Finish hydrogen. Then learn to approximate everything else.

Topics and mapped lessons

• From the separated angular equation to the associated Legendre equation ↗

  The course's one review topic: ψ = R(r) Y(θ, φ) and Φ = exp(i m φ) with integer m are recalled from Quantum Mechanics I, not re-derived. The new step is x = cos θ, which turns the polar equation into (1 - x²) P'' - 2x P' + [λ - m²/(1 - x²)] P = 0, with λ not yet known to be l(l + 1). Frobenius exponents of plus and minus |m|/2 at each pole, with a logarithm when m = 0, give one regular and one singular solution there; the singular one is not normalisable for |m| of 1 or more and has infinite ⟨L²⟩ for m = 0, so regularity is a physical demand.

• Series solution of Legendre's equation, and why it must terminate ↗

  For m = 0 a power series gives the two-step recursion a_(k+2) = [k(k + 1) - λ] aₖ / [(k + 1)(k + 2)]; left infinite, either branch has coefficients falling only as 1/k and diverges logarithmically at the poles. Re-expanding about x = 1 shows that no mixture of infinite branches is regular at both poles either. Regularity therefore forces λ = l(l + 1) with l a non-negative integer and the branch of the other parity set to zero: the l(l + 1) Quantum Mechanics I took from the ladder algebra now follows from the differential equation plus regularity, the equation alone admitting a solution for every λ.

• Legendre polynomials: Rodrigues' formula, orthogonality and norm ↗

  Pₗ(x) = (1/(2^l l!)) d^l/dx^l (x² - 1)^l is shown to solve Legendre's equation by differentiating (x² - 1) w' = 2l x w, with w = (x² - 1)^l, l + 1 times; Pₗ(1) = 1 fixes its scale and P₀ to P₃ are checked against the terminated recursion. Orthogonality with norm 2/(2l + 1) is derived by repeated integration by parts, and orthogonality again from the Sturm-Liouville form, whose boundary terms vanish at x = plus or minus 1 because 1 - x² does. Orthogonality also forces all l zeros of Pₗ to lie inside (−1, 1).

• The generating function and the multipole expansion of 1/|r - r'| ↗

  Expanding (1 - 2xt + t²)⁻¹⁄² in powers of t, for |t| < 1, generates coefficients shown to be the Pₗ(x), and differentiating it in t gives the recursion (2l + 1) x Pₗ = (l + 1) P_(l+1) + l P_(l-1). Setting t = r_</r_> gives 1/|r - r'| as the sum over l of (r_<^l / r_>^{(l + 1)}) Pₗ(cos γ), gamma the angle between r and r'. The series converges geometrically for |r| not equal to |r'| and is the expansion that evaluates the helium repulsion integral in Unit 10.

• Associated Legendre functions, and why |m| cannot exceed l ↗

  Writing P = (1 - x²)^(|m|/2) u leaves an equation for u whose series terminates only at degree l - |m|, and differentiating Legendre's equation |m| times identifies that polynomial as the |m|-th derivative of Pₗ, so Pₗ^|m|(x) = (1 - x²)^(|m|/2) d^|m|/dx^|m| Pₗ(x), regular at both poles. A degree cannot be negative, so |m| at most l follows from the differential equation, matching what Quantum Mechanics I got from the ladder algebra. The l - |m| polar nodes it quoted are the zeros of that derivative, counted by Rolle's theorem from the l zeros of Pₗ.

• Normalising Yₗ^m, and the Condon-Shortley phase ↗

  Integrating over d Ω = sin θ d θ d φ, with the Legendre norm found from N_(|m|+1) = (l + |m| + 1)(l - |m|) N_|m| by integration by parts, fixes the factor sqrt[(2l + 1)(l - |m|)! / (4 π (l + |m|)!)]. The sign (−1)^m for positive m is a convention, placed here in Yₗ^m rather than in Pₗ^|m| as some texts do, that makes the matrix elements of L₊ and L_- real and positive, as Quantum Mechanics I assumed; Yₗ^(−m) = (−1)^m [Yₗ^m]* follows. Y₀⁰ through the five l = 2 functions are built and checked against the tables Quantum Mechanics I quoted.

• The ladder route: solving L₊ Yₗ^l = 0 on the sphere ↗

  Transforming Lₓ plus or minus i L_y into angles gives Lₚₘ = plus or minus ℏ exp(plus or minus i φ)[d/d θ plus or minus i cot θ d/d φ], so L₊ Yₗ^l = 0 is a first-order equation solved by sin^l θ exp(i l φ). Lowering with the matrix elements of Quantum Mechanics I generates every Yₗ^m and reproduces Rodrigues' formula at m = 0; it fixes every sign within a multiplet but not the overall one, which the Condon-Shortley choice (−1)^l for Yₗ^l supplies, and so agrees with the series route for l = 2. Trying l = 1/2 gives a top state double-valued in φ and a lowered function, cos θ over √(sin θ), singular at both poles: one reason orbital l is an integer.

