Hydrogen Atom in 3D: Quantum Structure of the Simplest Atom

Table of Contents

1. Introduction
2. Importance of the Hydrogen Atom
3. Classical vs Quantum Models
4. The Coulomb Potential and Central Force Problem
5. Schrödinger Equation in 3D Spherical Coordinates
6. Separation of Variables
7. Radial and Angular Parts of the Wavefunction
8. Angular Solutions: Spherical Harmonics
9. Radial Equation and Effective Potential
10. Quantization of Energy Levels
11. Principal, Orbital, and Magnetic Quantum Numbers
12. Radial Wavefunctions and Their Nodes
13. Electron Orbitals and Probability Densities
14. Degeneracy and Symmetries
15. Fine Structure and Spin-Orbit Coupling (Preview)
16. Real-World Applications and Spectroscopy
17. Conclusion

1. Introduction

The hydrogen atom, consisting of a single proton and electron, serves as the foundation of atomic physics and quantum mechanics. Its study yields exact solutions to the quantum mechanical problem of a charged particle in a Coulomb potential and sets the stage for understanding more complex atoms and molecules.

2. Importance of the Hydrogen Atom

The hydrogen atom is the simplest bound quantum system. Its analytical solution:

• Validates quantum theory
• Predicts atomic spectra
• Helps define quantum numbers and orbitals
• Introduces tools like angular momentum, spherical harmonics, and radial equations

3. Classical vs Quantum Models

Classical View:

• Electron orbits nucleus in circular or elliptical path (Bohr model)
• Discrete orbits postulated without quantum basis

Quantum View:

• Electron described by wavefunction
• Energy levels and orbital shapes arise from solving Schrödinger equation

4. The Coulomb Potential and Central Force Problem

In atomic units, the electrostatic potential energy is:

\[
V(r) = -\frac{e^2}{4\pi\varepsilon_0 r}
\]

This spherically symmetric potential makes it a central force problem, ideal for spherical coordinates.

5. Schrödinger Equation in 3D Spherical Coordinates

The time-independent Schrödinger equation:

\[
-\frac{\hbar^2}{2\mu} \nabla^2 \psi(\vec{r}) + V(r)\psi(\vec{r}) = E\psi(\vec{r})
\]

Where:

• \( \mu \): reduced mass of electron-proton system
• \( V(r) \): Coulomb potential

In spherical coordinates \((r, \theta, \phi)\), \( \nabla^2 \) becomes:

\[
\nabla^2 = \frac{1}{r^2} \frac{\partial}{\partial r} \left( r^2 \frac{\partial}{\partial r} \right)
\frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta} \left( \sin\theta \frac{\partial}{\partial \theta} \right)
\frac{1}{r^2 \sin^2\theta} \frac{\partial^2}{\partial \phi^2}
\]

6. Separation of Variables

Assume solution:

\[
\psi(r, \theta, \phi) = R(r) Y(\theta, \phi)
\]

Insert into Schrödinger equation ⇒ separates into radial and angular parts.

7. Radial and Angular Parts of the Wavefunction

The equation separates into:

Angular Equation:

\[
\hat{L}^2 Y(\theta, \phi) = \hbar^2 \ell(\ell+1) Y(\theta, \phi)
\]

Radial Equation:

\[
\frac{d^2u}{dr^2} + \left[ \frac{2\mu}{\hbar^2} \left(E + \frac{e^2}{4\pi\varepsilon_0 r} \right) – \frac{\ell(\ell + 1)}{r^2} \right] u(r) = 0
\]

Where \( u(r) = r R(r) \) and \( \ell \): orbital angular momentum quantum number.

8. Angular Solutions: Spherical Harmonics

The angular part \( Y_\ell^m(\theta, \phi) \) are spherical harmonics, solutions of:

\[
\hat{L}z Y\ell^m = \hbar m Y_\ell^m, \quad \hat{L}^2 Y_\ell^m = \hbar^2 \ell(\ell+1) Y_\ell^m
\]

With:

• \( \ell = 0, 1, 2, \dots \)
• \( m = -\ell, \dots, \ell \)

These define the shapes and orientations of orbitals.

9. Radial Equation and Effective Potential

The radial equation contains an effective potential:

\[
V_{\text{eff}}(r) = -\frac{e^2}{4\pi\varepsilon_0 r} + \frac{\hbar^2 \ell(\ell + 1)}{2\mu r^2}
\]

The second term represents the centrifugal barrier due to angular momentum.

10. Quantization of Energy Levels

Solving the radial equation yields discrete energy levels:

\[
E_n = -\frac{\mu e^4}{2(4\pi\varepsilon_0)^2 \hbar^2 n^2}, \quad n = 1, 2, 3, \dots
\]

These depend only on the principal quantum number \( n \). This degeneracy reflects the high symmetry of the Coulomb potential.

11. Principal, Orbital, and Magnetic Quantum Numbers

• \( n \): Principal quantum number, \( n \ge 1 \)
• \( \ell \): Orbital quantum number, \( 0 \le \ell \le n-1 \)
• \( m \): Magnetic quantum number, \( -\ell \le m \le \ell \)

Each state is labeled \( |n, \ell, m\rangle \).

12. Radial Wavefunctions and Their Nodes

Radial functions are:

\[
R_{n\ell}(r) = \rho^\ell e^{-\rho/2} L_{n-\ell-1}^{2\ell+1}(\rho), \quad \rho = \frac{2r}{na_0}
\]

Where \( L_k^m \) are associated Laguerre polynomials, and \( a_0 \) is the Bohr radius.

• \( n – \ell – 1 \): number of radial nodes

13. Electron Orbitals and Probability Densities

The electron orbitals (e.g., 1s, 2p, 3d) are visual representations of \( |\psi(r, \theta, \phi)|^2 \).

• Shape dictated by angular part \( Y_\ell^m \)
• Size and radial structure by \( R_{n\ell}(r) \)
• Probability densities are used in atomic imaging and chemistry

14. Degeneracy and Symmetries

Each energy level \( E_n \) has a degeneracy:

\[
g_n = n^2 = \sum_{\ell=0}^{n-1} (2\ell + 1)
\]

Reflecting spherical symmetry and conservation of angular momentum.

15. Fine Structure and Spin-Orbit Coupling (Preview)

Real hydrogen energy levels split slightly due to:

• Relativistic corrections
• Spin-orbit coupling
• Quantum electrodynamic effects (Lamb shift)

These refinements are explained by Dirac theory and QED.

16. Real-World Applications and Spectroscopy

Hydrogen’s spectral lines (Lyman, Balmer series) match energy differences:

\[
\Delta E = E_n – E_{n’}
\]

Explains:

• Astronomical spectra
• Atomic clocks
• Quantum chemistry foundations

17. Conclusion

The 3D hydrogen atom is a landmark success of quantum theory. Its analytical solution explains atomic spectra, defines quantum numbers, and reveals orbital structures. Mastery of this system is essential for advanced quantum mechanics, spectroscopy, chemistry, and quantum computing.