page. IJulia allows you to run Julia in as a notebook in the browser. This way of working with Julia is similar to running Python using IPython on Jupiter Notebooks. The package installer Pkg of Julia also installs miniconda and Jupiter within Julia along with many other dependencies. Familiarize yourself with Pkg as you will need it to install many packages you need for your projects.

To open a IJulia notebook, run the command:

using IJulia; notebook()

This will open a window in your default browser where you can open or create an IJulia notebook. Happy coding!

The resulting probability distribution function in this case is a bell like curve with peak at x=0 along the x-axis and the width of the peak is twice of N​. In the limiting case of very large N, the distribution becomes a very common distribution is statistics called the normal distribution.

Why the distances travelled by a random walker in one dimension obey normal distribution with its mean at 0 and standard deviation square root of N? This is an important question because the random walk is one of the many phenomena where many small quantities with uniform random distributions add up to a final quantity that has normal distribution. When you add up a bunch of random numbers that were uniformly distributed, you find that their sum is normally distributed. This result is so fundamental that it has become a standard theorem in the science of statistics: the mean value theorem. The problem of random walk is similar but not identical. Yet, this problem is equally fundamental and the results can be viewed as a theorem or law.

We expect that the mean displacement of the random walkers is zero. That’s why most of them were strolling around the origin at the end of the walk. Also, you find that about half of the strollers are to the left of the origin and half are to the right. So, when you find the average value of these negative and positive distances, they add up to zero.

Let’s ask a slightly different question this time. What is the average of the distances |x| (without the + or – signs)? This time, the distance travelled to the left and to the right of the origin do not cancel out one another. The average distance is found to be proportional to the square root of N. For simplicity, let’s suppose that each step takes unit time. Then, the average distance travelled after time T is proportional to the square root of T. This relationship is sometimes called the Einstein’s law in the remembrance of the initial work done by Einstein on Brownian motion before the publication of the special theory of relativity. Einstein was studying the realistic movements of particles suspended in a liquid randomly bombarded by the its molecules from all around. However, the dependence of the average distance traveled by a particle on square root of time is a generic feature of random walk. To appreciate the full implications of random walk and the mathematics behind the Einstein’s law, we should consider the problem of random walk in two or more dimensions. We will explore these topics in the coming posts.

When there are many possible outcomes for an experiment like your random stroll on the number line, it is always instructive to repeat the experiment again and again (say, m=1000 times) and analyze all the outcomes. By repeating the experiment sufficiently large number of times, we expect that all that can happen will happen. Hence, a sample of all the outcomes will tell us what are all possible outcomes and how frequent a particular outcome was. In fact, we can plot the frequency distributions in the entire space of possible outcomes. In such a plot, we plot the outcomes on the x-axis and the number of times that outcome occurred on the y-axis. The sum of all such frequencies will be 1000, the total number of outcomes. 

To make this representation independent of the number m, we divide each frequency by m and call it probability. In our case of the random walk in one dimension, we can plot the displacement x after N=100 steps on the x-axis where x is an integer between -N and N, and we plot the probability p(x) of ending up at the point x  after N steps on the y-axis. The probability distribution in this example is discrete.

In the case of random walk in one dimension, we expect the probability distribution to be symmetric around the origin: the probability to end up at the point +x  is identical to the probability of ending up at the point -x. Moreover, we expect the probability distribution to be peaked at the origin. In the coming post, we will see how this distribution looks like and why.  

Here, I am extracting 25 lessons from the lecture. The points 1-5 are about having the right outlook towards research work. The points 6-15 are about how to approach the research problems. Hamming touches the issue of collaborative work in points 16-18. In the points 19-22, he talks about how to present our work and how to make a real contribution to our field. The concluding points 23-25 are about dealing with the larger system in which we work.

1. Drop the modesty and accept your aspiration to do significant work.
2. Luck favors the prepared mind.
3. Courage, commitment, and drive are more important than intelligence.
4. Knowledge and productivity are like money that grows with compound interest.
5. Make a significant shift in your area of research after every seven years.
6. Transform a defect to an asset by turning problems around a bit. Convert something difficult into something easy and still important.
7. Tolerate ambiguity and keep track of what doesn’t fit into the picture.
8. Keep your problem at the center of attention for a long duration so that it can access your subconscious sub-minds.
9. Work on important problems in your field that are attackable.
10. Contemplate about great ideas once a week. Understand the big problems in your field.
11. Have a list of 10 to 20 problems and keep looking for an attack to them. Whenever new ideas show up anywhere, find out what it bears for some of your problems and get after it.
12. Finding the way to get an already known answer in elegant or faster manner is also an important contribution.
13. Attack an isolated problem only if it is a characteristic of a class and others can stand on your shoulders.
14. Find the generalization of an isolated problem and then attack the whole class.
15. It is not how much but how you read that matters.
16. Keep your doors open for new ideas.
17. Brainstorm with people who are can reflect ideas back instead of just absorbing them.
18. It’s very valuable to have first-class people around.
19. Sell by writing clear papers and giving talks, both formal and informal.
20. Get credit for what you do.
21. Spend at least as much time in the presentation as you did in the original research.
22. Books that filter the essentials are the best long-term contributions.
23. Learn to work with the system and take advantage of all the system has to offer.
24. Conform to simple expectations of others like dressing when the price of asserting your ego is too high.
25. The day your vision, what you think needs to be done, is bigger than what you can do single-handedly, then you have to move toward management.

