# Spins, Rotations, and the Beauty of Complex Numbers

This is a simple quantum state ... |โ> = ฮฑ|โ> + ฮฒ|โ> ... built from an up |โ> state and a down state |โ>. ฮฑ and ฮฒ are complex numbers. The result |โ> is in the middle, oblique. The oblique state is a superposition or the up and down base states. Making a measurement, you … Continue reading Spins, Rotations, and the Beauty of Complex Numbers

# 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 isโthe theory of relativity, quantum mechanics, and other modern ideas. By the end of two years of our previous course, many … Continue reading Dirac’s Belt Trick

# 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. $latex e^{\left(-\frac{1}{x^{2}}\right)}&s=3$ As x tends to zero, the exponent tends to minus infinity. The function's value at zero tends to zero. It is … Continue reading Motivational Function

# 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 ofย  Hyperfunctions. A Hyperfunction is the difference of two complex functions: Each of the complex … Continue reading Integrating The Delta Function (Again and Again) – Penrose Version