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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2025

Paper 4

The orbital angular momentum operator is
$$
L_i=\varepsilon_{ik\ell}X_kP_\ell.
$$
Using $[X_i,X_j]=[P_i,P_j]=0$ and $[X_i,P_j]=i\hbar\delta_{ij}$,
$$
\begin{aligned}
[L_i,X_j]
&=\varepsilon_{ik\ell}X_k[P_\ell,X_j]\\
&=-i\hbar\varepsilon_{ikj}X_k
=i\hbar\varepsilon_{ijk}X_k,
\end{aligned}
$$
and
$$
\begin{aligned}
[L_i,P_j]
&=\varepsilon_{ik\ell}[X_k,P_j]P_\ell\\
&=i\hbar\varepsilon_{ik\ell}\delta_{kj}P_\ell
=i\hbar\varepsilon_{ij\ell}P_\ell.
\end{aligned}
$$
Thus the <rotation commutators for orbital angular momentum> are
$$
\boxed{[L_i,X_j]=i\hbar\varepsilon_{ijk}X_k},
\qquad
\boxed{[L_i,P_j]=i\hbar\varepsilon_{ijk}P_k}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C / a / Solution

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Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 33C / a / Solution

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