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## TTK Outreach: A Universe of Possibilities Probabilities

January 30th, 2018 | by

The universe may not be full of possibilities –most of it is dark and fatal– but what it does have in abundance are probabilities. Most of us know about Newton’s three laws of motions. Especially the third which, taken out of context, apparently makes for a good argument justifying revenge. For centuries, Newton’s laws made perfect sense: an object’s position and velocity specified at a certain time gives us complete knowledge of its future position and velocity aka its trajectory. Everything was neat and simple and well-defined. So imagine our surprise when we found out that Newton’s laws, valid as they are on large scales, completely break down, on smaller ones. We cannot predict with 100% certainty the motion of an atom in the same way that we can predict the motion of a car or a rocket or a planet. And the heart of this disagreement is quantum mechanics. So today let’s talk about two of the main principles of quantum mechanics: duality and uncertainty.

## Duality:

We begin with light. For a long time, no one seemed to be quite sure what light is. More specifically, we didn’t know if Light was a bunch of particles or a wave. Experiments verified both notions. We could see light interfering and diffracting much like two water waves would. At the same time, we had phenomena such as the photoelectric effect which could only be explained if Light was assumed to be made of particles. It is important to dwell on this dichotomy for a bit. Waves and particles lie on the opposite ends of a spectrum. At any given instant of time, a wave is spread out. It has a momentum, proportional to the speed with which it is traveling, but it makes no sense to talk of a definite, single position of a wave by its very definition. A particle, on the other hand, is localized. So the statement, ‘Light behaves as a wave and a particle’, is inherently non-trivial. It is equivalent to saying, ‘I love and hate pineapple on my pizza’, or ‘I love science fiction and hate Doctor Who.’

But nature is weird. And Light is both a particle and a wave, no matter how counter-intuitive this idea is to our tiny human brains. This is duality. And it doesn’t stop just at Light. In 1924, de Broglie proposed that everything exhibits a wave-like behavior. Only, as things grow bigger and bigger, their wavelengths get smaller and smaller and hence, unobservable. For instance, the wavelength of a cricket ball traveling at a speed of 50km/h is approximately 10-34 m.

And it is duality which leads us directly to the second principle of quantum mechanics.

## Uncertainty:

The idea of uncertainty, or Heisenberg’s Uncertainty principle, is simple: you can’t know the exact position and momentum of an object simultaneously. In popular science, this is often confused with something called the observer’s effect: the idea that you can’t make a measurement without disturbing the system in some unknowable way. But uncertainty is not a product of measurement, neither a limitation imposed by experimental inadequacy. It is a property of nature, derived directly from duality.

From our very small discussion about waves and particles above, we know that a wave has a definite momentum and a particle has a definite position. Let’s try to create a ‘particle’ out of a wave, or in other words, let’s try to localize a wave. It’s not that difficult actually. We take two waves of differing wavelengths (and hence differing momenta) and superimpose them. At certain places, the amplitudes of the waves would add up, and in others, they would cancel out. If we keep on adding more and more waves with slightly differing momenta, we would end up with a ‘wave-packet’, which is the closest we can get to a localized particle.

Image taken from these lecture notes.

Even now, there is a small, non-zero ‘spread’ in the amplitude of the wave-packet. We can say that the ‘particle’ exists somewhere in this ‘spread’, but we can’t say exactly where. Secondly, we’ve already lost information on the exact momenta of the wave and so there is an uncertainty there as well. If we want to minimize the position uncertainty, we’d have to add more waves, implying a larger momentum uncertainty. If we want a smaller momentum uncertainty, we would need a larger wave-packet and hence automatically increase the position uncertainty. This is what Heisenberg quantified in his famous equation:

### Δx Δp ≥ h/4π

And so we come to probabilities. At micro-scales statements such as, ‘the particle is in the box’, are meaningless. What we can say is, ‘the particle has a 99% probability of being in the box’. From Newton’s deterministic universe (which is still valid at large scales) we transition to quantum mechanics’ probabilistic one where impossible sounding ideas become reality.

The Doctor once said, “The universe is big, it’s vast and complicated, and ridiculous. And sometimes, very rarely, impossible things just happen and we call them miracles.” Or you know, at small enough scales, a manifestation of quantum mechanics. And that is fantastic.