Raising and Lowering Operators L±

The commutator algebra hands us a powerful pair of tools — the raising and lowering operators — that step a state up and down through the allowed values of $m$ without leaving the multiplet. They are the angular-momentum analogue of the harmonic-oscillator ladder operators.

Definitions

Define the non-Hermitian combinations

They are mutual adjoints, $\hat{L}_+^\dagger = \hat{L}_-$. Inverting, the Cartesian components are recovered by $\hat{L}_x = \tfrac{1}{2}(\hat{L}_+ + \hat{L}_-)$ and $\hat{L}_y = \tfrac{1}{2i}(\hat{L}_+ - \hat{L}_-)$.

Their key commutators

From the fundamental relations one finds

The first relation is what makes them "ladder" operators. If $|\ell, m\rangle$ is an eigenstate of $\hat{L}_z$ with eigenvalue $m\hbar$, then

So $\hat{L}_+|\ell,m\rangle$ is again an eigenstate of $\hat{L}_z$, now with eigenvalue $(m+1)\hbar$. Likewise $\hat{L}_-$ lowers $m$ by one. The second commutator shows neither operator changes the $\hat{L}^2$ eigenvalue, so they move within a fixed $\ell$.

The exact action

The full matrix elements (derived in the next lesson from $\hat{L}_\mp \hat{L}_\pm = \hat{L}^2 - \hat{L}_z^2 \mp \hbar\hat{L}_z$) are

Spin-1/2 example

For $\ell = \tfrac{1}{2}$ the multiplet has just two rungs, $m = \pm\tfrac{1}{2}$. As matrices (with $\hbar = 1$),

so $S_+|\tfrac12,-\tfrac12\rangle = |\tfrac12,+\tfrac12\rangle$ and $S_+|\tfrac12,+\tfrac12\rangle = 0$. The coefficient that raises the lower state is $\sqrt{\ell(\ell+1) - m(m+1)} = 1$, matching the general formula.

Try it

Use the ladder coefficient formula to compute the factor produced when $\hat{L}_+$ raises the spin-1/2 state $m = -\tfrac{1}{2}$ to $m = +\tfrac{1}{2}$. Return the number (it should be $1$).

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