Читать книгу Quantum Mechanics for Nuclear Structure, Volume 2 - Professor Kris Heyde - Страница 11

The Pauli spin matrices, σx,σy,σz, possess a number of useful properties. We redefine them by σj,σk,σl,(j,k,l)=(x,y,z). Then

σj2=σk2=σl2=Iˆ,(1.20)

σjσk+σkσj=0,forj≠k,(1.21)

i.e.

{σj,σk}=2δjkIˆ,(1.22)

where ‘{, }’ is an anticommutator bracket (also written ‘ [,]+’). Further,

[σj,σk]=2iεjklσl,(1.23)

where

εjkl≡εklj≡εljk≡1;εkjl≡εjlk≡εlkj≡−1.(1.24)

From equations (1.22) and (1.23)

σjσk=−σkσj=iσl.(1.25)

Also,

σj†=σj,(1.26)

det(σj)=−1,(1.27)

tr(σj)=0.(1.28)

For the three-dimensional Cartesian vector a⃗, σ⃗·a⃗ is a 2 × 2 matrix²:

σ⃗·a⃗≔σxax+σyay+σzaz,(1.29)

∴σ⃗·a⃗=azax−iayax+iay−az.(1.30)

This leads to the important identity:

(σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.31)

This can be obtained from equations (1.22) and (1.23):

(σ⃗·a⃗)(σ⃗·b⃗)=∑jσjaj∑kσkbk.(1.32)

∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k12{σj,σk}+12[σj,σk]ajbk,(1.33)

∴(σ⃗·a⃗)(σ⃗·b⃗)=∑j∑k(δjk+iεjklσl)ajbk,(1.34)

∴(σ⃗·a⃗)(σ⃗·b⃗)=a⃗·b⃗Iˆ+iσ⃗·(a⃗×b⃗).(1.35)

If the components of a⃗ are real then

(σ⃗·a⃗)2=∣a⃗∣2Iˆ,(1.36)

where ∣a⃗∣ is the magnitude of the vector a⃗.