Infinitesimal Rotations and Angular Momentum

D. Dubbers, H.‐J. Stöckmann

Gordon Baym

The orbital angular momentum of a particle in quantum mechanics is defined exactly as in classical mechanics. Before developing any properties of the angular momentum operators, students must know the commutation relations they satisfy with themselves, with the momentum operator, and with the position operator. The angular momentum operator is very closely related to rotations of the system. Students should compare the ability of the angular momentum operator to generate infinitesimal rotations, with the corresponding property of the linear momentum p. The chapter provides the theory of the orbital angular momentum sufficiently to study the problem of a particle in a central potential.

‎Louis Marchildon

G. S. Burra

Using noncommutative infinitesimal rotations, we work out a formulation for angular momentum which leads to an expression for nuclear magnetic resonance (NMR). We also indicate a cosmic footprint of these considerations, much like the 21 cm radio signal of cosmic Hydrogen. Here, the motivation is that there are a number of “second-order effects” which provide results over and above the usual well-known results.

Igal Talmi

Instead of considering the transformation properties of irreducible tensors under finite rotations, we will now consider transformations under infinitesimal rotations. Those rotations can be characterized by choosing a definite axis and a rotation around it by an infinitesimal angle δϕ . Consider rotations around the z-axis ( z′ = z ). Under a finite anticlockwise rotation of the frame of reference by an angle ϕ , the coordinates x′, y′ are given in terms of x, y by ( 6.1 ) x ′ = x cos ϕ + y sin ϕ x = x ′ cos ϕ − y ′ sin ϕ y ′ = − x sin ϕ + y cos ϕ y = x ′ sin ϕ + y ′ cos ϕ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780203739716/ff034a11-a369-4bc6-a470-f8cf1794dcb5/content/eq268.tif"/>