1.II.19D

Quantum Mechanics | Part IB, 2004

The angular momentum operators are L=(L1,L2,L3)\mathbf{L}=\left(L_{1}, L_{2}, L_{3}\right). Write down their commutation relations and show that [Li,L2]=0\left[L_{i}, \mathbf{L}^{2}\right]=0. Let

L±=L1±iL2,L_{\pm}=L_{1} \pm i L_{2},

and show that

L2=LL++L32+L3.\mathbf{L}^{2}=L_{-} L_{+}+L_{3}^{2}+\hbar L_{3} .

Verify that Lf(r)=0\mathbf{L} f(r)=0, where r2=xixir^{2}=x_{i} x_{i}, for any function ff. Show that

L3(x1+ix2)nf(r)=n(x1+ix2)nf(r),L+(x1+ix2)nf(r)=0L_{3}\left(x_{1}+i x_{2}\right)^{n} f(r)=n \hbar\left(x_{1}+i x_{2}\right)^{n} f(r), \quad L_{+}\left(x_{1}+i x_{2}\right)^{n} f(r)=0

for any integer nn. Show that (x1+ix2)nf(r)\left(x_{1}+i x_{2}\right)^{n} f(r) is an eigenfunction of L2\mathbf{L}^{2} and determine its eigenvalue. Why must L(x1+ix2)nf(r)L_{-}\left(x_{1}+i x_{2}\right)^{n} f(r) be an eigenfunction of L2\mathbf{L}^{2} ? What is its eigenvalue?

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