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§ Phase iii · The Hydrogen Atom · Chapter 02

Radial wavefunctions

A French analyst working on integration formulas in the 1870s left behind a peculiar family of polynomials that nobody in physics had ever heard of. Fifty years later, when Schrödinger went looking for the wavefunctions of the hydrogen atom, those same polynomials walked into his equation as if they had been waiting for him. The radial shape of every orbital you have ever seen drawn in a chemistry book is a Laguerre polynomial in disguise.

In the spherical harmonics chapter you saw how the angular piece of an atomic wavefunction sorts itself out. Once you write ψ(r, θ, φ) = R(r) · Y(θ, φ) and push the Laplacian through, the angular machinery factors out completely. The shapes of s, p, d, f orbitals (those famous lobes you have been drawing since high-school chemistry) live entirely in the Y(θ, φ) factor, regardless of which atom you are looking at. The angles do not care about the nucleus. They only care about the integer pair (ℓ, m).

But that is half the picture. An orbital has a size as well as a shape. A 2p orbital is larger than a 1s and smaller than a 3p, and the only place that “larger” or “smaller” can be encoded is in the radial factor R(r). That factor has to know about the nucleus. It has to know about the charge Z that pulls the electron in. It has to know the principal quantum number n that labels the energy. And, as you will see in this chapter, it has to do something rather odd at certain special radii: it has to pass through zero. The radial wavefunction is not a smooth, monotonic decay. It is a rippled thing, with nodes built into it. The number and location of those nodes are the fingerprint of a particular state.

To understand where the ripples come from you need to meet a polynomial family most physicists, and almost all chemists, have only the dimmest acquaintance with. The polynomials in question were studied in 1879 by a French mathematician named

Edmond Laguerre, an artillery officer turned analyst, working in Paris with nothing more cosmic in mind than the evaluation of certain definite integrals. His polynomials were elegant, they were orthogonal under a particular weighting, and like a great deal of nineteenth-century French analysis they sat for decades in textbooks of “special functions” without anyone outside of pure mathematics caring. Then in early 1926 Erwin Schrödinger separated his new equation in spherical coordinates and found, on the radial side, Laguerre’s polynomials staring back at him.

The story of this chapter is that meeting: a half-century-old piece of nineteenth-century algebra, brought face-to-face with the lightest atom in the universe, and forced to describe how an electron is distributed in three-dimensional space.

Start from the place we left off. After you separate variables in the time-independent Schrödinger equation for hydrogen, the radial factor R(r) obeys a one-dimensional ordinary differential equation. It looks more complicated than it is, and most of the complication is bookkeeping. The kinetic-energy operator in spherical coordinates produces a -(ℏ²/2m) · (1/r²) · d/dr(r² · dR/dr) term; the Coulomb attraction produces -Ze²/(4πε₀ · r); and the angular operator that already gave us Y(θ, φ) leaves behind a centrifugal-like contribution ℏ² · ℓ(ℓ+1)/(2m · r²). Add them up, set them equal to E · R, and you have a second-order linear ODE for R(r) on the half-line r > 0.

It is illuminating to look at the limits before trying to solve the whole thing. Near r = 0 the dominant term is the centrifugal piece (it blows up as 1/r²). The only way R(r) can survive at the origin without exploding is if it behaves like r^ℓ as r → 0. That is why s states (ℓ = 0) are finite at the nucleus and the higher-ℓ states vanish there. Far from the origin (r → ∞) the kinetic and potential terms scale together and the wavefunction must decay exponentially for the electron to be bound. The asymptotic form is R(r) ~ e^(-r/a) for some length scale a. Whatever is in the middle, between these two asymptotic forms, is what carries the structure of the orbital.

The standard move is to write R(r) = r^ℓ · e^(-Zr/(n·a₀)) · u(r) and ask what equation u(r) satisfies. The two prefactors take care of the two asymptotic limits exactly. Whatever is left over, u(r), must be a “well-behaved” function that grows slower than any exponential but is allowed to have features on the scale of a₀ (the Bohr radius). The shape of that leftover piece is what Schrödinger had to find. When he changed variables to a dimensionless radius ρ = 2Zr/(n·a₀) and chased the algebra, the equation for u collapsed onto a form Laguerre had already studied: the associated Laguerre differential equation x · y'' + (α + 1 − x) · y' + (n − α) · y = 0, with α = 2ℓ + 1 and degree n − ℓ − 1.

In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are nontrivial solutions of '''Laguerre's differential equation:' xy + (1 - x)y' + ny = 0,\ y = y(x) which is a second-order linear differential equation. This equation has nonsingular solutions only if n is a non-negative integer.

