Introduction to Dirac's constant: Difference between revisions

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{{Afd-mergeto|#REDIRECT [[Planck constant|Introduction#Reduced to Dirac'sPlanck constant|02 September 2009|date=September 2009}}]]

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'''Dirac's constant''' is also known as the "reduced Planck's constant." It is a natural constant that is involved in many [[quantum mechanics|quantum theoretical]] equations.

[[Image:Gallery SineWave Generation.jpg|thumb|right|300px|Relation between a cycle and a wave; half of a circle describes half of the cycle of a wave]]

Planck's constant was, originally, the factor of proportionality linking the energy that a light wave carries, and its frequency. A step in the development of this concept appeared in Bohr's work. Bohr was using a "planetary" or particle model of the electron, and at first could not understand why the factor 2π kept turning up in his experimentally derived algebraic formulas.

Later, [[de Broglie]] hypothesized that electrons, like photons, have frequencies. Moreover, the frequency of an electron must conform to the conditions for a [[standing wave]] that can exist in a certain orbit. That is to say, the beginning of one cycle of a wave at some point on the circumference of a circle (since that is what an orbit is) must coincide with the end of some cycle. There can be no gap, no length along the circumference that is not participating in the vibration, and there can be no overlap of cycles. So the [[circumference]] of the orbit, ''C'', must equal the wavelength, λ, of the electron multiplied by some positive [[integer]] (''n'' = 1, 2, 3...). Knowing the circumference one can calculate wavelengths that fit that orbit, and knowing the [[radius]], ''r'', of the orbit, one can calculate its circumference. In algebra:

::<math>C = 2 \pi r = n \lambda \,\!</math>

Solving for λ:

::<math> \lambda = 2 \pi r/n \,\!</math>

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.])

Now let the length of an arc of a circle equal the length of a radius of the circle. Connect the endpoints of this arc to the center of the circle. Then the angle between these two radii equals 1 [[radian]]. Hence both circles and wave cycles consist of 2π radians. Since one cycle equals 2π radians, when ''h'' is divided by 2π, the two "2 π" factors will cancel out, leaving just a variable measured in radians to contend with. So dividing ''h'' by 2π yields a constant that when multiplied by the frequency of a wave, gives the energy of the wave in joules per radian. The reduced Planck's constant, called "h bar," is written as:

::<math> \hbar = \frac{h}{2 \pi} \ </math>.

The reduced Planck's constant allows computing the energy of a wave in units per radian instead of in units per cycle. The constants ''h'' and ħ merely serve to convert frequency units into energy units.

The reduced Planck's constant appears more often than ''h'' in the algebra of QM for many reasons, one of which is that angular velocity or angular frequency is ordinarily measured in radians per second, so using ħ eliminates converting radians into degrees or vice-versa. Also, when QM equations are written in terms of ħ, the frequent 2π factors in numerator and denominator often cancel. However, in other cases, as in the orbits of the Bohr atom, ''h''/2π arises naturally from the algebra of orbital [[angular momentum]].

The numerical value of ''h'' depends on the choice of units in which energy and wavelength are measured. If energy is measured in [[electron volt]]s (eV, a common practice in [[particle physics]]) and wavelength is measured in [[ångström]]s (10⁻¹⁰[[meter|m]]), then the energy of a photon is approximately ''E''_{<small>eV</small>} = 12400/λ_ångström. This form is easily remembered and avoids the small values of SI units.<ref>A. P. French and Edwin F. Taylor, ''An Introduction to Quantum Physics,'', p. 18.</ref>

==References==

{{Reflist}}

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[[Category:Quantum mechanics]]