Saturday, 14 September 2019

quantum mechanics - Why does dot product equal to one? (Pauli spin matrices)


I was reading these lecture notes (NB: PDF):



For spin-1/2, the rotation operator R(s)α(n)=exp(iα2σˆn)

can be written as an explicit 2×2 matrix. This is accomplished by expanding the exponential in a Taylor series: exp(iα2σˆn)=1iα2(σˆn)+12!(iα2)2(σˆn)213!(iα2)3(σˆn)3+
Note that (σˆn)2=(σˆn)(σˆn)=ˆnˆn+iσ(ˆn׈n)=1
Thus, the Taylor series becomes exp(iα2σˆn)=1iα2(σˆn)+12!(iα2)2(σˆn)213!(iα2)3(σˆn)3+=[112!(α2)2+14!(α2)$+]iσˆn[(α2)13!(α2)3+]=cos(α2)iσˆnsin(α2)



However, the part I don't understand is:


(σˆn)2=(σˆn)(σˆn)=ˆnˆn+iσ(ˆn׈n)=1



Why is that equal to 1? Where do the dot-product and cross-product come from? Note that the σ are Pauli spin matrices.



Answer



To show that (σn)2=nn+iσ(n×n)

consider writing the above as (σa)(σb)=jσjajkσkbk=jk(12{σj,σk}+12[σj,σk])ajbk=jk(δjk+iϵjklσl)ajbk
where the 2nd line arises from using the anti-commutating and commutating relation for the matrices. In the third line, we have the Kronecker delta and Levi-Civita symbol. The result (1) follows from completing the math from (2) (that is, writing it in vector notation and replacing a and b with n).


The remainder is to show that this is equal to 1. For that, the following two hints should suffice:



  1. Note that for two vectors a and b, a×b=b×a. What requirement is needed if b=a: a×a=?

  2. For the unit vector, e.g. n=(1,0)T, what is the dot product?


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