Description of the Hydrogen Atom
A hydrogen atom is electrically neutral, containing a single proton and a single electron bound to the nucleus by the Coulomb force.
Learning Objective

Identify the unique features of the hydrogen atom that make it important for calculations in quantum mechanics
Key Points
 Atomic hydrogen comprises about 75 percent of the elemental mass of the universe.
 Hydrogen atoms are so reactive that they combine with almost all elements.
 Isolated hydrogen atoms called atomic hydrogen or monatomic hydrogen) are extremely rare; hydrogen is more commonly found in combination with other elements.
 The H–H bond is one of the strongest bonds in chemistry.
Terms

atom
The smallest possible amount of matter that still retains its identity as a chemical element. Atoms consist of a nucleus surrounded by electrons.

quantum field theory
Provides a theoretical framework for constructing quantum mechanical models of systems classically represented by an infinite number of degrees of freedom, that is, fields and manybody systems.

anisotropic
The property of being directionally dependent, as opposed to isotropy which implies identical properties in all directions.
Full Text
The hydrogen atom (consisting of one proton and one electron, not the diatomic form H_{2}) has special significance in quantum mechanics and quantum field theory as a simple twobody problem physical system that has yielded many simple analytical solutions in closedform.
Modeling the Hydrogen Atom
In 1914, Niels Bohr obtained the spectral frequencies of the hydrogen atom after making a number of simplifying assumptions. These assumptions, the cornerstones of the Bohr model, were not fully correct but did yield the correct energy answers. Bohr's results for the frequencies and underlying energy values were confirmed by the full quantummechanical analysis which uses the Schrödinger equation, as was shown in 1925–1926. The solution to the Schrödinger equation for hydrogen is analytical. From this, the hydrogen energy levels and thus the frequencies of the hydrogen spectral lines can be calculated. The solution of the Schrödinger equation goes much further than the Bohr model, because it also yields the shape of the electron's wave function (orbital) for the various possible quantummechanical states, thus explaining the anisotropic character of atomic bonds.
A model of the hydrogen atom
This model shows approximate dimensions for nuclear and electron shells (not drawn to scale). It shows a diameter about twice the radius indicated by the Bohr model.
The Schrödinger equation also applies to more complicated atoms and molecules, albeit they rapidly become impossibly difficult beyond hydrogen or other twobody type problems, such as helium cation He^{+}. In most such cases, the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made.
Solution of Schrödinger Equation: Overview of Results
The solution of the Schrödinger equation (wave equations) 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 determines the projection of the angular momentum on the (arbitrarily chosen) zaxis.
In addition to mathematical expressions for total angular momentum and angular momentum projection of wavefunctions, an expression for the radial dependence of the wave functions must be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, .... The principal quantum number in hydrogen is related to the atom's total energy. Note the maximum value of the angular momentum quantum number is limited by the principal quantum number: it can run only up to n − 1, i.e. ℓ = 0, 1, ..., n − 1.
Due to angular momentum conservation, states of the same ℓ but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, states of the same n but different ℓ are also degenerate (i.e. they have the same energy). However, this is a specific property of hydrogen and is no longer true for more complicated atoms that have a (effective) potential differing from the form 1/r, due to the presence of the inner electrons shielding the nucleus potential.
Calculated Energy Levels
The energy levels of hydrogen are given by solving the Schrödinger equation for the oneelectron atom:
Empirically, it is useful to group the fundamental constants into Rydbergs, which gives the much simpler equation below that turns out to be identical to that predicted by Bohr theory:
Further derivation can be performed to include fine structure, which is given by:
