Quantum - 1

I’ve been reading a bit of popular science articles on quantum mechanics recently. In this post, I’d like to share a bit of what I understood, because I found it really exciting. If any of you find inaccuracies in this post, let me know. I will fix them

   There’s so much jargon and equations involved in learning quantum mechanics, which makes approaching the subject scary. Half of what I read goes over my head. But, the physics underlying those equations seems beautiful. Let’s begin this post by answering a question, ‘What’s the difference between classical physics & quantum physics?’

   Let’s take a bike moving at the speed of 80KMPH from Bengaluru to Chennai. We’re assuming that the road is perfectly straight without speed bumps, there’s no traffic and the bike is going to maintain its speed forever. How do we describe the bike? We can say, the bike is currently(at 10PM on 16/09/2021) 30KM away from Bengaluru going towards Chennai at the speed of 80KMPH(speed + direction gives us velocity) with no acceleration. If someone asks us where the bike will be at 11PM, we can say that it is going to be 110KM away from Bengaluru on its way towards Chennai. There’s a certainty here. 

  In Quantum mechanics, there’s a loss of certainty. It’s a bit like the real world. In a real world, the bike will never be able to maintain a constant 80KMPH. There’re speed bumps, there’re toll gates, there’s traffic, there’s friction, the road isn’t perfectly straight, there’s the minor issue of driver fatigue too. So, Quantum mechanics gives us the probability of the rider position. It can say, the driver will be 110KM away from Bangalore on way to Chennai with probability 0.33, 100KM away with probability 0.25 and so on. If the rider makes a huge number of trips from Bengaluru to Chennai, we can test out if the probability values are accurate or not. In Quantum mechanics, the state of particle is given by wave function and it indirectly tells us about the probability.

   Secondly, how do we calculate the wave function? To do that, we make some assumptions, like boundary conditions. For example, we could assume that a school going kid will be present within the premises of school on a working day. So, we assume that the probability of kid present outside of school is 0. Similarly, if we add up all the different probabilities of the kid present in each place in school, we should get 1. In simple terms, it means that we’re assuming, the kid will be present in school and not anywhere else. Similarly, we try to characterise the system and solve its hamiltonian. Basically, we solve some equations by placing some more constraints of energy and use that to get the wave function. We will see how this will be used later in the post. 

   Thirdly, the most important principle of Quantum mechanics in so far as I understood it is hidden within name. It’s quantisation. What’s Quantisation? Let’s say someone told you, I’ve some notes of money in my wallet and asked you to guess a note. What would you guess? You could guess that I’ve a 10 rupee note, or a 20 rupee note, a 50, 100, 200, 500 or a 2000 rupee note. You’ll never guess that I’ve a 72 rupee note in my wallet. Because, we all know that RBI doesn’t print a 72 rupee note. So, that’s an illegal note. Similarly, quantum mechanics postulates that the observables of system are quantised. For example, it states that the energy/momentum/position/velocity or any other observable can only take one of the allowed values for that observable. It never takes illegal values. If we know the wave function of the particle and characteristics of the system, we can calculate the probability for the observable to be one of the quantised value. If we recall our high school physics, we can find some examples. The angular momentum of electron is quantised. Similarly, the energy of an electron is quantised.

  How do we calculate what quantised values does a measurable like energy take? Usually, we characterise the system in form of a Matrix. For example, to calculate energy of a particle in hydrogen atom, we first characterise the system as a Hamiltonian Matrix(My understanding is that most of the research in quantum mechanics deals with figuring out these matrices for various observables under various conditions). The eigen values of this matrix are the quantised values that our electron is allowed to take. This is another postulate of quantum mechanics.

  Okay, now that we’ve quantised values, how do we calculate the probability that electron’s energy is particular eigen value? All we need to do is to write wave function as a linear combination of its eigen vectors. The square of normalised coefficients of these eigen vectors gives us the probability corresponding to that eigen value. To give an inexact parallel, we can think of it as a factorisation in powers of primes. For example, 56 can be written as 8 * 7 which is (2^3 * 7). So, when we reduce 56 as multiples of powers of primes, we get that it has 2 raised to power 3 times, and 7. Here 2 and 7 are similar to eigen vectors. Their powers, 3 and 1 can be treated as their coefficients or the terms which help us determine the probability that the observable takes a particular quantised value. Their squares would be the odds of that particular value. For example, it could take value corresponding to 2 as 3^2 I.e 9/10 times. And value corresponding to 7, 1 out of 10 times