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Quantum Mechanics: Hydrogen Atom and Electron Spin

By Dragica Vasileska1, Gerhard Klimeck2

1. Arizona State University 2. Purdue University

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Abstract

A hydrogen atom is an atom of the chemical element hydrogen. The electrically neutral atom contains a single positively-charged proton and a single negatively-charged electron bound to the nucleus by the Coulomb force. The most abundant isotope, hydrogen-1, protium, or light hydrogen, contains no neutrons; other isotopes contain one or more neutrons. This article primarily concerns hydrogen-1.

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.

In 1913, 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 quantum-mechanical analysis which uses the Schrödinger equation, as was shown in 1925/26. 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 however, because it also yields the shape of the electron's wave function ("orbital") for the various possible quantum-mechanical states, thus explaining the anisotropic character of atomic bonds.

The Schrödinger equation also applies to more complicated atoms and molecules. However, in most such cases the solution is not analytical and either computer calculations are necessary or simplifying assumptions must be made. Solution of the Schrodinger equation for the hydrogen atom is provided below:

  • Slides on the Solution of the Schrodinger equation for Hydrogen atom
  • In physics and chemistry, spin refers to a non-classical kind of angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. A particle's spin is essentially the direction a particle turns along a given axis, which in turn can be used to determine the particle's magneticism.[1] Although this special property is only explained in the relativistic quantum mechanics of Paul Dirac, it plays a most-important role already in non-relativistic quantum mechanics, e.g., it essentially determines the structure of atoms.

    In classical mechanics, any spin angular momentum of a body is associated with self rotation, e.g., the rotation of the body around its own center of mass. For example, the spin of the Earth is associated with its daily rotation about the polar axis. On the other hand, the orbital angular momentum of the Earth is associated with its annual motion around the Sun.

    In fact, in classical theories there is no analogue to the quantum mechanical property meant by the name spin. The concept of this nonclassical property of elementary particles was first proposed in 1925 by Ralph Kronig, George Uhlenbeck, and Samuel Goudsmit; but the name related to the phenomenon of spin in physics is Wolfgang Pauli.

    Applications of Spin in nanoelectronics are given in the presentation slides below:

  • The story of the two spins
  • Cite this work

    Researchers should cite this work as follows:

    • Dragica Vasileska; Gerhard Klimeck (2008), "Quantum Mechanics: Hydrogen Atom and Electron Spin," http://nanohub.org/resources/4995.

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    In This Series

    1. Quantum Mechanics: Hydrogen Atom

      09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

      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...

    2. Quantum Mechanics: The story of the electron spin

      09 Jul 2008 | Teaching Materials | Contributor(s): Dragica Vasileska, Gerhard Klimeck

      One of the most remarkable discoveries associated with quantum physics is the fact that elementary particles can possess non-zero spin. Elementary particles are particles that cannot be divided into any smaller units, such as the photon, the electron, and the various quarks. Theoretical and...

    nanoHUB.org, a resource for nanoscience and nanotechnology, is supported by the National Science Foundation and other funding agencies. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.