Category: Classical mechanics

Why are states identified by position and momentum?

Why are states identified by position and momentum?

TL;DR – States are identified by position and momentum because this is the space that allows us to define coordinate-invariant densities. As we saw in many of our posts, Hamiltonian mechanics describes the deterministic and reversible evolution of states. This leaves one question: why are states fully described by position and momentum? Why not only …

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CPT-like theorem for classical particles

CPT-like theorem for classical particles

TL;DR – Inverting the direction of time and space is equivalent to inverting the direction of the parametrization. In previous posts we have seen the classical equivalent for anti-particles. Here we see something that looks similar to the CPT theorem. Naturally, it’s not a complete analogue because we are not really looking at fields at …

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What are Poisson brackets?

What are Poisson brackets?

TL;DR – The Poisson bracket tells how a quantity changes under a transformation generated by another. It also tells us the state count of a cell of phase space identified by the two variables. Another operator of particular importance in Hamiltonian mechanics is the Poisson bracket: $\{f,g\}=\frac{\partial f}{\partial x}\frac{\partial g}{\partial p} – \frac{\partial f}{\partial p} …

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Classical spin

Classical spin

TL;DR – The Hamiltonian description of a direction in space is the classical version of the spin of a particle. Now we turn our attention to the classical version of another quantum concept: spin. What we show here is that Hamiltonian motion of a spatial direction is qualitatively the same as the evolution of spin …

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Classical free particles (i.e. Klein-Gordon)

Classical free particles (i.e. Klein-Gordon)

TL;DR – The extended Hamiltonian for a classical free particle is the classical version of the Klein-Gordon equation. After having seen the classical version of antiparticles we will see the classical version of the Klein-Gordon equation. This is the essentially the extended Hamiltonian for the free particle. Let’s look at the details. 1. Extended Hamiltonian for …

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Classical antiparticles

Classical antiparticles

TL;DR – Antiparticle states are those for which time is a decreasing function $t(s)$ of the trajectory parametrization instead of an increasing function (i.e. time and trajectory parametrization are anti-aligned). In a previous post we introduced the extended Hamiltonian equations. In the next few posts we will see that they allow us to create parallels …

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