Deriving Equation (5.61) via Laguerre Polynomials

⚡ Key Takeaways

Equation (5.61) is the normalization constant of the hydrogen atom radial wavefunction, derived from the orthogonality and normalization integrals of the associated Laguerre polynomials. The key result: the radial function is $R_{n\ell}(r) = -\sqrt{\left(\frac{2}{na}\right)^3 \frac{(n-\ell-1)!}{2n[(n+\ell)!]}}\, e^{-r/na}\left(\frac{2r}{na}\right)^\ell L^{2\ell+1}_{n-\ell-1}(2r/na)$ , applicable in nonrelativistic quantum mechanics of the hydrogen atom (Griffiths, Introduction to Quantum Mechanics). The normalization stems from a standard Laguerre integral first cataloged in the 19th century.

💡 The Core Insight

The radial wavefunction is a Laguerre polynomial times an exponential and a power. Its squared norm reduces to the integral

\int_0^\infty x^{p+1} e^{-x} \left[L^p_q(x)\right]^2 dx = \frac{(2q+p+1)\,(q+p)!}{q!}.

The single nontrivial number here — the factor $(2q+p+1)$ — is what makes the normalization constant of Eq. (5.61) look the way it does. Everything else is bookkeeping of substitutions $x = 2r/na$ .

DefinitionAssociated Laguerre Polynomials

The associated (generalized) Laguerre polynomials are defined by Rodrigues' formula

1

\quad L^p_q(x) = \frac{e^x\,x^{-p}}{q!}\,\frac{d^q}{dx^q}\left(e^{-x} x^{q+p}\right),

Derivation hint: Apply the $q$ -fold Leibniz rule to $e^{-x}x^{q+p}$ ; the $e^x x^{-p}$ prefactor cancels the surviving exponential and power, leaving a degree- $q$ polynomial. equivalently by the finite sum

2

\quad L^p_q(x) = \sum_{k=0}^{q} (-1)^k \frac{(q+p)!}{(q-k)!\,(p+k)!\,k!}\,x^k.

Derivation hint: Expand $\dfrac{d^q}{dx^q}\!\left(e^{-x}x^{q+p}\right)$ with Leibniz, differentiate $x^{q+p}$ exactly $q-k$ times and $e^{-x}$ exactly $k$ times, then reindex. Here $q = n-\ell-1 \ge 0$ is the degree and $p = 2\ell+1$ is the order for the hydrogen problem. (Caution: a competing convention writes $L_n^k$ with different factorial normalization. Griffiths uses [1]–[2]; physicists must always check which convention multiplies their normalization integral. See Laguerre Polynomial Conventions.)

DefinitionThe Hydrogen Radial Equation in Dimensionless Form

With $a = 4\pi\epsilon_0\hbar^2/me^2$ the Bohr radius, the bound-state energies are $E_n = -\big[m e^4/(32\pi^2\epsilon_0^2\hbar^2)\big]/n^2$ . Defining $\kappa = \sqrt{-2mE_n}/\hbar = 1/(na)$ and $\rho = 2\kappa r = 2r/(na)$ , the radial function written as $R = u/r$ satisfies a reduced equation whose normalizable solution is

3

\quad u(\rho) = C\,\rho^{\ell+1} e^{-\rho/2}\, v(\rho), \qquad v(\rho) = L^{2\ell+1}_{n-\ell-1}(\rho).

Derivation hint: Strip the asymptotic behavior — $\rho^{\ell+1}$ from the $\rho\to 0$ centrifugal term, $e^{-\rho/2}$ from the $\rho\to\infty$ free-fall term — substitute into the radial equation, and obtain Kummer/Laguerre's ODE for $v$ , whose polynomial (terminating) solutions exist only when $n=\ell+1+(\text{degree})$ is a positive integer.

TheoremEquation (5.61): The Normalized Radial Wavefunction

The normalized hydrogen radial wavefunctions are

4

\quad R_{n\ell}(r) = -\sqrt{\left(\frac{2}{na}\right)^{3}\frac{(n-\ell-1)!}{2n\,[(n+\ell)!]}}\; e^{-r/na}\left(\frac{2r}{na}\right)^{\ell} L^{2\ell+1}_{n-\ell-1}\!\left(\frac{2r}{na}\right),

Derivation hint: Insert [3] into the normalization $\int_0^\infty R_{n\ell}^2\, r^2 dr = 1$ , change variables to $\rho = 2r/na$ , and evaluate using the master Laguerre integral [7] below; solve for $C$ . normalized so that

5

\quad \int_0^\infty \big|R_{n\ell}(r)\big|^2\, r^2\, dr = 1.

