The solutions of the hydrogen atom radial Schrödinger equation are expressed in terms of a special family of functions known as Laguerre polynomials. These functions ensure that the radial wavefunctions are finite at the origin, decay properly at large distances, and are normalizable.
Laguerre polynomials
The Laguerre polynomials, denoted \(L_n(x)\), arise as solutions of a second-order differential equation on the interval \([0,\infty)\). They form a complete and orthogonal set with respect to an exponential weighting function.
One way to define the Laguerre polynomials is through their generating function:
\[ \frac{e^{-xt/(1-t)}}{1-t} = \sum_{n=0}^{\infty} L_n(x)\,t^n. \]
From this definition, the first few Laguerre polynomials can be generated.
\[ \begin{aligned} L_0(x) &= 1, \\ L_1(x) &= 1 - x, \\ L_2(x) &= \tfrac{1}{2}(x^2 - 4x + 2). \end{aligned} \]
Recursion relation
Laguerre polynomials satisfy a useful recursion relation:
\[ (n+1)L_{n+1}(x) = (2n+1-x)L_n(x) - nL_{n-1}(x), \]
which allows higher-order polynomials to be generated from lower-order ones.
Orthogonality
The Laguerre polynomials are orthogonal with respect to the weighting function \(e^{-x}\):
\[ \int_0^{\infty} L_n(x)\,L_m(x)\,e^{-x}\,dx = \delta_{nm}. \]
This orthogonality property is essential for ensuring that different hydrogen atom wavefunctions are orthogonal.
Associated Laguerre polynomials
The hydrogen atom radial equation does not involve the simple Laguerre polynomials directly, but rather the associated Laguerre polynomials, denoted \(L_n^{(\alpha)}(x)\).
These are defined by
\[ L_n^{(\alpha)}(x) = (-1)^{\alpha} \frac{d^{\alpha}}{dx^{\alpha}} L_{n+\alpha}(x), \]
where \(\alpha\) is a nonnegative integer. In the hydrogen atom problem, \(\alpha = 2\ell + 1\).
Radial wavefunctions of the hydrogen atom
Using the associated Laguerre polynomials, the normalized hydrogen atom radial wavefunctions can be written as
\[ R_{n\ell}(r) = N_{n\ell} \left(\frac{2r}{na_0}\right)^{\ell} e^{-r/(na_0)} L_{n-\ell-1}^{(2\ell+1)} \!\left(\frac{2r}{na_0}\right), \]
where:
- \(a_0\) is the Bohr radius,
- \(n\) is the principal quantum number,
- \(\ell\) is the orbital angular momentum quantum number,
- \(N_{n\ell}\) is a normalization constant.
The polynomial, power-law, and exponential factors work together to ensure that \(R_{n\ell}(r)\) is finite at \(r=0\) and decays exponentially as \(r\to\infty\).
Interactive: visualize hydrogen atom radial functions
Choose \(n\) and \(\ell\) (with \(\ell \le n-1\)) and plot hydrogen radial functions. The horizontal axis is \(r/a_0\).
Tips: Try plotting \(P(r)=r^2|R|^2\) to see where the electron is most likely to be found. Overlaying curves helps you compare nodes and how probability shifts as \(n\) or \(\ell\) changes.
Big idea: the structure of the hydrogen atom radial wavefunctions is determined by mathematical properties of Laguerre polynomials, which enforce normalizability and quantization of the atomic energy levels.
An important feature of hydrogen atom wavefunctions is the presence of radial nodes—values of \(r\) at which the radial wavefunction (and therefore the probability density) is zero. The number of radial nodes is determined entirely by the quantum numbers \(n\) and \(\ell\).
Definition of a radial node
A radial node is a value of \(r > 0\) for which \(R_{n\ell}(r)=0\). At a radial node, the probability of finding the electron at that radius is zero.
Radial nodes arise from the polynomial part of the radial wavefunction, not from the exponential decay or power-law prefactor.
Counting radial nodes
The hydrogen atom radial wavefunctions contain the associated Laguerre polynomial
\[ L_{n-\ell-1}^{(2\ell+1)}\!\left(\frac{2r}{na_0}\right). \]
The order of this polynomial is \(n-\ell-1\), which is exactly the number of times the polynomial can cross zero.
Therefore, the number of radial nodes is
\[ \boxed{\text{Number of radial nodes} = n - \ell - 1.} \]
Implications
- For fixed \(n\), increasing \(\ell\) reduces the number of radial nodes.
- States with \(\ell = n-1\) have no radial nodes.
- States with \(\ell = 0\) (s orbitals) have the maximum number of radial nodes for a given \(n\).
Although all states with the same \(n\) have the same energy in the hydrogen atom, they can have very different spatial structures because the number of radial nodes depends on \(\ell\).
Big idea: the quantum numbers \(n\) and \(\ell\) together determine the nodal structure of the hydrogen atom radial wavefunctions, revealing how orbitals with the same energy can have very different shapes.
Your turn
| Problem 1 | |
|---|---|
| What is the primary reason Laguerre polynomials appear in the hydrogen atom radial wavefunctions? | |
| They describe angular momentum eigenstates | |
| They ensure the radial wavefunction is normalizable | |
| They arise from relativistic corrections | |
| They eliminate the Coulomb potential | |
| Problem 2 | |
|---|---|
| Which mathematical function ensures that the hydrogen atom radial wavefunction decays as \(r \to \infty\)? | |
| The exponential factor \(e^{-r/(na_0)}\) | |
| The power-law factor \(r^\ell\) | |
| The associated Laguerre polynomial | |
| The normalization constant | |
| Problem 3 | |
|---|---|
| For a given principal quantum number \(n\), how many radial nodes does the hydrogen atom wavefunction have? | |
| \(n\) | |
| \(n+\ell\) | |
| \(\ell\) | |
| \(n-\ell-1\) | |
| Problem 4 | |
|---|---|
| What physical quantity is represented by the radial probability distribution \(P(r)=r^2|R(r)|^2\)? | |
| The total energy of the electron | |
| The probability of finding the electron between \(r\) and \(r+dr\) | |
| The angular distribution of the electron | |
| The expectation value of \(r\) | |
| Problem 5 | |
|---|---|
| For fixed \(n\), how does increasing \(l\) affect the radial wavefunction? | |
| It shifts probability away from the nucleus | |
| It increases the total energy | |
| It removes radial nodes | |
| It changes the Rydberg constant | |
Key points (one glance)
- The hydrogen atom wavefunction separates into a radial and an angular part: \(\psi(r,\theta,\phi)=R_{n\ell}(r)Y_\ell^{m_\ell}(\theta,\phi)\).
- The radial wavefunctions are built from exponential, power-law, and polynomial factors involving associated Laguerre polynomials.
- The exponential factor ensures decay of the wavefunction as \(r\to\infty\), while the power-law factor ensures finiteness at \(r=0\).
- Radial nodes occur at values of \(r>0\) where \(R_{n\ell}(r)=0\).
- The number of radial nodes is determined by the quantum numbers: \(\text{nodes}=n-\ell-1\).
- For a fixed \(n\), increasing \(\ell\) reduces the number of radial nodes and shifts probability away from the nucleus.
- States with the same \(n\) but different \(\ell\) have the same energy in hydrogen but different spatial structures.
Big picture: the mathematical structure of Laguerre polynomials determines the nodal patterns and shapes of hydrogen atom orbitals, revealing how quantum numbers encode spatial information beyond energy alone.