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Nitty-Gritty of Quantum Mechanics from a Rubberneck’s POV (Chapter 1)

Quantum Mechanics- a hard nut to crack? Let’s break it down to its root words- Quantum and Mechanics. Let’s start with Mechanics; this is related to the mathematical formalism of things- from a grain of sand to the moon and so on- with which the whole universe can be described. Quantum basically refers to the smallest discrete unit of any physical property, such as energy or matter. So, a combination of these two words results in something that describes how everything in this world works; it is like a panacea for everything in this universe. In other words, QM describes how and why atoms and sub-atomic particles behave in certain manners. Moreover, it is a great way to perceive how the microscopic world affects the macroscopic world- the world in which we live and see. Elucidating all aspects of Quantum Mechanics is beyond the scope of this article, rather this article might act as an introductory lesson for beginners.

In short, QM is the study of atoms and sub-atomic particles which can describe the macroscopic effects that we are able to witness so as to why they are occurring in the first place. QM is “A projection of this beautiful, witnessable world from an electron’s or an atom’s POV”.

Let’s ease that out with an example. QM deals with probabilities, not certainties. Conversely, classical mechanics is all about certainties; like what is the value of an object’s velocity, momentum, or what is its exact position if it is kept inside a closed-lid box. If the object is near the bottom-left edge of the box then we know its position, and likewise, we can evaluate its instantaneous velocity and other physical properties. QM, being the polar opposite of Classical Mechanics, does not reveal any certain values for these properties unless or until these properties are measured. This seems very counter-intuitive but trust me, all of this will start making sense when we gradually dive deep into the realm of Quantum Mechanics.

A quantum-mechanical meme

Squirrel in a Box:

Let’s consider a box inside of which is a squirrel. Classical Physics can accurately measure its position, momentum, acceleration, or any other physical properties. Say, the box having a length of 6 meters (from X = 0 to X = 5) is occupying a squirrel at X = 1 and after 5 seconds, it moves to X = 2; that means we can define its final velocity, acceleration using Newton’s laws of motion, given some adequate known initial properties. But in Quantum Physics, the behavioral action of that very squirrel cannot be easily explained or the physical properties cannot be measured since this branch of science refers to only probabilities of possible states; that is, that squirrel has a probability (say) of being at X = 1 is 0.03 and likewise, at the other positions the probabilities will vary according to certain conditions. Wait? Yes, you read it right! The squirrel can dance at all the positions inside the box simultaneously- at least QM approves that, even with mathematical proof. So, unless measured, the position of the squirrel can be at any of those locations (from X = 0 to X = 5)

A quantum-mechanical system: Box-bound Squirrel
After observation, the squirrel’s position lies only on one value. x = 2

Classical Physics is deterministic and Quantum Physics is probabilistic

What is the Reason Behind this Quantum Weirdness?

One-liner: Classical Physics deals with particles while QM is all about waves, and waves have, not necessarily always, an up-and-down nature which can easily be expressed as probability functions where an up-state means a higher probability than the down-state (there's a caveat here which I will explain in a bit).
Detailed: Notions provided by QM go back in time when electrons exhibited an interference pattern in the double-slit experiment, so did regular wave functions. But, electrons are particles so how do they behave like waves? Or, how do these electrons exhibit wave-like properties when they are particles? In addition to this, the light also exhibited particle-like properties (light going through only one slit and not exhibiting an interference pattern unlike in the case of two slits where photons resulted in an interference pattern). So, photons can also act like particles in a single-slit environment and waves in double-slit environments. So did an electron. Thus, they both have wave-like and particle-like properties. This is not so intuitive for classical physics proponents. Also, waves are more fundamental than particles since it is easier to express particles by waves than waves by particles. So, by QM logic, everything in this universe can be described by waves. And these waves very readily give out the probabilities of all possible states within a quantum system. So, the squirrel can be in all possible positions until its position is measured. Only then, we can say its position with certainty and then the wave function collapses- the measured position has a probability of 100% while all other probabilities become 0% at the moment of measuring the squirrel’s position. For the electron case, it starts as a particle, then while it goes through the two slits, it acts like a wave (only explanation as to why there exists an interference pattern), and finally, when it hits the surface where it leaves a dot after hitting, it again becomes a particle. So, the sequence for an electron’s nature goes like this:

Particle (Initially)> Wave (Until impinging on the detector)> Particle (after hitting the detector)

Now, going back to the box-bound squirrel to have a deeper look at it. So, the squirrel can be at all the positions and this weird behavior is described by its wave function. Yes, the cyan curve is the squirrel’s wave function. It does not mean the trajectory of this squirrel inside the box, unlike in the case if you throw this squirrel in the air under the influence of gravity. Then, Newton’s equations of motion will act as a significant factor as the trajectory (parabola) will define the squirrel’s actual path. We can get where this squirrel will be after 5 seconds and we can get accurate measurements of position, velocity. But, for the box-bound squirrel, this is not the case. The curve does not signify its actual path, rather it gives a probabilistic idea of where the squirrel can be. But, we can see, from the picture below, that there are negative values of probabilities at x = 0.5, 2, 4. We all know the probability is just a number that ranges between 0 and 1. There’s no room for negative probability. So, a good way to make these negative values as significant as the non-negative (positive and zero) ones, the values are squared so that we get to have all the values- this is known as the probability density function. This denotes the possibility of finding the squirrel at each position; the higher the peak, the more chances the squirrel will be found at that location. Only the location is our concern, we can use this exact concept for other physical properties, such as velocity, momentum, and so on.

Squirrel’s Wave Function
Wave Function and Probability Density

What happens to the squirrel’s wave function when it is observed?

A Wave function, before observation, acts as a probability curve. It acts as a certainty curve only after it is observed. Because, before measurement, the squirrel is in a state that is a superposition of all other states. But, when we use a method to measure its position, then it becomes a certainty curve which indicates the measured position with certainty.

Before Observation

In the above image, we can see there was a higher probability that the squirrel is likely to be found at x = 1 than x =2. But, when we measure this, like in the image below, it is revealed that the squirrel is at x = 2. It is said that the squirrel has interacted with its environment at the instance when the measurement occurred.

After Observation

In the next chapter, we will go through some mathematical formalism of quantum science. I am new to this topic and never had any prior courses offered to me when I was doing my undergrad. So, any kind of opinion is appreciated. Correct me if I missed out on any crucial information or somehow confounded anything. I have attached herewith a link to my google drive where I have uploaded these images that I have drawn in Adobe Illustrator CC 2019 so that you can tune these images to your satisfaction.

Google Drive Link: https://drive.google.com/drive/folders/19seII1ipd0Y-jE_yU9W2CpuTty9ayw1N?usp=sharing

Thank you for your time. Best regards for your patience.

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