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Question

Given a basis of angular momentum eigenstates | l , m >, where m is the component of

angular momentum along the z-axis and l is angular momentum operator:


(a) Show that the matrix elements < l', m' | L± | l, m> are non-zero only if l' = l and m' = m ± 1. (L± is the raising and lowering angular momentum operators)


(b) Using the relation L±L∓ = L2 − Lz2 ± (hbar)Lz, show that < l, m ± 1| L± | l, m> = hbar[l(l + 1) − m(m ± 1)]1/2


(c) Using the relation L± = Lx± iLy, derive the matrix representations for Lx = Sx, Ly = Sy, and Lz = Sz for a spin 1/2 system. Combine these operators together to get a matrix representation for L2 = S2 .

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IMG_20191102_013956.jpg

(0 ) we know,
L + le, my = Jeceti ) - mim+1) t le, mtly
and L_ le, my = eletD-mlmy) tile, my.
so, 2 + le, my = ele + 1 )- m( mil ) tile, m+1 )
So, &lt; &lt;m'/ L + 1 1,my
= &lt;e, m' / / ele+1 ) -...

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