Spins, Rotations, and the Beauty of Complex Numbers

This is a simple quantum state … |➚> = α|↑> + β|↓> … built from an up |↑> state and a down state |↓>. α and β are complex numbers. The result |➚> is in the middle, oblique. The oblique state is a superposition or the up and down base states. Making a measurement, you…

Integrating the Delta Function (Again) – Dirac Version

The Delta Function is, roughly speaking, shaped like an infinitely tall and infinitely thin needle. It’s discovery – or invention – is commonly attributed to Paul Dirac[*]. Dirac needed a function like this to work with integrals that are common on quantum mechanics, a generalization of a matrix that has 1’s in the diagonal and…

Poetry: Dynamical Variables and Observables

The lines of the following poem are phrases selected from consecutive pages of the second chapter of Paul Dirac’s Principles of Quantum Mechanics, Fourth Edition (Revised), Dynamical Variables and Observables. we may look upon the passage for the triple product We therefore make the general rule in spite of this fundamental difference which conforms with…

Poetry: The Principle of Superposition

The lines of the following poem are phrases selected from consecutive pages of the first chapter of Paul Dirac’s Principles of Quantum Mechanics, Fourth Edition (Revised), The Principle of Superposition. ~ one would be inclined to think There must certainly be some internal motion from general philosophical grounds we cannot expect to find any causal…

Entropy and Dimensions (Following Landau and Lifshitz)

Some time ago I wrote about volumes of spheres in multi-dimensional phase space – as needed in integrals in statistical mechanics. The post was primarily about the curious fact that the ‘bulk of the volume’ of such spheres is contained in a thin shell beneath their hyperspherical surfaces. The trick to calculate something reasonable is…

Learning Physics, Metaphors, and Quantum Fields

In my series on Quantum Field Theory I wanted to document my own learning endeavors but it has turned into a meta-contemplation on the ‘explain-ability’ of theoretical physics. Initially I had been motivated by a comment David Tong made in his introductory lecture: Comparing different QFT books he states that Steven Weinberg‘s books are hard reads because at…