where α is the finestructure constant and j is a number which is the total angular momentum eigenvalue; that is, ℓ ± 1/2 depending on the direction of the electron spin. The quantity in square brackets arises from relativistic (spinorbit) coupling interactions.
The value of 13.6 eV is called the Rydberg constant and can be found from the Bohr model and is given by:
where m_{e} is the mass of the electron, q_{e} is the charge of the electron, h is the Planck constant, and ε_{0} is the vacuum permittivity.
The Rydberg constant is connected to the finestructure constant by the relation:
This constant is often used in atomic physics in the form of the Rydberg unit of energy:
Key Term Reference
 Hamiltonian
 angular momentum
 Appears in these related concepts: Angular Quantities as Vectors, Angular vs. Linear Quantities, and Conservation of Angular Momentum
 bond
 Appears in these related concepts: Factors Affecting the Price of a Bond, Current Maturities of LongTerm Debt, and Preferred Stock
 cation
 Appears in these related concepts: Ionic Radius, The Shielding Effect and Effective Nuclear Charge, and Predicting Precipitation Reactions
 chemistry
 Appears in these related concepts: MasstoMole Conversions, The Study of Chemistry, and General Trends in Chemical Properties
 coulombs
 Appears in these related concepts: Nuclear Fusion, Millikan's Oil Drop Experiment, and Electrolysis Stoichiometry
 degenerate
 Appears in these related concepts: Crystal Field Theory, The Central Dogma: DNA Encodes RNA and RNA Encodes Protein, and Types of Conic Sections
 diatomic
 Appears in these related concepts: Homonuclear Diatomic Molecules, Properties of Hydrogen, and Heteronuclear Diatomic Molecules
 eigenfunctions
 electrically neutral
 Appears in these related concepts: Naming Ionic Compounds, Formulas of Ionic Compounds, and Mass Spectrometry to Measure Mass
 electron
 Appears in these related concepts: Periods 1 through 3, Electrolytic Properties, and Microscopy
 element
 Appears in these related concepts: The Periodic Table, Elements and Compounds, and The Periodic Table
 elemental
 Appears in these related concepts: Extractive Metallurgy, The Law of Definite Composition, and Properties of Nitrogen
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 frequency
 Appears in these related concepts: Frequency of Sound Waves, Characteristics of Sound, and Sound
 group
 Appears in these related concepts: Writing Lewis Symbols for Atoms, Secondary Groups, and Primary Groups
 molecule
 Appears in these related concepts: Molecules, Levels of Organization of Living Things, and Chemical Reactions and Molecules
 momentum
 Appears in these related concepts: Inelastic Collisions in Multiple Dimensions, The Second Law: Force and Acceleration, and Differentiation and Rates of Change in the Natural and Social Sciences
 monatomic
 Appears in these related concepts: Electron Configuration of Cations and Anions, The Noble Gases (Group 18), and Photochemical Smog
 nucleus
 Appears in these related concepts: Charge Separation, Electric Charge in the Atom, and Clusters of Neuronal Cell Bodies
 orbital
 Appears in these related concepts: Periodic Table Position and Electron Configuration, Linear Combination of Atomic Orbitals (LCAO), and Electron Orbitals
 proton
 Appears in these related concepts: Overview of Atomic Structure, Development of the Periodic Table, and Cationic ChainGrowth Polymerization
 quanta
 Appears in these related concepts: The Bottom of the Periodic Table, Distribution of Molecular Speeds and Collision Frequency, and The Uncertainty Principle
 quantum
 Appears in these related concepts: The de Broglie Wavelength, Carbonyl Compounds, and Enone Reactions
 quantum number
 Appears in these related concepts: Quantum Numbers, Wave Equation for the Hydrogen Atom, and Diamagnetism and Paramagnetism
 solution
 Appears in these related concepts: Electrolyte and Nonelectrolyte Solutions, Turning Your Claim Into a Thesis Statement, and What is an Equation?
 system
 Appears in these related concepts: Free Energy Changes for Nonstandard States, Definition of Management, and Comparison of Enthalpy to Internal Energy
 wave
 Appears in these related concepts: Properties of Waves and Light, Waves, and Atomic Structure
 wavefunction
 Appears in these related concepts: Particle in a Box, The Pauli Exclusion Principle, and The Phase of Orbitals
Sources
Boundless vets and curates highquality, openly licensed content from around the Internet. This particular resource used the following sources: