To describe the hydrogen atom (and hydrogen-like systems), we begin by constructing the Hamiltonian operator. Unlike the rigid rotor or harmonic oscillator, the hydrogen atom involves a charged particle moving in a central electrostatic potential.
Electrostatic potential energy
The hydrogen atom consists of an electron of charge \(-e\) attracted to a nucleus of charge \( +e \). The electrostatic interaction between these charges is described by the Coulomb potential:
\[ V(r) = -\frac{e^2}{4\pi\varepsilon_0 r}, \]
where \(r\) is the distance between the electron and the nucleus. The negative sign indicates an attractive interaction.
Because the potential depends only on the distance \(r\), the hydrogen atom is a central-force problem.
Kinetic energy and reduced mass
As discussed in the review of the Rydberg constant, the electron and nucleus form a two-body system. This can be reduced to an effective one-body problem by replacing the electron mass with the reduced mass \(\mu\):
\[ \mu = \frac{m_e M}{m_e + M}, \]
where \(M\) is the nuclear mass. The kinetic energy operator is therefore
\[ \hat{T} = -\frac{\hbar^2}{2\mu}\nabla^2. \]
The hydrogen atom Hamiltonian
Because the hydrogen atom is a three-dimensional problem, the wavefunction depends on all three spherical polar coordinates: \(r\), \(\theta\), and \(\phi\). The correct form of the wavefunction is therefore \(\psi(r,\theta,\phi)\).
The time-independent Schrödinger equation for the hydrogen atom is
\[ \hat{H}\psi(r,\theta,\phi) = E\,\psi(r,\theta,\phi), \]
or explicitly,
\[ \left[ -\frac{\hbar^2}{2\mu}\nabla^2 - \frac{e^2}{4\pi\varepsilon_0 r} \right] \psi(r,\theta,\phi) = E\,\psi(r,\theta,\phi). \]
Solving this equation yields the allowed energy levels and wavefunctions of the hydrogen atom.
Big idea: the hydrogen atom Hamiltonian combines kinetic energy with a Coulomb potential, leading to a central-force Schrödinger equation whose solutions explain atomic spectra and quantized electronic structure.
The hydrogen atom Schrödinger equation can be solved by exploiting the spherical symmetry of the Coulomb potential. Because the potential depends only on the radial coordinate \(r\), the wavefunction can be separated into radial and angular parts.
Separation of variables
We assume a separable solution of the form
\[ \psi(r,\theta,\phi) = R(r)\,Y(\theta,\phi), \]
where:
- \(R(r)\) describes the radial dependence,
- \(Y(\theta,\phi)\) describes the angular dependence.
The Laplacian operator
In spherical polar coordinates, the Laplacian can be written as the sum of two operators:
\[ \nabla^2 = \frac{1}{r^2}\frac{d}{dr} \left( r^2\frac{d}{dr} \right) - \frac{\hat{L}^2}{\hbar^2 r^2}. \]
The first term is a purely radial operator. The second term contains the angular momentum operator \(\hat{L}^2\), which acts only on the angular variables \(\theta\) and \(\phi\).
Angular equation and spherical harmonics
Substituting the separated wavefunction into the Schrödinger equation leads to an angular equation involving \(\hat{L}^2\). Its eigenfunctions are the spherical harmonics, \(Y_\ell^{m_\ell}(\theta,\phi)\).
These functions satisfy the eigenvalue equation
\[ \hat{L}^2\,Y_\ell^{m_\ell}(\theta,\phi) = \hbar^2\,\ell(\ell+1)\,Y_\ell^{m_\ell}(\theta,\phi), \]
where \(\ell = 0,1,2,\ldots\) is the orbital angular momentum quantum number.
The radial Schrödinger equation
Using the angular eigenvalue \(\hbar^2\ell(\ell+1)\), the full Schrödinger equation reduces to a radial differential equation for \(R(r)\):
\[ -\frac{\hbar^2}{2\mu} \left[ \frac{1}{r^2}\frac{d}{dr} \left( r^2\frac{dR}{dr} \right) - \frac{\ell(\ell+1)}{r^2}R \right] - \frac{e^2}{4\pi\varepsilon_0 r}\,R = E\,R. \]
Solving this equation yields the allowed radial wavefunctions and the quantized energy levels of the hydrogen atom. And because the quantum number l appears in the differential equation, we expect to see a constraint placed on the values l can take due to the solutions to the differential equation.
Big idea: separating variables reduces the three-dimensional hydrogen atom problem to an angular equation (with known solutions) and a single radial equation whose solutions determine atomic energies and orbital shapes.