Angular momentum

Apologies for being so quiet lately. I thought that I’d share this little blog post examining sayabiki (the pulling back of the scabbard with the left hand to draw the Japanese katana with the right hand) from the physicist’s perspective of conservation of angular momentum:

Sayabiki and Angular Momentum

This is an interesting point, although I think that it is more relevant to, say, the dynamics of a throw in Aikidō than drawing the sword in Iaidō. While I don’t doubt that using sayabiki to exploit the conservation of angular momentum does allow a more powerful draw, I think that the primary reason for sayabiki is simple mechanics: if one does not retract the scabbard, then the sword will not come cleanly out of the scabbard mouth — if the sword comes out at all, then its cutting tip will damage the scabbard and indeed the Iaidōka’s hand!

Annihilation, creation, and ladder operators

These are some notes, mostly for my own benefit, on annihilation, creation, and ladder operators in quantum mechanics, with a few remarks towards the end on angular momentum, spin and Clebsch–Gordan coefficients.

First, the abstract definition: if T, LV → V are linear operators on a vector space V over a field K, then L is said to be a ladder operator for T if there is a scalar cK such that the commutator of T and L satisfies

\displaystyle [T, L] := TL - LT = cL.

The operator L is called a raising operator for T if c is real and positive, and a lowering operator for T if c is real and negative.

The motivation behind this definition is that if (λv) ∈ K × V is an eigenpair for T (i.e. Tv = λv), then a quick calculation reveals that (λ + cLv) is an eigenpair for T:

\displaystyle T(Lv) = (TL)v = (LT - [T,L])v = LTv + cLv = (\lambda + c) (L v).

Ladder operators come up in quantum mechanics because many of the elementary operations on quantum systems act as ladder operators and increase or decrease the eigenvalues of other operators. Those eigenvalues often encode important information about the system, and the increments and decrements provided by the ladder operators often come in discrete, rather than continuous, values. Annihilation and creation operators are a prime example of this phenomenon.

Continue reading “Annihilation, creation, and ladder operators”