Basics of Gauge Theories
David Bailin · 2024
I assume everybody has encountered Schrödinger’s equation, which determines the time evolution of the wave function ψ describing a non-relativistic system. One way to obtain it is by means of the correspondence principle in which one makes the substitutions ( 1.1 ) https://www.w3.org/1998/Math/MathML" display="block"> E → i ℏ ∂ ∂ t https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003575924/18a74078-c2b9-45b1-8f31-1a452039284e/content/eq109.tif "/> and ( 1.2 ) https://www.w3.org/1998/Math/MathML" display="block"> p a → - i ℏ ∇ a https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003575924/18a74078-c2b9-45b1-8f31-1a452039284e/content/eq110.tif "/> where E is the total energy, p α is the momentum conjugate to the coordinate r a of the α th particle, (and ħ is Planck’s constant divided by 2π.) So for the simplest case of a single free particle we have ( 1.3 ) https://www.w3.org/1998/Math/MathML" display="block"> E = p 2 / 2 m https://www.w3.org/1999/xlink" xlink:href=" https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003575924/18a74078-c2b9-45b1-8f31-1a452039284e/content/eq111.tif "/>