Franz Schwabl Quantum Mechanics 4th Edit · Problem 8.4

Spin

Physics — Quantum Mechanics · angular-momentum-matrix-representation

Problem

Calculate the matrix representation of the angular momentum operators LxL_x, LyL_y, LzL_z, and L2\boldsymbol{L}^2 for the values l=1/2, 1, 3/2l = 1/2,\, 1,\, 3/2, and 22 by using the formulae ⟨l′,m′∣L2∣l,m⟩=ℏ2δll′δmm′l(l+1)  ,\langle l', m'|L^2|l, m\rangle = \hbar^2 \delta_{ll'} \delta_{mm'} l(l+1) \;, ⟨l′,m′∣Lz∣l,m⟩=ℏδll′δmm′m  ,\langle l', m'|L_z|l, m\rangle = \hbar \delta_{ll'} \delta_{mm'} m \;, ⟨l′,m′∣L−∣l,m⟩=ℏ(l−m+1)(l+m) δll′ δm−1,m′  ,\langle l', m'|L_-|l, m\rangle = \hbar\sqrt{(l-m+1)(l+m)}\,\delta_{ll'}\,\delta_{m-1,m'} \;, ⟨l′,m′∣L+∣l,m⟩=ℏ(l+m+1)(l−m) δll′ δm+1,m′  ,\langle l', m'|L_+|l, m\rangle = \hbar\sqrt{(l+m+1)(l-m)}\,\delta_{ll'}\,\delta_{m+1,m'} \;, −l≤m≤l  .-l \leq m \leq l \;.

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