Let us read the answer Schrödinger arrived at and decode it piece by piece. The bound radial wavefunctions of the hydrogenic atom are:

R_\{n,ℓ\}(r) = N_\{n,ℓ\} · (2Zr/(n·a₀))^ℓ · e^(−Zr/(n·a₀)) · L^\{2ℓ+1\}_\{n−ℓ−1\}(2Zr/(n·a₀))

There is a lot in that line, so go slow. N_\{n,ℓ\} is a normalisation constant, a number that ensures the total probability of finding the electron somewhere in space integrates to one. The factor (2Zr/(n·a₀))^ℓ enforces the right behaviour at the origin: it vanishes like r^ℓ so that high-angular-momentum states stay away from the nucleus. The exponential e^(−Zr/(n·a₀)) enforces the right behaviour at infinity: it falls off fast enough to keep the integral finite. And in the middle, doing all the interesting work, sits the associated Laguerre polynomial L^\{2ℓ+1\}_\{n−ℓ−1\}(ρ), evaluated at the scaled radius ρ = 2Zr/(n·a₀). Its degree is n − ℓ − 1. Its parameter is 2ℓ + 1. Both of those numbers are non-negative integers built out of the two quantum numbers you already know.

That polynomial is the source of the ripples. A polynomial of degree k has at most k real roots, and the radial wavefunction inherits those roots as nodes: places where R_\{n,ℓ\}(r) crosses zero. The number of radial nodes in the orbital (n, ℓ) is exactly n − ℓ − 1. Read that again. It is one of the cleanest results in atomic physics. The principal quantum number n controls the size of the orbital. The orbital quantum number ℓ controls the angular shape. And the difference, n − ℓ − 1, controls how many times the radial wavefunction crosses zero as you walk outward from the nucleus.

The total number of nodes in any hydrogenic orbital is n − 1, and the split between radial and angular obeys a tidy bookkeeping rule. There are n − ℓ − 1 radial nodes (the polynomial’s zeros) and ℓ angular nodes (the planes through the origin on which the spherical harmonic Y vanishes). Add them together: (n − ℓ − 1) + ℓ = n − 1. The bookkeeping is exact for every orbital. Increase ℓ to put more nodes in the angles, and you get fewer radial wiggles. Decrease ℓ to get a smooth angular face (s orbitals are spherically symmetric), and the radial direction takes up the slack. It is a conservation law, in a way, for the complexity of an orbital.

This is not a random coincidence. It follows from the structure of the differential equation. The principal quantum number n enters as an eigenvalue label for the bound-state energies, and the Laguerre polynomial that solves the equation has degree exactly one less than the difference between n and the angular index ℓ + 1. A more compact way to say it is: the radial Schrödinger equation for hydrogen has exactly n − ℓ linearly independent bound solutions for a fixed ℓ, and they are indexed by the radial quantum number n_r = 0, 1, 2, …, n − ℓ − 1. The integer n_r counts the nodes directly. Schrödinger himself, in his first long paper of 1926, derived the result this way and remarked, with a kind of audible relief, that the integers had finally come out of mathematics rather than being imposed from outside.

A node is not just a curiosity. It has real, measurable consequences. Take the 2s orbital. Because n = 2 and ℓ = 0, it has one radial node and zero angular nodes; the wavefunction R_\{2,0\}(r) changes sign once as you walk outward. The electron probability density, which goes like |R|² times 4πr² for a spherical shell, ends up having two distinct peaks. There is an inner peak, close to the nucleus, sitting inside the radial node, and an outer peak, further out, sitting beyond it. The inner peak is the famous “penetration” of the 2s state. An s electron can be found near the nucleus in a way that no higher-ℓ electron can, because high-ℓ wavefunctions are killed near the origin by the r^ℓ factor. The 2p orbital, by contrast, has zero radial nodes (because n − ℓ − 1 = 0) and a single broad peak. Comparing 2s to 2p, both states are nominally at the same energy in the pure Coulomb problem, but the 2s electron penetrates the inner core more efficiently, and as a result it feels a less-screened nuclear charge in many-electron atoms. The order in which orbitals fill across the periodic table (the famous 2s before 2p, 4s before 3d) is driven directly by the node structure of the Laguerre polynomials. Penetration is not a metaphor. It is the mathematical statement that low-ℓ orbitals have an inner peak and high-ℓ orbitals do not.

There is one more number you should know about, and it is closely related to the picture above. The location of the most likely radius, the peak of 4πr²|R|², scales roughly as n² · a₀ / Z. This is a remarkable formula. It says the size of an atom grows quadratically with the principal quantum number and shrinks inversely with the nuclear charge. A hydrogen atom in its n = 3 state is nine times the linear size of an atom in its ground state; a singly ionised helium ion (Z = 2, n = 1) is half the size of hydrogen. The factor a₀ itself, the Bohr radius (about 0.529 angstroms), is the natural length scale that drops out of the equation once you have set ℏ, e, and the electron mass.

introduced it in 1913 as the radius of his first quantised orbit; Schrödinger’s wave mechanics reinterpreted it as the most-likely radius of the 1s ground state. The integer that Bohr put in by hand reappeared as the eigenvalue of a Laguerre-typed differential equation. Same number. Deeper provenance.

