Hydrogen atom


A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral hydrogen atom contains a single positively charged proton in the nucleus, and a single negatively charged electron bound to the nucleus by the Coulomb force. Atomic hydrogen constitutes about 75% of the baryonic mass of the universe.
In everyday life on Earth, isolated hydrogen atoms are extremely rare. Instead, a hydrogen atom tends to combine with other atoms in compounds, or with another hydrogen atom to form ordinary hydrogen gas, H2. "Atomic hydrogen" and "hydrogen atom" in ordinary English use have overlapping, yet distinct, meanings. For example, a water molecule contains two hydrogen atoms, but does not contain atomic hydrogen.
Atomic spectroscopy shows that there is a discrete infinite set of states in which a hydrogen atom can exist, contrary to the predictions of classical physics. Attempts to develop a theoretical understanding of the states of the hydrogen atom have been important to the history of quantum mechanics, since all other atoms can be roughly understood by knowing in detail about this simplest atomic structure.

Isotopes

The most abundant isotope, protium, or light hydrogen, contains no neutrons and is simply a proton and an electron. Protium is stable and makes up 99.985% of naturally occurring hydrogen atoms.
Deuterium contains one neutron and one proton in its nucleus. Deuterium is stable, makes up 0.0156% of naturally occurring hydrogen, and is used in industrial processes like nuclear reactors and nuclear magnetic resonance spectroscopy.
Tritium contains two neutrons and one proton in its nucleus and is not stable, decaying with a half-life of 12.32 years. Because of its short half-life, tritium does not exist in nature except in trace amounts.
Heavier isotopes of hydrogen are only created artificially in particle accelerators and have half-lives on the order of 10−22 seconds. They are unbound resonances located beyond the neutron drip line; this results in prompt emission of a neutron.
The formulas below are valid for all three isotopes of hydrogen, but slightly different values of the Rydberg constant must be used for each hydrogen isotope.

Hydrogen ion

Lone neutral hydrogen atoms are rare under normal conditions. However, neutral hydrogen is common when it is covalently bound to another atom, and hydrogen atoms can also exist in cationic and anionic forms.
If a neutral hydrogen atom loses its electron, it becomes a cation. The resulting ion, which consists solely of a proton for the usual isotope, is written as "H+" and sometimes called hydron. Free protons are common in the interstellar medium, and solar wind. In the context of aqueous solutions of classical Brønsted–Lowry acids, such as hydrochloric acid, it is actually hydronium, H3O+, that is meant. Instead of a literal ionized single hydrogen atom being formed, the acid transfers the hydrogen to H2O, forming H3O+.
If instead a hydrogen atom gains a second electron, it becomes an anion. The hydrogen anion is written as "H–" and called hydride.

Theoretical analysis

The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple two-body problem physical system which has yielded many simple analytical solutions in closed-form.

Failed classical description

by Ernest Rutherford in 1909 showed the structure of the atom to be a dense, positive nucleus with a tenuous negative charge cloud around it. This immediately raised questions about how such a system could be stable. Classical electromagnetism had shown that any accelerating charge radiates energy, as shown by the Larmor formula. If the electron is assumed to orbit in a perfect circle and radiates energy continuously, the electron would rapidly spiral into the nucleus with a fall time of:
where is the Bohr radius and is the classical electron radius. If this were true, all atoms would instantly collapse. However, atoms seem to be stable. Furthermore, the spiral inward would release a smear of electromagnetic frequencies as the orbit got smaller. Instead, atoms were observed to emit only discrete frequencies of radiation. The resolution would lie in the development of quantum mechanics.

