The hydrogen atom is the simplest atomic system, making it a perfect starting point for understanding atomic structure. By examining its energy levels, wave functions, and orbitals, we can grasp the fundamentals of quantum mechanics applied to atoms.

Hydrogenic systems extend this knowledge to one-electron ions with higher nuclear charges. This helps us bridge the gap between the hydrogen atom and more complex multi-electron atoms, setting the stage for deeper atomic structure studies.

Bohr Model and Schrödinger Equation

Bohr's Model of the Atom

• Bohr model proposed that electrons orbit the nucleus in fixed circular paths called stationary states or orbits
• Each orbit has a specific energy associated with it and electrons can only transition between these discrete energy levels by absorbing or emitting photons
• The energy of an electron in a particular orbit is given by $E_n = -\frac{R_H}{n^2}$, where $R_H$ is the Rydberg constant and $n$ is the principal quantum number (1, 2, 3, ...)
• While successful in explaining the hydrogen atom spectrum, the Bohr model fails to accurately describe more complex atoms and the behavior of electrons in molecules

Schrödinger Equation and Quantum Mechanics

• Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior of a quantum system, such as an electron in an atom
• It is written as $\hat{H}\Psi = E\Psi$, where $\hat{H}$ is the Hamiltonian operator, $\Psi$ is the wave function, and $E$ is the energy of the system
• The wave function $\Psi$ contains all the information about the quantum system and its square modulus $|\Psi|^2$ represents the probability density of finding the particle at a given point in space
• Solving the Schrödinger equation for the hydrogen atom yields the energy levels and wave functions that describe the behavior of the electron

Energy Levels and Quantum Numbers

• Energy levels in the hydrogen atom are quantized and determined by the principal quantum number $n$, which can take integer values (1, 2, 3, ...)
• Each energy level is associated with a specific set of quantum numbers that describe the state of the electron:
  • Principal quantum number ($n$): determines the energy and size of the orbital
  • Azimuthal quantum number ($l$): determines the shape of the orbital (s, p, d, f, ...)
  • Magnetic quantum number ($m_l$): determines the orientation of the orbital in space
  • Spin quantum number ($m_s$): describes the intrinsic angular momentum of the electron (spin up or spin down)
• The energy levels in the hydrogen atom are degenerate, meaning that multiple states with different quantum numbers can have the same energy (e.g., 2s and 2p orbitals)

Wave Functions and Orbitals

Radial and Angular Wave Functions

• The wave function for the hydrogen atom can be separated into radial and angular components: $\Psi(r,\theta,\phi) = R(r)Y(\theta,\phi)$
• The radial wave function $R(r)$ describes the probability of finding the electron at a certain distance from the nucleus and depends on the principal and azimuthal quantum numbers ($n$ and $l$)
• The angular wave function $Y(\theta,\phi)$ describes the angular distribution of the electron and depends on the azimuthal and magnetic quantum numbers ($l$ and $m_l$)
• The angular wave functions are called spherical harmonics and are denoted as $Y_l^{m_l}(\theta,\phi)$

Orbitals and Electron Probability Density

• Orbitals are three-dimensional regions in space where there is a high probability of finding an electron
• They are obtained by squaring the wave function: $|\Psi(r,\theta,\phi)|^2 = |R(r)|^2|Y(\theta,\phi)|^2$
• The shape of an orbital depends on the azimuthal quantum number $l$:
  • s orbitals ($l=0$) are spherically symmetric
  • p orbitals ($l=1$) have two lobes with a node at the nucleus
  • d orbitals ($l=2$) have more complex shapes with multiple lobes and nodes
• The electron probability density is the square of the wave function and represents the probability of finding the electron at a given point in space
• Regions with high electron probability density correspond to the lobes of the orbitals, while nodes are regions where the probability density is zero

Hydrogenic Systems

Hydrogenic Ions and Their Properties

• Hydrogenic systems are one-electron ions with a single electron orbiting a positively charged nucleus, such as He$^+$, Li$^{2+}$, Be$^{3+}$, etc.
• The energy levels and wave functions for hydrogenic ions can be obtained by solving the Schrödinger equation with a modified potential energy term that accounts for the increased nuclear charge
• The energy levels of hydrogenic ions are given by $E_n = -\frac{Z^2R_H}{n^2}$, where $Z$ is the atomic number (nuclear charge) and $R_H$ is the Rydberg constant
• As the nuclear charge increases, the energy levels become more negative, and the electron is more tightly bound to the nucleus
• The orbitals of hydrogenic ions have the same shapes as those of the hydrogen atom but are smaller in size due to the increased nuclear charge
• Hydrogenic ions are important in understanding the behavior of electrons in multi-electron atoms and in the study of atomic spectra