The angular equation separates once more as Y = ΘΦ. The azimuthal equation is solved by complex exponentials, while the polar equation becomes the associated Legendre equation after the substitution x = cos θ.
Single-valuedness after a complete rotation requires
The integer \(m\) controls the phase winding around the z-axis. Its name anticipates the magnetic splitting discussed later; here it is produced entirely by the boundary condition in \(\phi\).
With \(x=\cos\theta\), the polar equation becomes
Regularity on \(-1\le x\le1\) requires
For each \(l\), there are \(2l+1\) allowed values of \(m\).
The equation \(\Phi''+m^2\Phi=0\) permits any real m locally. The global condition \(\Phi(\phi+2\pi)=\Phi(\phi)\) requires \(e^{i2\pi m}=1\), so m must be an integer. A noninteger phase would assign two amplitudes to the same physical direction after one full rotation. Quantization therefore comes from single-valuedness on a closed coordinate.
The associated Legendre equation has singular coefficients at \(x=\pm1\), corresponding to the poles. Most formal solutions diverge there. Requiring finiteness and square integrability forces l to be a nonnegative integer and the series to terminate. The derivative construction of \(P_l^m\) vanishes when \(|m|>l\), leaving exactly \(2l+1\) magnetic substates.
Use the following chain when working with Azimuthal and polar equations. First identify the symmetry, prepared state and observable being discussed; this determines which basis and quantum numbers are meaningful. Next write the operator or differential equation before substituting eigenvalues. Apply boundary conditions, commutators or normalization one step at a time, keeping dimensions and allowed quantum-number ranges visible. Only then simplify to the highlighted result. Finally translate the mathematics into a measurement statement: specify which outcomes are possible, which quantities remain uncertain, how degeneracy is counted, or where probability is concentrated. This sequence prevents a correct formula from becoming disconnected from the physical assumptions that justify it.
Self-check: Periodic boundary conditions, the magnetic quantum number and the associated Legendre equation. The final answer should explicitly connect this objective to at least one equation on the page and one experimentally meaningful prediction.
Periodicity quantizes \(m\); convergence and normalizability at the poles quantize \(l\) and restrict \(|m|\le l\). Keep these logical origins distinct.
This page is designed to support short guided exercises on: Periodic boundary conditions, the magnetic quantum number and the associated Legendre equation.
Use these anchors to design compact exercises. The exercise should be answerable from this page plus standard algebra, while longer derivations, full worked examples and broader context should point back to the original book.