QM

Background

The history of Science can be told as a story of revolutions. We had the Copernican Revolution in Astronomy, and the Darwinian Revolution in Biology. Physics has witnessed two revolutions that transformed the very foundations of the discipline: Newtonian Mechanics, which describes the "classical" world, and Quantum Mechanics (QM), which describes physical systems with small dimensions and small masses, e.g., atoms and molecules.

In 1964, Physicist Richard Feynman said: "I think I can safely say that nobody understands Quantum Mechanics" (take a quick read of this link which contains a short introduction to a lecture he gave). This sentiment is equally true today. For a theory that had seen unparalleled empirical success at predicting and accounting for the outcome of high precision experiments, the embarrassing truth is that Physicists cannot claim to have a very good understanding of what the theory actually is. We do, however, understand an enormous amount about the theory. This is the deepest and most fundamental picture of the world we now have.

Consider a hydrogen atom. When we visualize it in our head, we tend to imagine an electron orbiting around a proton, much like a planet in the Solar System orbits around the Sun. This is the Rutherford Model of the atom. It's also... wrong. As we learned from black body radiation, the photoelectric effect, and radiation from excited atoms, small objects obey the laws of quantum mechanics.

In Classical Mechanics, if you know the state of the system you can predict with certainty what any measurement outcome would be. In QM, the state of the system is a superposition of all the possible measurement outcomes, known as "the wave function" of the system. The wave function is a combination of every result you could get by doing an observation, with different probabilities for each possibility. The state of an electron in an atom, for example, would be a superposition of all the allowed orbits with fixed energies. QM is a profound change from Classical Mechanics, where the outcomes of an experiment are not perfectly predictable even if we know the current system state exactly. QM tells us the probability for seeing any particular outcome when we observe a quantum system. The lack of perfect predictability is not because we don't have enough information about the system, it's just the best that QM allows us to do. When we say that a quantum state is a superposition, we don't mean that "it could be any one of several different possibilities, but we're not sure which". We mean that "it is a weighted combination of all the possibilities at the same time".

If it makes your brain hurt, you're not alone. QM took a long tome to put together, and we're still arguing about what it all means. The Schrödinger Equation describes how a quantum system evolves over time when we're not observing it. The left hand side of the equation asks how does the state of the system evolve, and the right hand side provides the answer: by performing some operation (the Hamiltonian Operator) on the wave function. It's parallel to Newton's famous F=ma equation, in which forces determine how the system changes. The system evolution according to QM is similar to Classical Mechanics in a way that it too is smooth, reversible, and completely deterministic. If that was the entire story, QM wouldn't have been problematic. But there's also an entirely different way in which quantum states can evolve: when it is observed. In that case, the wave function "collapses", and we obtain some particular possible outcome. The collapse is sudden and non-deterministic: knowing what the state was before, you can't perfectly predict what the state will be afterwards; all you have are probabilities. Despite the appearance of probabilities, QM is extraordinarily precise.

Spin multiplicity