Derivation hint: This is the definition of normalization in spherical coordinates, where the radial volume element is $r^2\,dr$ (the angular part $\int|Y_\ell^m|^2 d\Omega = 1$ is already separately normalized). The leading minus sign is a phase convention (Griffiths' choice) and carries no physical content.

Figure

Radial probability density

r^2 R_{n\ell}^2

for the hydrogen ground state

n{=}1,\ell{=}0

, showing the Bohr-radius peak

ProofFrom Normalization Integral to Equation (5.61)

Step 1 — Set up the norm. Write $R_{n\ell} = N\, \rho^{\ell} e^{-\rho/2} L^{2\ell+1}_{n-\ell-1}(\rho)$ with $\rho = 2r/(na)$ , so $r = \tfrac{na}{2}\rho$ and $dr = \tfrac{na}{2}\,d\rho$ . Then

6

\quad \int_0^\infty R_{n\ell}^2\, r^2\, dr = N^2 \left(\frac{na}{2}\right)^3 \int_0^\infty \rho^{2\ell} e^{-\rho}\,\big[L^{2\ell+1}_{n-\ell-1}(\rho)\big]^2\, \rho^2\, d\rho.

Derivation hint: Substitute $r$ and $r^2 dr = (na/2)^3 \rho^2 d\rho$ ; the two factors of $\rho^\ell$ combine with $\rho^2$ to give $\rho^{2\ell+2}$ , and the two $e^{-\rho/2}$ combine to $e^{-\rho}$ .

Step 2 — Identify the master integral. The remaining integral has the form $\int_0^\infty x^{p+1} e^{-x}[L^p_q(x)]^2 dx$ with $p = 2\ell+1$ , $q = n-\ell-1$ , since $2\ell+2 = p+1$ . The exact value is

7

\quad \int_0^\infty x^{p+1} e^{-x}\big[L^p_q(x)\big]^2\, dx = \frac{(2q+p+1)\,(q+p)!}{q!}.

Derivation hint: Use the recurrence $x L_q^p = (2q+p+1)L_q^p - (q+1)L_{q+1}^p - (q+p)L_{q-1}^p$ , multiply by $x^p e^{-x} L_q^p$ , integrate, and apply the basic orthogonality $\int_0^\infty x^p e^{-x} L_q^p L_{q'}^p\,dx = \delta_{qq'}(q+p)!/q!$ ; only the diagonal $(2q+p+1)$ term survives.

Step 3 — Plug in the hydrogen indices. With $q = n-\ell-1$ and $p = 2\ell+1$ ,

8

\quad 2q+p+1 = 2(n-\ell-1)+(2\ell+1)+1 = 2n, \qquad (q+p)! = (n+\ell)!,\quad q! = (n-\ell-1)!.

Derivation hint: Pure arithmetic — the $-2\ell$ and $+2\ell$ cancel; $q+p = (n-\ell-1)+(2\ell+1) = n+\ell$ .

Step 4 — Combine. Therefore

9

\quad \int_0^\infty R_{n\ell}^2\, r^2\, dr = N^2\left(\frac{na}{2}\right)^3 \frac{2n\,(n+\ell)!}{(n-\ell-1)!} = 1.

Derivation hint: Substitute [7]–[8] into [6]; the integral equals $2n\,(n+\ell)!/(n-\ell-1)!$ .

Step 5 — Solve for $N$ . Inverting,

10

\quad N = \sqrt{\left(\frac{2}{na}\right)^3 \frac{(n-\ell-1)!}{2n\,(n+\ell)!}}.

Derivation hint: Take the reciprocal of the bracketed product in [9] and the square root; the $(na/2)^3$ inverts to $(2/na)^3$ . Reinstating $\rho^\ell = (2r/na)^\ell$ and $e^{-\rho/2} = e^{-r/na}$ , and choosing the conventional overall minus sign, yields exactly Equation (5.61), i.e. [4]. $\blacksquare$

∎

⚠️ The Integral is Not the Orthogonality Relation

A frequent fatal error is to use the orthogonality integral

11

\quad \int_0^\infty x^{p} e^{-x}\,L^p_q(x)\,L^p_{q'}(x)\, dx = \frac{(q+p)!}{q!}\,\delta_{qq'}

Derivation hint: This is the standard weight- $x^p e^{-x}$ orthogonality on $[0,\infty)$ ; the diagonal value follows from Rodrigues' formula and $q$ integrations by parts. directly in Step 2. But the radial measure $r^2 dr$ produces $x^{p+1}$ , one power higher than the orthogonality weight $x^p$ . That extra $x$ is precisely what generates the factor $2n = 2q+p+1$ . Forgetting it loses the $2n$ and gives a wrong normalization.

