Quantum Mechanics | Part IB, 2001

(a) Write down the angular momentum operators L1,L2,L3L_{1}, L_{2}, L_{3} in terms of xix_{i} and

pi=ixi,i=1,2,3p_{i}=-i \hbar \frac{\partial}{\partial x_{i}}, i=1,2,3

Verify the commutation relation

[L1,L2]=iL3\left[L_{1}, L_{2}\right]=i \hbar L_{3}

Show that this result and its cyclic permutations imply

[L3,L1±iL2]=±(L1±iL2)[L2,L1±iL2]=0\begin{aligned} &{\left[L_{3}, L_{1} \pm i L_{2}\right]=\pm \hbar\left(L_{1} \pm i L_{2}\right)} \\ &{\left[\mathbf{L}^{2}, L_{1} \pm i L_{2}\right]=0} \end{aligned}

(b) Consider a wavefunction of the form ψ=(x32+ar2)f(r)\psi=\left(x_{3}^{2}+a r^{2}\right) f(r), where r2=x12+x22+x32r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}. Show that for a particular value of a,ψa, \psi is an eigenfunction of both L2\mathbf{L}^{2} and L3L_{3}. What are the corresponding eigenvalues?

Typos? Please submit corrections to this page on GitHub.