推 yyc2008 :謝謝JohnMash大 07/21 20:52
※ 引述《yyc2008 ()》之銘言：
: 想請教一個證明
: 卡了有點久
: 想不出來...
: If three 2 by 2 matrices, M_1, M_2, and M_3, satisfy the following relation,
: [M_i,σ_j] = 2iε_ijk M_k where σs are Pauli matrices,
: prove (M_1, M_2, M_3) = a (σ_1, σ_2, σ_3) a is some constant.
Suppose M1 is an arbitrary 2*2 matrix
M1=aI+bσ1+cσ2+dσ3, where a,b,c,d are COMPLEX
Similarly, M2=eI+fσ1+gσ2+hσ3
[M1,σ2]=[bσ1+cσ3,σ2]=2ibσ3-2icσ1=2iM3
[M2,σ1]=[gσ2+hσ3,σ1]=-2igσ3-2ihσ2=-2iM3
hence, M3 = k3 σ3,
Similarly, M2 = k2 σ2, M1 = k1 σ1
but [σi,σj] = 2εijk σk
hence, k1 = k2 =k3
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