• Parity, completeness, and the recursion for cos θ Yₗ^m ↗

  Under r → −r, θ → π - θ and φ → φ + π; Pₗ^|m| supplies (−1)^{(l + |m|)} and exp(i m φ) supplies (−1)^m, so Yₗ^m gains (−1)^l, derived rather than quoted. Differentiating the Legendre recursion |m| times, and removing the term in the (|m| - 1)-th derivative with P'_(l+1) - P'_(l-1) = (2l + 1) Pₗ from the generating function, gives (2l + 1) x Pₗ^|m| = (l - |m| + 1) P_(l+1)^|m| + (l + |m|) P_(l-1)^|m|, so cos θ connects Yₗ^m only to Y_(l+1)^m and Y_(l-1)^m: the first sighting of the rule Unit 5 derives from commutators. Completeness on the sphere is quoted from Sturm-Liouville theory, and cos² theta is expanded on Y₀⁰ and Y₂⁰ by projection.

• The addition theorem for spherical harmonics ↗

  Pₗ(cos γ) = [4 π/(2l + 1)] times the sum over m of Yₗ^m*(θ', phi') Yₗ^m(θ, φ), gamma the angle between the two directions, is verified for l = 1, where it is the identity cos γ = cos θ cos θ' + sin θ sin θ' cos(φ - φ'). The general proof is outlined: the m-sum is unchanged by a common rotation of both directions, a property of rotation operators quoted here and justified in Unit 5, so it can be evaluated with one direction on the z axis, where only m = 0 survives. Substituted into the multipole expansion it writes 1/|r - r'| as a double sum over l and m that separates the directions of r and r' but not the radii, the form Unit 10 uses.

Learning outcomes

• Transform the separated polar equation into the associated Legendre equation in x = cos θ, and show from its Frobenius exponents and the Legendre recursion that regularity at both poles forces λ = l(l + 1) with l a non-negative integer.
• Derive |m| at most l from the degree of the polynomial factor in Pₗ^|m|, and predict the l - |m| polar nodes and the (−1)^l parity of any Yₗ^m before plotting it.
• Construct and normalise any Yₗ^m with l at most 3 by both the series route and the ladder route, and reconcile the two results to the same Condon-Shortley phase.
• Use the associated Legendre recursion to state which Yₗ'^m' the product cos θ Yₗ^m contains, and expand a given angular function on the Yₗ^m by projection.
• Expand 1/|r - r'| in Legendre polynomials and, through the addition theorem, in spherical harmonics, and use orthogonality to reduce an integral over two sets of angles to the few multipoles that survive, the l = 0 term alone when either distribution is spherically symmetric.
• Say for each property of the Yₗ^m whether this unit derives it, as for λ, the bound on m, normalisation and parity, or quotes it, as for completeness and the general addition theorem.

Prerequisite thread

• The separated central-potential equations of Quantum Mechanics I: ψ = R(r) Y(θ, φ), Φ = exp(i m φ) with integer m from single-valuedness, and the polar equation with its separation constant
• The angular-momentum ladder algebra: the L² and L_z eigenvalues, Lₚₘ, and the matrix elements ℏ √(l(l + 1) - m(m pm 1))
• L_z and L² as operators on the angles in spherical coordinates, and normalisation over d Ω = sin θ d θ d φ
• Power-series solution of an ordinary differential equation whose recursion must terminate, as done for the harmonic oscillator and the Hermite polynomials
• Orthogonality of eigenfunctions of a Hermitian operator, integration by parts with boundary terms, and the Leibniz rule for repeated derivatives of a product
• Binomial and Taylor expansions, and the law of cosines for the angle between two vectors

Laboratory directions

Problem practice

Deriving each property of the Yₗ^m from the polar equation or the ladder algebra rather than reading it off a table, and carrying one phase convention from the first line of a calculation to the last

Capstone: Take l = 3: derive the allowed λ from the Legendre recursion, build Y₃¹ once by differentiating P₃ and once by lowering Y₃³ twice, normalise both and reconcile their phases, confirm the parity and polar-node count, and use the cos θ recursion to name the two Yₗ¹ that cos θ Y₃¹ contains.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge

Topics and mapped lessons

• Centre of mass, relative coordinates, and the reduced mass ↗

  Changing variables in the two-body Schrodinger equation separates two particles interacting through V(|r₁ - r₂|) into free motion of the centre of mass and one particle of reduced mass μ = mₑ mₚ/(mₑ + mₚ) in V(r). The separation is exact for any central interaction. Replacing mₑ by mu raises every hydrogen level by about 1 part in 1836, and the resulting 0.18 nm hydrogen-deuterium splitting of Balmer-α is computed. Positronium, with μ = mₑ/2 and levels at minus 6.8 eV over n-squared, is where the correction is largest; both particles are still treated as non-relativistic point charges.

