Hydrogen Atom Quantum Solution

The hydrogen atom is the Rosetta Stone of quantum mechanics—a real-world system whose precise solution validates the entire theoretical framework. By solving the Schrödinger equation for the Coulomb potential, you don't just get numbers; you derive the very architecture of the periodic table and the origin of light itself. This journey from a differential equation to the explanation of atomic spectra is a cornerstone of modern physics, connecting abstract mathematics to observable reality.

The Schrödinger Equation in Spherical Coordinates

The problem begins with the time-independent Schrödinger equation $\hat{H}\psi =E\psi$ for an electron of reduced mass $\mu$ in the Coulomb potential of a proton, $V(r)=-\frac{e^2}{4\pi {ϵ}_{0}r}$. Because the potential is central (depending only on the distance $r$), it is natural to work in spherical coordinates $(r,\theta ,\varphi )$. The Hamiltonian operator transforms into a more complex form:

$$\hat{H}=-\frac{{\hslash }^2}{2\mu }{\nabla }^2-\frac{e^2}{4\pi {ϵ}_{0}r}$$

The Laplacian ${\nabla }^2$ in spherical coordinates contains terms for radial and angular derivatives. The key breakthrough is the assumption that the wavefunction $\psi (r,\theta ,\varphi )$ can be separated into a product of functions, each depending on a single coordinate: $\psi (r,\theta ,\varphi )=R(r)Y(\theta ,\varphi )$. Substituting this product into the Schrödinger equation and carefully separating variables leads to two distinct, simpler eigenvalue equations: one purely radial and one purely angular.

The angular equation recovers the spherical harmonics $Y_l^m(\theta ,\varphi )$, which are solutions to the angular momentum eigenvalue problem. This separation introduces a constant of separation, which turns out to be $l(l+1){\hslash }^2$, where $l$ is the azimuthal quantum number. The remaining radial equation for $R(r)$ then becomes:

$$-\frac{{\hslash }^2}{2\mu }\frac{1}{r^2}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right)+\left[\frac{l(l+1){\hslash }^2}{2\mu r^2}-\frac{e^2}{4\pi {ϵ}_{0}r}\right]R=ER$$

This equation contains the effective potential, which combines the attractive Coulomb potential with a repulsive centrifugal barrier term proportional to $l(l+1)/r^2$.

Quantum Numbers and the Form of the Wavefunctions

Solving the radial equation requires a systematic series solution method (Frobenius method). The requirement for the wavefunction to be normalizable—to not blow up at infinity—forces the series to terminate. This termination condition is the source of quantization. It leads to the principal result for the energy levels:

$$E_n=-\frac{\mu e^4}{32{\pi }^2{ϵ}_0^2{\hslash }^2}\frac{1}{n^2}=-\frac{13.6\text{ eV}}{n^2}$$

Here, $n$ is the principal quantum number, a positive integer ($n=1,2,3,...$). The energy depends only on $n$, not on the other quantum numbers emerging from the solution.

The complete hydrogen atom wavefunction is characterized by three quantum numbers:

• $n$ (Principal Quantum Number): Determines the energy level and the overall spatial extent of the electron's probability cloud. $n\ge 1$.
• $l$ (Azimuthal/Angular Momentum Quantum Number): Determines the magnitude of orbital angular momentum, $L=\sqrt{l(l+1)}\hslash$. For a given $n$, $l$ can range from $0$ to $n-1$.
• $m_l$ (Magnetic Quantum Number): Determines the z-component of orbital angular momentum, $L_z=m_l\hslash$. For a given $l$, $m_l$ ranges from $-l$ to $+l$ in integer steps.

The full wavefunction is ${\psi }_{nl{m}_l}(r,\theta ,\varphi )=R_{nl}(r)Y_l^{m_l}(\theta ,\varphi )$. The radial wavefunction $R_{nl}(r)$ is a product of an exponential decay $e^{-r/(n{a}_0)}$, a polynomial (the associated Laguerre polynomial), and a power of $r$. The Bohr radius $a_0=\frac{4\pi {ϵ}_0{\hslash }^2}{\mu e^2}\approx 0.529\text{ A˚}$ naturally emerges as the characteristic length scale. For example, the ground state ($n=1,l=0,m_l=0$) wavefunction is ${\psi }_{100}=\frac{1}{\sqrt{\pi a_0^{3/2}}}{e}^{-r/a_0}$, spherically symmetric and exponentially decaying.

The angular wavefunctions are the spherical harmonics $Y_l^{m_l}$. They dictate the shape and orientation of the orbital. For $l=0$ (s-orbitals), $Y_0^0$ is a constant, leading to spherical symmetry. For $l=1$ (p-orbitals), the shapes are dumbbell-like, oriented along the x, y, or z axes depending on $m_l$.

