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…

Dirac’s Belt Trick

Is classical physics boring? In his preface to Volume 1 of The Feynman Lectures on Physics, Richard Feynman worries about students’ enthusiasm: … They have heard a lot about how interesting and exciting physics is—the theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many…

Motivational Function

Deadly mutants are after us. What can give us hope? This innocuous-looking function is a sublime light in the dark. It proves you can always recover. If your perseverance is infinite. As x tends to zero, the exponent tends to minus infinity. The function’s value at zero tends to zero. It is a zero value…

Integrating The Delta Function (Again and Again) – Penrose Version

I quoted Nobel prize winner Paul Dirac’s book, now I will quote this year’s physics Nobel prize winner Roger Penrose. In his book The Road to Reality Penrose discusses not-so-well-behaved functions like the Delta Function: They belong in the category of  Hyperfunctions. A Hyperfunction is the difference of two complex functions: Each of the complex…