Interestingly, when we solve this equation for an electron around a nucleus, it doesn’t tell us how the position of the electron changes with time. Instead, this equation throws on us a function that simply contains information about the chance of finding the electron at a place for a given time. Where is the electron? We do not know. What we can calculate instead is a cloud of probabilities for the electron. If one could take thousands of pictures of the electron moving around the nucleus and superimpose all of them on a photographic plate, the final picture that would emerge will match with the cloud of probabilities calculated from the Schrodinger equation. So, the electron exists not like a dense billiard ball but like a diffused cloud: you cannot pinpoint where it is. It is everywhere simultaneously in the cloud!

The electron cloud can change with time as things change around the atom. Suppose that two atoms come closer to one another. The electron clouds around them might start attracting or repelling one another. The problem is that it is very difficult to develop a common sense about how does this picture of the electron as a cloud of probabilities changes as it interacts with a given potential (the technical name for the landscape in which the electron moves). The cloud analogy does not help us much in this task.

There is another way to imagine this cloud of probabilities that is more intuitive. We can picture the potential in which the atom moves as a landscape with mountains, valleys, and ridges. We can picture the electron cloud as a small pool of liquid in the landscape. Where will the liquid settle after we pour it in the landscape? It will seek the deepest valley and settle there making a little pond. It tries to minimize its energy which decreases with the altitude in the landscape. But, the same thing would happen with a billiard ball. So this is just a classical picture based upon Newton’s laws. Our pond of water is still behaving in the Newtonian way.

How does quantum mechanics come into this picture? Quantum mechanics attributes some strange properties to this liquid. It can not only flow downhill, but it can also flow uphill as well. Think of the downhill motion as the average motion obeying the laws of Newton and the uphill motion as some strange quantum property of this liquid to persistently explore its environment by flowing against the dictums of Newton. So, if you trap it in a shallow pond, it will not just stay there but will occasionally get impatient and shoot out the boundaries of the pond and then return back. If you trap it in a deep well, it will spill out flowing uphill in all directions! If it is rolling downhill and encounters a mountain ahead, don’t think that it will climb the same height from which it came down. It will sore to greater heights and may even jump to the other side of the mountain. A billiard ball obeying Newton can never do that. This climbing over the heights which is impossible in the Newtonian framework is called the quantum tunneling. If you trap it in a large rectangular tub, it will move back and forth forming nice waves. 

This strange liquid is not an imagination. It is how an electron behaves when it moves in an atomic landscape. The strangeness of this liquid (its ability to spread around and move uphill) is measured by a fundamental constant of nature: Planck constant. It is called a constant because it is the same everywhere for all objects at all times. But, we can imagine a faraway universe on which this constant is very large; so large that all objects have this quantum strangeness. How strange that universe would be? It is a good topic for some science fiction. In our real world, this strangeness is small and manifest only at the atomic scale. But it is essential for life to grow out of electrons, atoms, and molecules. It is also essential for the formation of galaxies in the evolution of the universe. 

How did these textbooks achieved to teach intricacies of quantum mechanics to kids that are in their final years in schools? By depicting a wave function as an electron cloud. When we were visualizing, for example, the wave function of the hydrogen molecule, we were instead creating a 3D picture of not the wavefunction of the molecule but some other function calculated out of the wavefunction – the probability density. The probability density is like taking the square of the wave function. Yet, it is a more workable quantity. You can easily start intuiting about it. This three-dimensional picture of the probability density function is like a cloud of electron and was called the orbital in our textbooks.

You will object, “how can two electrons of the hydrogen molecule make a cloud?” Yes, it is possible because of the miracle of quantum mechanics, our textbooks told us. If you take a snapshot of the electron at a given time, you will find it at a point near to protons of the hydrogen moment. Take a snapshot at the next moment and it is gone somewhere else. The successive snapshots, when developed on a single photographic plate, will give the final picture of how the electron is behaving. The picture will look like a cloud which is denser around the two protons and less dense in the middle – like a dumbbell.

