The solution of the Schrödinger equation (wave equation) for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it is radially symmetric in space and only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the orbitals) are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: the eigenstates of the Hamiltonian (that is, the energy eigenstates) can be chosen as simultaneous eigenstates of the angular momentum operator. This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, ℓ and m (both are integers). The angular momentum quantum number ℓ = 0, 1, 2, ... determines the magnitude of the angular momentum. The magnetic quantum number m = −ℓ, ..., +ℓ determines the projection of the angular momentum on the (arbitrarily chosen) z-axis. 6.1: The Schrodinger Equation for the Hydrogen Atom Can Be Solved Exactly This page explains the modeling of a hydrogen atom as a system of an electron and a proton, utilizing reduced mass for simplifying calculations around the center of mass. It discusses the time-independent Schrödinger equation which combines kinetic and potential energy through the Coulomb potential. 6.2: The Wavefunctions of a Rigid Rotator are Called Spherical Harmonics This page discusses the solutions to the hydrogen atom Schrödinger equation, detailing wavefunctions dependent on quantum numbers \(n\), \(l\), and \(m_l\), which define electron properties and probability densities. It covers angular momentum orientations, atomic orbitals, and the corresponding wavefunctions, emphasizing their visualization and implications in molecular interactions. 6.3: The Three Components of Angular Momentum Cannot be Measured Simultaneously with Arbitrary Precision This page explores the measurement and quantization of orbital angular momentum in quantum mechanics, starting from classical definitions and extending to operator formalism. It covers commutation relations of angular momentum operators, demonstrating how they affect measurement uncertainty and ensure consistent states. 6.4: Hydrogen Atomic Orbitals Depend upon Three Quantum Numbers This page covers wavefunctions of the hydrogen atom, highlighting the role of quantum numbers \(n\), \(l\), and \(m_l\) in determining electron position and probability density. It discusses normalization, orthogonality, allowed values for the quantum numbers, and the radial behavior of wavefunctions related to atomic number \(Z\). 6.5: s-orbitals are Spherically Symmetric This page discusses hydrogen atom wavefunctions, highlighting atomic orbitals like the 1s and 2s. It explains how quantum numbers determine the number of orbitals and their charge distributions, which influence chemical reactivity. Concepts such as radial wavefunctions, radial nodes, and radial distribution functions (RDF) illustrate electron charge distribution and its probability around the nucleus. 6.6: Orbital Angular Momentum and the p-Orbitals This page discusses the relationship between classical and quantum angular momentum for electrons in atoms, emphasizing quantum numbers \(l\), \(n\), and \(m_l\) that define electron behavior. It distinguishes between spherical and non-spherical distributions, specifically analyzing p- and d-orbitals in context with their respective quantum numbers. The text also mentions contributors involved in the topic, acknowledging individuals from McMaster University and Stack Exchange members. 6.7: The Helium Atom Cannot Be Solved Exactly This page addresses the complexities of solving Schrödinger equations for multi-electron atoms like helium, which lacks an analytic solution unlike hydrogen. It discusses the non-separable nature of multi-electron Hamiltonians, the orbital approximation for wavefunctions, and the effects of electron-electron interactions. Key components of the helium Hamiltonian are explored, along with the significance of neutron variations on electron energies. 6.E: The Hydrogen Atom (Exercises) This page covers various mathematical evaluations related to Chebyshev polynomials and quantum mechanics, including integrals, radial functions for hydrogen atom orbitals, and properties of Hermitian operators like the Hamiltonian. It highlights the significance of quantum numbers, magnetic field effects on atomic transitions, and the variational method for calculating ground-state energy. Thumbnail: Hydrogen atom. (Public Domain; Bensaccount via Wikipedia )