mathematician · Edmond Laguerre | 1834–1886

Edmond Nicolas Laguerre, a French artillery officer turned mathematician at the École Polytechnique, introduced the polynomials now bearing his name in his 1879 study of the integral $\int_x^\infty e^{-t}/t\,dt$ . He was a master of analytic geometry and continued fractions. The associated (generalized) Laguerre polynomials $L^p_q$ were systematized later (and by Sonine/Sonin in Russia, hence "Sonine polynomials"); their starring role in quantum mechanics came only in 1926 when Schrödinger's wave equation for hydrogen demanded exactly these terminating polynomial solutions. Laguerre never saw the atomic application, dying four decades before the quantum theory that immortalized his name in every chemistry classroom.

🧠 Intuition

Think of the normalization constant as the answer to a probability bookkeeping question: "How much amplitude must I dial in so that the electron is somewhere with total probability one?" Three competing scales fight inside Eq. (5.61):

The $(2/na)^3$ factor is dimensional — it turns the dimensionless $\rho$ -integral back into a physical $r$ -integral with units of (length) $^{-3}$ , so that $|R|^2 r^2 dr$ is dimensionless.
The $(n-\ell-1)!/(n+\ell)!$ ratio is the "shape penalty": higher-order Laguerre polynomials oscillate more and spread the amplitude over many radial lobes, so the per-lobe height shrinks.
The $1/2n$ factor reflects how spread out high- $n$ orbits are. The electron in a Rydberg state ( $n$ large) lives far from the nucleus over a huge volume, so the wavefunction must be small everywhere — and $2n$ in the denominator under the square root enforces that.

The deep point: all the "physics" of normalization is encoded in the single number $2q+p+1 = 2n$ from the recurrence relation. The factorials are just the orthogonality baseline.

ExampleGround State

n=1,\ell=0

Given: $n=1$ , $\ell=0$ , so $q = n-\ell-1 = 0$ , $p = 2\ell+1 = 1$ .

Step 1. $L^1_0(x) = 1$ (the lowest associated Laguerre polynomial, from [2] with $q=0$ ).

Step 2. Compute the constant from [10]:

N = \sqrt{\left(\frac{2}{1\cdot a}\right)^3 \frac{0!}{2\cdot 1\cdot (1)!}} = \sqrt{\frac{8}{a^3}\cdot\frac{1}{2}} = \sqrt{\frac{4}{a^3}} = \frac{2}{a^{3/2}}.

Step 3. Assemble: $R_{10}(r) = N\, e^{-r/a}(2r/a)^0 \cdot 1$ (ignoring the overall sign).

Answer:

\boxed{R_{10}(r) = \dfrac{2}{a^{3/2}}\, e^{-r/a}}\quad\text{[units: m}^{-3/2}\text{]}

Check: $\int_0^\infty (4/a^3) e^{-2r/a} r^2 dr = (4/a^3)\cdot \tfrac{2!}{(2/a)^3} = (4/a^3)\cdot \tfrac{a^3}{4} = 1.$ ✓

ExampleState

n=2,\ell=1

Given: $n=2$ , $\ell=1$ , so $q = 2-1-1 = 0$ , $p = 2(1)+1 = 3$ .

Step 1. $L^3_0(x) = 1$ .

Step 2. Constant from [10] with $na = 2a$ :

N = \sqrt{\left(\frac{2}{2a}\right)^3 \frac{0!}{2\cdot 2\cdot (3)!}} = \sqrt{\frac{1}{a^3}\cdot\frac{1}{4\cdot 6}} = \sqrt{\frac{1}{24 a^3}} = \frac{1}{\sqrt{24}\,a^{3/2}}.

Step 3. Assemble with $(2r/na)^\ell = (2r/2a)^1 = r/a$ :

R_{21}(r) = \frac{1}{\sqrt{24}\,a^{3/2}}\, e^{-r/2a}\,\frac{r}{a}.

Answer:

\boxed{R_{21}(r) = \dfrac{1}{\sqrt{24}\,a^{3/2}}\,\dfrac{r}{a}\, e^{-r/2a}}\quad\text{[units: m}^{-3/2}\text{]}

Check: $\int_0^\infty \tfrac{1}{24 a^3}\tfrac{r^2}{a^2}e^{-r/a} r^2 dr = \tfrac{1}{24 a^5}\int_0^\infty r^4 e^{-r/a}dr = \tfrac{1}{24 a^5}\cdot 24 a^5 = 1.$ ✓

⚠️ Convention-Dependence of the Polynomial

The numerical coefficients in Eq. (5.61) are valid only for the Rodrigues convention [1]. Some texts (and Mathematica's `LaguerreL[q,p,x]`) use a different normalization in which $L^p_q$ differs by a factor of $(q+p)!$ or $q!$ . If you import a polynomial from one convention into Griffiths' normalization constant, your $|R|^2$ will not integrate to 1. Always self-check by verifying [5] numerically for $n=1$ .