• The dimensionless radial equation for the Coulomb potential ↗

  With κ = √(−2 μ E)/ℏ, ρ = κ r and ρ₀ = μ e²/(2 π ε₀ ℏ² κ), the radial equation for u = rR, taken from Quantum Mechanics I, becomes d²u/d ρ² = [1 - ρ₀/ρ + l(l + 1)/ρ²] u. Bound states need E < 0 so that kappa is real, and the unknown energy now hides entirely in the single number ρ₀, so finding the spectrum means finding which ρ₀ are allowed. Replacing e² by Z e² covers any one-electron ion. States with E > 0 form the continuum and are not treated here.

• Behaviour at infinity and at the origin, and the equation for v(ρ) ↗

  At large ρ, u ~ exp(−ρ); the growing exp(ρ) is discarded because it cannot be normalised. Near the origin, u ~ ρ^{(l + 1)}; the ρ^(−l) branch is discarded as non-normalisable for l at least 1, and for l = 0 because R ~ 1/r would put a δ function into the Laplacian. Writing u = ρ^{(l + 1)} exp(−ρ) v(ρ) defines v and assumes nothing, and gives ρ v'' + 2(l + 1 - ρ) v' + [ρ₀ - 2(l + 1)] v = 0. In the variable 2 rho this is Kummer's confluent hypergeometric equation, whose solution regular at the origin, M(l + 1 - ρ₀/2, 2l + 2, 2 ρ), is quoted, so the condition at infinity is the only one left to impose.

• The power series for v(ρ) and its recursion ↗

  Substituting v = sum of cⱼ ρ^j gives the one-step recursion c_(j+1) = [2(j + l + 1) - ρ₀] cⱼ / [(j + 1)(j + 2l + 2)], fixed up to normalisation by c₀. It yields a solution for every ρ₀, so the equation alone quantises nothing. For large j the ratio approaches 2/j, so an unterminated series grows like exp(2 ρ) times some power of ρ, and u like exp(ρ), which cannot be normalised. The crude ratio argument cannot fix that power; the quoted large-ρ form of Kummer's function gives u ~ ρ^(−ρ₀/2) exp(ρ), not the ρ^{(l + 1)} exp(ρ) often written, and confirms that exponential growth sets in whenever the series fails to end.

• Truncation: n = nᵣ + l + 1 and the Bohr energies derived ↗

  The series must stop at some j = nᵣ, which requires ρ₀ = 2(nᵣ + l + 1) = 2n, so n is a positive integer and l runs from 0 to n - 1. Solving for E gives κ = 1/(na), Eₙ = -[μ/(2 ℏ²)](e²/(4 π ε₀))²/n² and a = 4 π ε₀ ℏ²/(μ e²): Quantum Mechanics I's minus 13.6 eV over n-squared, now derived from normalisability, with its Z-squared scaling and radii a/Z following from Z e². Counting l and m gives n-squared states per level, an l-degeneracy the derivation leaves unexplained and topic 02.09 explains. These non-relativistic, spinless, point-nucleus energies are corrected for relativity and spin in Units 8 and 9 and for nuclear size in Unit 6.

• The associated Laguerre polynomials as the terminated series ↗

  The terminated v is proportional to L_(n-l-1)^(2l+1)(2 ρ), with 2 ρ = 2r/(na). It is built from the Rodrigues formula L_q(x) = (e^x/q!) d^q/dx^q (e^(−x) x^q) and from L_q^p(x) = (−1)^p d^p/dx^p L_(q+p)(x), then checked against the recursion for n up to 3. Texts differ by factorials, signs and index labelling: the older physics convention writes the same polynomial as L_(n+l)^(2l+1), which is minus (n + l)! times the one used here. Every formula in this unit names its convention, and every tabulated Rₙₗ is checked by direct normalisation rather than trusted.

• Normalised radial functions Rₙₗ for n up to 3 ↗

  R₁₀ = 2 a⁻³⁄² exp(−r/a), R₂₀, R₂₁ and the three n = 3 functions are built from the recursion and normalised so that the integral of R² r² dr is 1; the general constant is quoted. Each is checked for n - l - 1 radial nodes and r^l behaviour at the origin, and functions of different n at equal l are shown orthogonal; radial functions of different l need not be, since the Yₗ^m already are. States with l = n - 1 are nodeless, with radial density peaking at r = n² a, the circular-orbit limit. The contact density |ψₙ₀₀(0)|² = 1/(π n³ a³), checked to n = 3, is recorded for Units 6, 8 and 9.

