Quantum Mechanical Description of Atomic Orbitals

Introduction

For years the nature of the atom was poorly understood. Through spectroscopy one could see the subtle patterns of light excited atoms emitted, but a precise explanation for their being eluded scientists. Niels Bohr was the first to suggest something akin to our modern theory of quantum mechanics. In Bohr's model, an atom's electrons circled the nucleus, much like planets orbit the sun, but were restricted to only a few specific energies. The transitions between these restricted orbits accounted for the emission and absorption lines of elements. However, this quantization was not justified, and though empirically sound, proved unsatisfactory. It was not until Erwin Schrodinger introduced his wave equation that the quantized nature of the atom was fully understood.

http://commons.wikimedia.org/wiki/File:Niels_Bohr_1935.jpg

http://en.Wikipedia.org/wiki/File:Visible_spectrum_of_hydrogen.jpg

The visible spectrum of hydrogen, with its characteristic distinct lines. Thicker lines indicate a greater number of available transitions - the hyper-fine structure.

Development of Quantum Theory

en.Wikipedia.org/wiki/File:Er...chrödinger.jpg

Although Schrodinger's equation is inherently non-relativistic and many improvements have been made to his original theory, it still is a powerful predictive and pedagogical tool. According to quantum mechanical theory, all the measurable properties of a particle can be calculated from its wave function, whose evolution is dictated by the equation

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In this new theory, probabilistic solutions supersede the determinism of classical mechanics. The solutions to this equation can only be used to determine a probability distribution for the measurables of a particle, position, momentum, etc. Hence unlike planetary orbits, electrons bound by a nucleus take on a much fuzzier, cloud-like existence, first called orbitals by Mulliken. Even the closest and most tightly bound electron in a hydrogen atom may be, however unlikely, thousands of miles away. This implausible, but very real physical result highlights the probabilistic nature of the solutions.

http://en.Wikipedia.org/wiki/File:HAtomOrbitals.png Creative CommonsAttribution ShareAlike 3.0

Atomic orbitals are described by three parameters n, l, and m, called quantum numbers. These correspond to the state of the electron in question. Additionally, the Dirac equation, a revised version of Schrodinger's equation which incorporates relativity, adds a fourth quantum number, s, the spin quantum number.They obey the following relationships:

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Together, these 4 numbers describe all possible states of an electron bound to an atom, and the Pauli exclusion principle also dictates that no two electrons share the same quantum numbers. The addition of the spin quantum number explained anomalies such as the Zeeman effect and the hyperfine splitting of hydrogen observed by astronomers.

Below is a graphical representation of the wave functions of a bound electron with different quantum numbers. The color indicates phase.

http://en.Wikipedia.org/wiki/File:Neon_orbitals.JPG

Although much of the literature concerns itself with the hydrogen atom - being the simplest and mathematically tractable case - the physical results can be extended to higher Z atoms and even complex molecules. Perhaps one of the greatest triumphs of quantum mechanics is that quantization arises naturally in the mathematical solution of the equations, rather than an artificial constraint, the elegance of which is difficult to ignore.

External Links

• Orbital Viewer http://www.orbitals.com/orb/index.html
• Wikipedia article on Atomic Orbital http://en.Wikipedia.org/wiki/Atomic_orbital
• A very informative chart of hydrogen-like orbitals en.Wikipedia.org/wiki/Atomic_orbital#Orbitals_table

References

1. Levine, Ira (2000). Quantum Chemistry (5 ed.). Prentice Hall. ISBN 0-13-685512-1.
2. Nagaoka, Hantaro (May 1904). "Kinetics of a System of Particles illustrating the Line and the Band Spectrum and the Phenomena of Radioactivity". Philosophical Magazine 7: 445–455.
3. Bohr, Niels (1913). "On the Constitution of Atoms and Molecules". Philosophical Magazine 26 (1): 476.
4. Mulliken, Robert S. (July 1932). "Electronic Structures of Polyatomic Molecules and Valence. II. General Considerations". Phys. Rev.41 (1): 49–71.

Problems

1. What were some problems with the Bohr model?

2. Briefly describe the orbitals in relation to Schrodinger's equation.

3. Do you think quantization is represented in Schrodinger's equation? If so, describe how. (You do not need to delve into mathematics.)

4. How is quantum theory successful in explaining the observed spectrum of hydrogen?

5. What is the hyper-fine structure?

Answers

1. One particularly troublesome aspect of the Bohr model is that if electrons physically orbit nuclei, then Maxwell's well-established theory of electromagnetism predicts the electrons would radiate their energy away, eventually spiraling into the nucleus. A makeshift solution is to require that only some select state transitions be allowed, but this rather adhoc assumption is naturally explained by quantum theory.

2. Orbitals are solutions to Schrodinger's equation with different quantum numbers. Each corresponds to a possible electron state. Unlike classical trajectories, orbitals are distribution functions which may be used to calculate probabilities.

3. The equation itself displays no obvious sign of quantization, however once a particular potential field is introduced, the eigenfunctions that solve the equation are naturally quantized. For an illustrative example see en.Wikipedia.org/wiki/Particle_in_a_box .

4.The solutions for electron orbitals quantize the allowed energies, the transitions of which correspond to the lines we see in the hydrogen spectrum.

5. In the same shell there are many available electron configurations with different energies. The different transitions can cause the blurring and splitting of spectral lines. See Zeeman effect.