CounterexampleUsing

r\,dr

Instead of

r^2\,dr

Suppose a student mistakenly normalizes with the two-dimensional measure $\int_0^\infty |R|^2 r\, dr$ (perhaps confusing it with a 2D radial problem). For the ground state this gives

\int_0^\infty (4/a^3)e^{-2r/a} r\, dr = (4/a^3)\cdot \frac{1!}{(2/a)^2} = (4/a^3)\cdot\frac{a^2}{4} = \frac{1}{a} \ne 1.

The result carries units of m $^{-1}$ , a dead giveaway that the wrong volume element was used. In 3D spherical coordinates the radial Jacobian is always $r^2$ , which is why the Laguerre integral [7] needs the power $p+1$ , not $p$ .

connection

Equation (5.61) sits at a crossroads of several theories:

Associated Laguerre Polynomials supply the polynomial factor and the master integral [7]; the same polynomials appear in the 3D isotropic harmonic oscillator.
Hydrogen Atom Schrödinger Equation produces the radial ODE [3]; the energy quantization $E_n \propto -1/n^2$ emerges from the termination condition that makes $v(\rho)$ a polynomial.
<a class="wiki-link" href="/wiki/spherical-harmonics">Spherical Harmonics</a> carry the angular dependence; the full eigenfunction is $\psi_{n\ell m} = R_{n\ell}(r)\,Y_\ell^m(\theta,\phi)$ , with each factor independently normalized.
Rodrigues' Formula is the unifying tool for evaluating both the orthogonality [11] and the master integral [7] via integration by parts.
SO(4) Symmetry of Hydrogen explains the otherwise-accidental $\ell$ -degeneracy that makes the energy depend only on $n$ , the same $n$ appearing as $2n$ in the normalization.

🔬 Where this appears

⏳ History

- 1879 — Edmond Laguerre introduces the basic Laguerre polynomials while studying the exponential-integral function.

- 1880s — Nikolai Sonine generalizes them to the associated/generalized polynomials

L^p_q

(hence "Sonine polynomials" in older Russian literature).

- 1885 — Rodrigues-type and orthogonality properties consolidated within the emerging theory of classical orthogonal polynomials.

- 1926 (Jan) — Erwin Schrödinger publishes the wave equation and solves hydrogen, recognizing the radial solutions as associated Laguerre polynomials in his first Annalen der Physik communication.

- 1926 — Wolfgang Pauli derives the hydrogen spectrum independently via matrix mechanics and the Runge–Lenz vector, foreshadowing the SO(4) explanation of degeneracy.

- 1929 — Condon and Shortley begin systematizing phase conventions (the source of the leading minus sign debate) in their atomic-spectra work, culminating in their 1935 treatise.

- 1935 — The Theory of Atomic Spectra (Condon & Shortley) fixes standard normalization and phase conventions used by physicists thereafter.

- 1957 — Bethe and Salpeter's Quantum Mechanics of One- and Two-Electron Atoms tabulates the radial functions and matrix-element integrals in the modern reference form.

- 1995/2005 — Griffiths' Introduction to Quantum Mechanics assigns the now-canonical "Equation (5.61)" numbering used in countless undergraduate courses.

open-problem

Several finer points reward reflection:

Convention sensitivity means there is no single "right" Eq. (5.61) across textbooks — the factor structure depends on the chosen Laguerre normalization, the phase, and even whether reduced mass is used. A definitive cross-text reference must always state its convention explicitly.
The continuum ( $E>0$ ) states are not covered by [4]; for scattering states the Laguerre polynomials become non-terminating confluent hypergeometric functions and require delta-function ("energy") normalization instead of [5]. The smooth matching of bound to continuum normalization remains pedagogically delicate.
Relativistic corrections (Dirac equation) replace the simple Laguerre structure with a coupled pair of radial functions whose normalization involves Laguerre polynomials of non-integer order — a genuine generalization where [7] no longer applies verbatim.
Finite nuclear size and QED self-energy shift the very levels whose wavefunctions [4] describes; whether the nonrelativistic normalization remains a useful zeroth-order basis for high-precision (Lamb-shift) calculations is a practical question of perturbation-theory convergence.

Deriving Equation (5.61) via Laguerre Polynomials

Formal Statement

Why It Holds / Derivation

Building Intuition

Worked Examples

Common Misconceptions

Connections

Real-World Applications

Historical Development

Subtleties & Limits

See Also