↘ Derive: ⟨r⟩ in closed form for any hydrogenic state

The expectation value of the radial coordinate in a state |n, ℓ, m⟩ is the integral

⟨r⟩_\{n,ℓ\} = ∫₀^∞ r · |R_\{n,ℓ\}(r)|² · r² dr

(the r² comes from the spherical volume element). Substitute the explicit Laguerre form for R_\{n,ℓ\}, change variables to ρ = 2Zr/(n·a₀), and use the standard integral identity for the associated Laguerre polynomials,

∫₀^∞ ρ^(2ℓ+2) · e^(−ρ) · [L^\{2ℓ+1\}_\{n−ℓ−1\}(ρ)]² · ρ dρ = (2n) · (n + ℓ)! / (n − ℓ − 1)! · [3n² − ℓ(ℓ+1)] / (4n²) · …

(the algebra is unforgiving; either work through Pauling and Wilson §21 or trust the result). The integral collapses to a closed form valid for every bound state of every hydrogenic ion:

⟨r⟩_\{n,ℓ\} = (n² · a₀ / Z) · [1 + (1/2) · (1 − ℓ(ℓ + 1)/n²)]

Read off three consequences:

1. Quadratic scaling in n. ⟨r⟩ grows like n². The atom inflates rapidly as you climb. A Rydberg state with n = 100 has a mean radius about ten thousand a₀, around half a micron, which is why Rydberg atoms can actually be photographed under the right conditions.
2. Inverse scaling in Z. Heavier nuclei pull the electron in tighter. Hydrogen-like uranium (Z = 92) has a 1s mean radius of (3/2) · a₀ / 92, deep inside the inner shells of a real uranium atom.
3. Weak ℓ dependence. For fixed n, increasing ℓ slightly shrinks ⟨r⟩ via the −ℓ(ℓ+1)/n² correction. The 3s state has ⟨r⟩ = 27/2 · a₀ ≈ 13.5 a₀; the 3d state has ⟨r⟩ = 21/2 · a₀ ≈ 10.5 a₀. Subtler than the n² effect, but it is responsible for the ordering of multi-electron screening.

All of these numbers are baked into the Laguerre polynomial. Change the polynomial and you change the geometry of the atom.

This is the deeper point of the chapter. The radial wavefunctions are not arbitrary functions you happen to remember from a homework problem. They are the answer to a question pure mathematicians had already asked, in the abstract, half a century before there was an atomic theory worth answering it with. Laguerre, working with continued fractions and Gauss-style quadrature in 1879, had no idea he was describing the radial geometry of hydrogen. Schrödinger, separating his new wave equation in 1926, recognised the polynomials in the textbook Weyl had pointed him to and turned them over to physics. The transfer was quiet and complete. The roots of the polynomial became the nodes of the orbital. The orthogonality relation of the polynomials became the orthogonality of the bound states. The degree of the polynomial became the radial quantum number n_r = n − ℓ − 1. Every fact about the radial structure of the hydrogen atom is a fact about Laguerre polynomials translated into the language of physics.

There is a humility that comes from this story which is worth pausing on. It is tempting, especially when you are first learning quantum mechanics, to imagine that the Schrödinger equation was a leap into a darkness no one had ever seen before. It was not. By 1926 the mathematics of self-adjoint differential operators on bounded domains was a mature subject; Courant and Hilbert had just published the encyclopaedia of it. What Schrödinger did was recognise that the hydrogen atom was an instance of that mathematics, and Bohr’s empirical integers were the eigenvalue index. The physical leap was the wave equation. The mathematical machinery for solving it was waiting on the shelf. The radial part of the answer, the Laguerre polynomials, had been sitting in nineteenth-century textbooks for fifty years before anyone needed them for atoms.

This is the rhythm you should expect from now on. Each piece of quantum mechanics will arrive in two parts: a physical insight (often radical, often controversial) and a mathematical structure (usually older, often quite classical) that turns the insight into a calculation. In the next chapter we will see how the three integers labelling these orbitals (n, ℓ, m) coalesce into something larger. They are not three separate numbers happening to satisfy three separate constraints. They are a coordinate system for the state space of a single electron bound to a nucleus, and the way they emerge tells you something about the symmetry of the Coulomb potential that Bohr’s older model never made visible.

§

Three integers fall out of the hydrogen problem: a size, a shape, and an orientation. Each one is forced on you by a separate piece of the equation, but together they label every bound state of the simplest atom. The next chapter follows the trail from n, ℓ, m back to the symmetry that produced them.

next chapter → Quantum numbers emerge