Bohr–Sommerfeld model

In 1913, Niels Bohr obtained the energy levels and spectral frequencies of the hydrogen atom after making a number of simple assumptions in order to correct the failed classical model. The assumptions included:
  1. Electrons can only be in certain, discrete circular orbits or stationary states, thereby having a discrete set of possible radii and energies.
  2. Electrons do not emit radiation while in one of these stationary states.
  3. An electron can gain or lose energy by jumping from one discrete orbit to another.
Bohr supposed that the electron's angular momentum is quantized with possible values:
where
and is Planck constant over. He also supposed that the centripetal force which keeps the electron in its orbit is provided by the Coulomb force, and that energy is conserved. Bohr derived the energy of each orbit of the hydrogen atom to be:
where is the electron mass, is the electron charge, is the vacuum permittivity, and is the quantum number. Bohr's predictions matched experiments measuring the hydrogen spectral series to the first order, giving more confidence to a theory that used quantized values.
For, the value
is called the Rydberg unit of energy. It is related to the Rydberg constant of atomic physics by
The exact value of the Rydberg constant assumes that the nucleus is infinitely massive with respect to the electron. For hydrogen-1, hydrogen-2, and hydrogen-3 which have finite mass, the constant must be slightly modified to use the reduced mass of the system, rather than simply the mass of the electron. This includes the kinetic energy of the nucleus in the problem, because the total kinetic energy is equivalent to the kinetic energy of the reduced mass moving with a velocity equal to the electron velocity relative to the nucleus. However, since the nucleus is much heavier than the electron, the electron mass and reduced mass are nearly the same. The Rydberg constant RM for a hydrogen atom is given by:
where is the mass of the atomic nucleus. For hydrogen-1, the quantity is about 1/1836. For deuterium and tritium, the ratios are about 1/3670 and 1/5497 respectively. These figures, when added to 1 in the denominator, represent very small corrections in the value of R, and thus only small corrections to all energy levels in corresponding hydrogen isotopes.
There were still problems with Bohr's model:
  1. it failed to predict other spectral details such as fine structure and hyperfine structure
  2. it could only predict energy levels with any accuracy for single–electron atoms
  3. the predicted values were only correct to, where is the fine-structure constant.
Most of these shortcomings were resolved by Arnold Sommerfeld's modification of the Bohr model. Sommerfeld introduced two additional degrees of freedom, allowing an electron to move on an elliptical orbit characterized by its eccentricity and declination with respect to a chosen axis. This introduced two additional quantum numbers, which correspond to the orbital angular momentum and its projection on the chosen axis. Thus the correct multiplicity of states was found. Further, by applying special relativity to the elliptic orbits, Sommerfeld succeeded in deriving the correct expression for the fine structure of hydrogen spectra. However, some observed phenomena, such as the anomalous Zeeman effect, remained unexplained. These issues were resolved with the full development of quantum mechanics and the Dirac equation. It is often alleged that the Schrödinger equation is superior to the Bohr–Sommerfeld theory in describing hydrogen atom. This is not the case, as most of the results of both approaches coincide or are very close, and in both theories the main shortcomings result from the absence of the electron spin. It was the complete failure of the Bohr–Sommerfeld theory to explain many-electron systems which demonstrated its inadequacy in describing quantum phenomena.

Schrödinger equation

The Schrödinger equation is the standard quantum-mechanics model; it allows one to calculate the stationary states and also the time evolution of quantum systems. Exact analytical answers are available for the nonrelativistic hydrogen atom. Before we go to present a formal account, here we give an elementary overview.
Given that the hydrogen atom contains a nucleus and an electron, quantum mechanics allows one to predict the probability of finding the electron at any given radial distance. It is given by the square of a mathematical function known as the "wavefunction", which is a solution of the Schrödinger equation. The lowest energy equilibrium state of the hydrogen atom is known as the ground state. The ground state wave function is known as the wavefunction. It is written as:
Here, is the numerical value of the Bohr radius. The probability density of finding the electron at a distance in any radial direction is the squared value of the wavefunction:
The wavefunction is spherically symmetric, and the surface area of a shell at distance is, so the total probability of the electron being in a shell at a distance and thickness is
It turns out that this is a maximum at. That is, the Bohr picture of an electron orbiting the nucleus at radius corresponds to the most probable radius. Actually, there is a finite probability that the electron may be found at any radius, with the
probability indicated by the square of the wavefunction. Since the probability of finding the electron somewhere in the whole volume is unity, the integral of is unity. Then we say that the wavefunction is properly normalized.
As discussed below, the ground state is also indicated by the quantum numbers. The second lowest energy states, just above the ground state, are given by the quantum numbers,, and. These states all have the same energy and are known as the and states. There is one state:
and there are three states:
An electron in the or state is most likely to be found in the second Bohr orbit with energy given by the Bohr formula.