Terence Tao

There's also an object called Hamiltonian.

It's a different type of object.

It's what's called an operator rather than a function.

But again, once you specify it, you specify the entire dynamics.

So there's something called Schrodinger's equation.

that tells you exactly how quantum systems evolve once you have the Hamiltonian.

So side by side, they look completely different objects.

One involves particles, one involves waves, and so forth.

But with this centrality, you could start actually transferring a lot of intuition and facts from classical mechanics to quantum mechanics.

For example, in classical mechanics, there's this thing called Noether's theorem.

Every time there's a symmetry in a physical system, there is a conservation law.

So the laws of physics are translation invariant.

Like if I move 10 steps to the left, I experience the same laws of physics as if I was here.

And that corresponds to conservation momentum.

If I turn around by some angle, again, I experience the same laws of physics.

This corresponds to the conservation of angular momentum.

If I wait for 10 minutes, I still have the same laws of physics.

So this time transition invariance, this corresponds to the law of conservation of energy.

So there's this fundamental connection between symmetry and conservation.

And that's also true in quantum mechanics, even though the equations are completely different.