Introduction to Dirac's constant: Difference between revisions

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This formula is expressed in terms of the radius ''r'', when what is relevant to determining allowed frequencies and wavelengths is the circumference ''C''. Hence 2π recurs in QM because it is the factor of proportionality linking the radius of any circle to its circumference.<ref>J. P. McEvoy and Oscar Zarate, ''Introducing Quantum Theory'', pp. 114, 118.</ref>

In [[1925]], when [[Werner Heisenberg]] was developing his wave formulation of full quantum theory, calculations involving [[Fourier series]] were fundamental, and the factor 2π is ubiquitous in the algebra of Fourier series. Adopting the "reduced" version of Planck's constant (''h''/2π) eliminated most appearances of 2π from wave analysis algebra. A few years later, the reduced Planck's constant appeared naturally in Dirac's equation, and for this reason it was named "Dirac's constant." We now say more about this constant, even though the theories for which it was more convenient than Planck's constant have yet to be discussed.

As noted above, the energy of any wave is given by its frequency multiplied by Planck's constant. A wave is made up of peaks and troughs. A complete cycle for a wave is the time required for a wave to return to some chosen initial position. For example, starting from a peak, a wave is complete when it has its next peak. A cycle is mathematically related to a circle, and both have 360 degrees. A degree is a unit of measure for the amount of turn needed to produce an [[arc (geometry)|arc]] equal to 1/360 of the circumference. A point on the circumference of a circle traces out a [[sine curve]] as the circle rotates. (For a demonstration, see [http://www.math.utah.edu/~cherk/ccli/bob/Rotation/sin12.swf Rotation Applet.])