Degeneracy and Underlying Symmetry

A striking feature of the hydrogen atom solution is its high degeneracy. Many different quantum states share the same energy. For a given principal quantum number $n$, the allowed values of $l$ are $0,1,2,...,n-1$. For each $l$, there are $(2l+1)$ values of $m_l$. The total number of distinct states with energy $E_n$ is the sum:

$$\text{Degeneracy of level }n=\sum _{l=0}^{n-1}(2l+1)=n^2$$

This $n^2$ degeneracy is "accidental" in the sense that it is greater than the $(2l+1)$ degeneracy expected for any central potential (which comes from rotational symmetry and conservation of angular momentum). The extra degeneracy, where states with different $l$ but the same $n$ have the same energy, is a special property of the pure $1/r$ Coulomb potential. It hints at a hidden symmetry (described by the SO(4) group) related to the conservation of the Laplace-Runge-Lenz vector, a quantum-mechanical analogue of a conserved quantity in the classical Kepler problem.

Selection Rules and the Origin of Spectral Series

The hydrogen atom's quantized energy levels directly explain its line spectrum. When an electron transitions from a higher energy level $E_i$ to a lower one $E_f$, a photon is emitted with energy $h\nu =E_i-E_f$. However, not every possible transition between levels occurs with significant probability. The interaction strength with the electromagnetic radiation field is governed by the transition dipole moment, an integral involving the wavefunctions of the initial and final states.

This integral imposes selection rules that dictate which transitions are "allowed" (strong) and which are "forbidden" (very weak). For electric dipole radiation in hydrogen, the rules are:

1. $\Delta l=\pm 1$ (The azimuthal quantum number must change by exactly 1).
2. $\Delta m_l=0,\pm 1$ (The magnetic quantum number can change by 0 or ±1).

The selection rule on $l$ is crucial. An electron in a 2s state ($n=2,l=0$) cannot make an electric dipole transition directly to the 1s ground state ($n=1,l=0$) because $\Delta l=0$. Such a transition, if it occurs, must proceed via other, slower mechanisms. This explains the metastability of the 2s state.

These quantized energies and selection rules generate the famous spectral series, named for the principal quantum number of the lower energy level in the transition:

• Lyman Series: Transitions to $n=1$ ($\Delta l=+1$ for the photon-emitting electron). In the ultraviolet.
• Balmer Series: Transitions to $n=2$. Several lines are in the visible spectrum (e.g., H-alpha, H-beta).
• Paschen, Brackett, Pfund Series: Transitions to $n=3,4,5$, respectively, in the infrared.

The formula for the wavelength $\lambda$ of any line follows directly from the energy formula:

$$\frac{1}{\lambda }=R_H\left(\frac{1}{n_f^2}-\frac{1}{n_i^2}\right)$$

where $R_H$ is the Rydberg constant and $n_i>n_f$.

Common Pitfalls

1. Confusing the roles of quantum numbers: A common error is to think the principal quantum number $n$ determines angular momentum. Remember, $n$ sets the energy and size, but $l$, which is constrained to be less than $n$, determines the angular momentum magnitude. A high-$n$ state can still have $l=0$ (a circular orbit is not required).
2. Misapplying selection rules to all systems: The selection rule $\Delta l=\pm 1$ is derived for electric dipole transitions in a system with a single electron in a central potential (like hydrogen). It does not hold universally for many-electron atoms or for other types of transitions (e.g., magnetic dipole, electric quadrupole), which have their own, different selection rules.
3. Misinterpreting the wavefunction plot: When visualizing $|\psi {|}^2$, the radial probability density $P(r)=r^2|R_{nl}(r){|}^2$ is what gives the probability of finding the electron between $r$ and $r+dr$. Simply plotting $R_{nl}(r)$ or $|\psi {|}^2$ without the $r^2$ volume factor can be misleading. For example, the $R_{10}(r)$ function is maximum at $r=0$, but $P(r)$ for the ground state is maximum at $r=a_0$ (the Bohr radius).

Summary

• The hydrogen atom is solved by separating the Schrödinger equation in spherical coordinates, yielding a radial equation for $R(r)$ and an angular equation solved by spherical harmonics $Y_l^m(\theta ,\varphi )$.
• Quantization arises from boundary conditions, producing discrete energy levels $E_n=-13.6\text{ eV}/n^2$ and three quantum numbers: $n$ (energy, size), $l$ (angular momentum magnitude), and $m_l$ (angular momentum orientation).
• The complete wavefunction is ${\psi }_{nl{m}_l}=R_{nl}(r)Y_l^{m_l}(\theta ,\varphi )$, where the radial part contains associated Laguerre polynomials and defines orbital nodes.
• The $n^2$ degeneracy of each energy level exceeds that expected from simple rotational symmetry, revealing a hidden symmetry unique to the $1/r$ potential.
• Photon emission/absorption spectra are explained by transitions between these levels, governed by selection rules ($\Delta l=\pm 1$, $\Delta m_l=0,\pm 1$) that give rise to the distinct Lyman, Balmer, and other spectral series.