• Radial expectation values: the virial theorem and Kramers' relation ↗

  In a stationary state <[H, r . p]> = 0, which gives the three-dimensional virial theorem 2⟨T⟩ = <r . grad V>; for the Coulomb potential that means ⟨V⟩ = 2Eₙ and <1/r> = 1/(n² a), with no integral done. Kramers' relation, derived from the radial equation by integrating by parts, links ⟨r^s⟩, ⟨r^(s-1)⟩ and ⟨r^(s-2)⟩ for s above -(2l + 1). It gives ⟨r⟩ = (a/2)[3n² - l(l + 1)], returning Quantum Mechanics I's 1.5a for 1s, and ⟨r²⟩ = (n² a²/2)[5n² + 1 - 3l(l + 1)]. It cannot close on <1/r²>, which waits for the Feynman-Hellmann theorem in Unit 6; its s = −1 case then gives the <1/r³>, finite only for l above zero, that fine structure needs in Unit 8.

• The l-degeneracy explained: the quantum Runge-Lenz vector ↗

  The classical Laplace-Runge-Lenz vector, conserved only for an inverse-square force, is symmetrised into the Hermitian A = (p x L - L x p)/(2 μ) - k r-hat with k = e²/(4 π ε₀), and [H, A] = 0 and A . L = 0 are checked by commutator algebra. A commutes with H but not with L², so it connects states of equal n whose l differ by one: within n = 2, A_z carries 2s into 2p₀ and Aₓ and A_y carry it into 2p_{(plus or minus 1)}, and the combinations (2s plus or minus 2p₀)/√(2) that A_z selects are the states of Unit 9's linear Stark effect. Pauli's so(4) solution, which recovers Eₙ and the n-squared count from A² = k² + (2H/μ)(L² + ℏ²), is outlined with its algebra quoted.

Learning outcomes

• Separate a two-body problem with a central interaction into centre-of-mass and relative motion, and compute the reduced-mass shift of a hydrogen, deuterium or positronium line.
• Reduce the Coulomb radial equation to dimensionless form, justify each asymptotic factor removed at the origin and at infinity, and derive the recursion for the remaining power series.
• Show that an unterminated series cannot be normalised, and derive from its termination n = nᵣ + l + 1, the bound l at most n - 1, Eₙ and the Bohr radius with the reduced mass and nuclear charge Z included.
• Construct and normalise Rₙₗ for n up to 3 from the recursion, identify each with an associated Laguerre polynomial in a stated convention, and check its nodes, its r^l behaviour near the origin and its orthogonality.
• Evaluate <1/r>, ⟨r⟩ and ⟨r²⟩ for any n and l from the virial theorem and Kramers' relation without doing an integral, and confirm one of them by direct integration.
• Verify that the quantum Runge-Lenz vector commutes with H and is perpendicular to L, and explain from it why states of equal n and different l share an energy, saying which part of the so(4) argument is quoted.

Prerequisite thread

• The power-series solution of the harmonic oscillator: an asymptotic factor peeled off, a recursion for the remainder, and termination forced by normalisability rather than by a wall
• The separated central-potential problem: ψ = R(r) Yₗ^m(θ, φ), the radial equation for u = rR with its centrifugal term and u(0) = 0, and the hydrogen results Quantum Mechanics I quoted from it
• The spherical harmonics as derived in Unit 1, with the separation constant l(l + 1) and the bound |m| at most l established rather than assumed
• Integrals of r^k exp(−br) from 0 to infinity as k!/b^{(k + 1)}, and normalisation with the volume element r² dr d Omega
• Commutators among xᵢ, pⱼ and Lₖ, and cross products of vector operators whose components do not commute
• Centre-of-mass and relative coordinates, and the reduced mass, from classical two-body mechanics

Laboratory directions

Problem practice

Carrying one derivation from the two-body equation to normalised radial functions without skipping a step, and saying at each step which condition is being imposed and which result is being quoted

Capstone: Take a Coulomb-bound pair other than hydrogen, such as positronium, muonic hydrogen or the He+ ion. Carry it from the two-body equation to its normalised 1s and 2p radial functions, derive its energies and length scale with the correct reduced mass and charge, and evaluate ⟨r⟩ and <1/r> by Kramers' relation, confirming one of them by direct integration. Finally, for each assumption of the derivation, non-relativistic motion, spinless particles and a point nucleus, estimate by scaling whether the correction it neglects is larger or smaller, relative to the binding energy, than in hydrogen.

• Concept checks that require an explanation, not only a numerical answer
• Multi-step quantitative modelling with unit and limiting-case checks
• A calculus or data-interpretation challenge