TL;DR – Events with low probability happen all the time, even ones with probability zero. When I was in high school, a friend of mine vehemently told me: “Me with him? It’s more likely that donkeys start raining. It’s more likely that hell freezes over. It’s more likely that I enter a convent and never …
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} …
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 …
TL;DR – Statistics applies to populations, not individuals. There is no such thing as the probability of one individual having some characteristic. I remember talking to a friend of mine a long time ago. “I wonder what will I do when I grow up?” “What are the chances that I become a doctor?” “What are …
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 …
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 …