Table of Contents

1. Introduction
2. Central Potentials and Radial Schrödinger Equation
3. Separation of Variables in Spherical Coordinates
4. Angular Momentum and the Centrifugal Term
5. Redefining the Radial Equation
6. Effective Potential and Physical Interpretation
7. Principal Quantum Number \( n \)
8. Orbital Angular Momentum Quantum Number \( \ell \)
9. Radial Nodes and Quantum Number Relationships
10. Solving the Radial Equation for Hydrogen
11. Radial Wavefunctions and Laguerre Polynomials
12. Bohr Radius and Atomic Length Scales
13. Radial Probability Density
14. Quantum Numbers and Degeneracy
15. Extensions to Multi-Electron Atoms
16. Conclusion

---

1. Introduction

In atomic quantum mechanics, understanding how the electron behaves within a central potential is key to predicting its energy levels and spatial distribution. The radial equation, derived from the 3D Schrödinger equation, contains essential physics and introduces important quantum numbers that govern atomic structure.

---

2. Central Potentials and Radial Schrödinger Equation

A central potential depends only on the distance from the origin:

\[
V(\vec{r}) = V(r)
\]

This symmetry allows the use of spherical coordinates \((r, \theta, \phi)\), leading to separable solutions and well-defined angular momentum properties.

---

3. Separation of Variables in Spherical Coordinates

Assume a total wavefunction:

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

Substituting into the time-independent Schrödinger equation and separating variables yields:

• An angular equation, solved by spherical harmonics \( Y_\ell^m(\theta, \phi) \)
• A radial equation, which governs the energy and radial probability

---

4. Angular Momentum and the Centrifugal Term

From the angular solution, we get:

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

This leads to a term in the radial equation:

\[
\frac{\ell(\ell+1)\hbar^2}{2\mu r^2}
\]

Called the centrifugal barrier, it prevents low-\( \ell \) states from collapsing at the origin and shapes the effective potential.

---

5. Redefining the Radial Equation

Define a new function \( u(r) = r R(r) \). The radial equation becomes:

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

This is a second-order differential equation whose solutions provide the allowed radial wavefunctions and energy levels.

---

6. Effective Potential and Physical Interpretation

Rewriting the radial equation:

\[
\frac{d^2 u}{dr^2} + \frac{2\mu}{\hbar^2} \left[E – V_{\text{eff}}(r) \right] u(r) = 0
\]

Where:

\[
V_{\text{eff}}(r) = V(r) + \frac{\hbar^2 \ell(\ell+1)}{2\mu r^2}
\]

• The second term acts like a repulsive force
• Influences the turning points and radial confinement

---

7. Principal Quantum Number \( n \)

The principal quantum number \( n \) labels energy levels:

\[
n = 1, 2, 3, \dots
\]

It arises from boundary conditions ensuring normalizable and finite solutions. In hydrogen-like atoms:

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

---

8. Orbital Angular Momentum Quantum Number \( \ell \)

The quantum number \( \ell \) arises from solving the angular equation:

• Determines the shape of the orbital
• Allowed values: \( \ell = 0, 1, 2, \dots, n-1 \)

It also appears in the centrifugal term, shaping the radial solution.

---

9. Radial Nodes and Quantum Number Relationships

The number of radial nodes \( n_r \) (zero crossings in \( R(r) \)) is:

\[
n_r = n – \ell – 1
\]

Thus:

• Ground state (\( n = 1, \ell = 0 \)): 0 nodes
• Higher \( n \): more nodes, more oscillatory behavior

---

10. Solving the Radial Equation for Hydrogen

For hydrogen’s Coulomb potential:

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

The radial solutions involve associated Laguerre polynomials:

\[
R_{n\ell}(r) = \rho^\ell e^{-\rho/2} L_{n – \ell – 1}^{2\ell + 1}(\rho), \quad \rho = \frac{2r}{n a_0}
\]

Where \( a_0 \) is the Bohr radius.

---

11. Radial Wavefunctions and Laguerre Polynomials

Radial functions are:

\[
R_{n\ell}(r) = N_{n\ell} \left( \frac{r}{a_0} \right)^\ell e^{-r/(na_0)} L_{n-\ell-1}^{2\ell+1} \left( \frac{2r}{na_0} \right)
\]

These are orthogonal and form a complete basis for bound states.

---

12. Bohr Radius and Atomic Length Scales

The Bohr radius is:

\[
a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{\mu e^2}
\]

It defines the natural length scale of hydrogenic orbitals.

Typical orbital sizes increase with \( n \), as \( \langle r \rangle \sim n^2 a_0 \).

---

13. Radial Probability Density

The radial probability density is:

\[
P(r) = |R(r)|^2 r^2
\]

Gives the likelihood of finding the electron at a distance \( r \). Plots of \( P(r) \) show peaks and nodes, visualizing electron shell structure.

---

14. Quantum Numbers and Degeneracy

Each energy level \( n \) has a degeneracy of:

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

This reflects the number of allowed \( \ell \) and \( m \) values per \( n \).

---

15. Extensions to Multi-Electron Atoms

While hydrogen allows exact solutions, in multi-electron atoms:

• The radial equation is modified by electron shielding
• \( n \) and \( \ell \) remain, but energies depend on screened nuclear charge
• Leads to fine structure splitting and shell structures

---

16. Conclusion

The radial equation lies at the heart of quantum atomic theory. It introduces key quantum numbers that define the structure, energy, and behavior of atomic orbitals. Understanding how radial functions and quantum numbers interplay provides a deeper insight into electron configuration, spectroscopy, and the periodic table.

---

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.

---