The most intriguing aspect of this picture is that the electron is not moving from one point to another in a well-defined trajectory. If you quickly project the snapshots on the screen like an animation film instead of developing them on a single photographic plate, don’t expect a nice movie of an electron moving from one point to another in a continuous manner. You will see the electron far away from both the protons at one instant and at the next instant it will be in the middle of the two protons. You will be seeing a dot appearing and disappearing on the screen at random points – like a shooter shooting bullets on a screen. However, if you watch long enough, you will discover a pattern in the dots on the target screen. You will find that the shooter is aiming for two points – the two protons! So, most of his aims are distributed around those two points.

We were told that there are only a few standard shapes these clouds can acquire, each shape having its own energy. When two atoms come close enough, their electrons clouds rearrange into new shapes of lower energies. These new clouds were called molecular orbitals. A few qualitative rules about these molecular orbits enabled us to make sense of the complicated world of molecular structure.

Let’s estimate the probability of a team winning the toss N times consequently. When N=1, we have just two possible toss outcomes for our favorite team: win or loss. Let us denote a win by W and a loss by L. So, the possible outcomes for a single match are {W, L}. This simply means that the probability of winning toss one time is 1/2. When N=2, the possible outcomes of the toss can be {WW, WL, LW, LL}. So, the possibility of winning the toss consequently two times by our favorite team would be 1/4 as there is one such event in a possible set of four events. When N=3, the possible outcomes are {WWW, WWL, WLW, LWW, WLL, LWL, LLW, LLL}.

The probability of our team winning three tosses consequently is clearly 1/8. Looking at this pattern, we can say that the probability of a team winning N times consequently in a series of N matches is 1/2^N. However, the probability of the other team winning the toss in all the N matches is also 1/2^N. So, the probability of any of the two teams winning the toss consequently in N matches is 1/2^N+1/2^N which is 2/2^N or 2^(N-1).  So, or a series with 4 matches,  probability of one team always winning the toss is 1/8. For 5 matches, this probability is 1/16.

Most series in cricket are of 5 matches. So, on average, in every sixteenth of such series, one of the teams should win the toss in all the matches. It’s not that rare! Once in a year or so, we should see this happening.

If we are interested the probability of a team loosing toss only once in a series of N matches, the probability will be even more higher: 2N/2^N. (How to do such a calculation is a topic for another post.) For, N=5, this probability is  5/16, a fraction very close to 1/3. So, in every third cricket series of 5 matches on average, a team should loose toss only once. This is quite a headline!

I like to explain this principle using a loose biological analogy. I like to view light a small organism trying to reach its food in shortest possible time by adapting its trajectory in a medium where its movement varies from place to place. The organism does not want to be trapped in the denser regions where it can be trapped for longer times. It want to make the maximum use of the rarer regions where it can move relatively faster. Of course, the analogy is not literal in the sense that light senses its medium and adopts a path that minimises the time taken to reach its destination. The analogy is appropriate only in the sense that it describes the optimisation going on in the propagation of light in a language we can easily understand.

In classical mechanics, a similar principle, called principle of least action, dictates the trajectory of a particle under the influence of a force field. However, the allowed path does not minimise time but a quantity called action. The path that minimises action is always consistent with Newton’s law of motion. So, the principle of least action is mathematically equivalent to Newton’s law of motion.

In the quantum mechanics, electrons and other elementary particles do a similar thing: they sense all the neighbouring paths so that they can move on the fastest one. Yet, this sensing is not perfect in quantum mechanics and you may find the electron straying into regions not allowed classically. Yet, if you let hundreds of particles move from one point to another, most of them will be roaming around the classical path.

Feynman was not happy with the way his contemporaries were visualising the quantum mechanics. So, he invented a formulation of quantum mechanics which assigns probabilities to all the various paths a particle can take while going from one point to another point. It turns out that the probability for the path dictated by Newton’s law comes out to be the maximum while the probabilities for all other alternate paths is heavily suppressed for almost all large-scale motions we encounter in our daily lives. However, for motions of elementary particles like electrons at small scales, the probabilities for alternate paths are also appreciable. The Schrodinger equation simply tells those probabilities. 

But, when a microscopic particle like electron moves between two points crossing some hindrances, it behaves in a fashion similar to the small organism drifting in search of rarer regions in a fluid. Just like the living organism, the response of the electron is not deterministic in quantum mechanics. You make it run again, and it will follow a different trajectory. As always, there will be many trajectories close to the classical path. But, we will see more drastic deviations more frequently as we probe motions on smaller and smaller scale.  Of course, the analogy doesn’t go too far. Electrons are much simpler than living organisms.  They simply obey Schrödinger equation.