Abstract. Much of the argument for the quantum-mechanical uncertainty principle can be cast in terms of classical physics. For any classical variable, there is a limit to how localized it can be in both time and frequency. The quantum connection between the energy of the photon and its frequency then yields the uncertainty principle.

The best known qualitative feature of quantum physics is the uncertainty principle. For example, the Oxford English Dictionary [1] provides an amazingly detailed definition of Heisenberg's uncertainty principle:

"certain pairs of observables (e.g., ... the energy and lifetime of a quantum level) cannot both be precisely and simultaneously known ... The principle is usually stated as an inequality such that the product of the uncertainties of the pair of observables cannot be less than a quantity of the order of Planck's constant."

**
It often hard to understand why any particular idea seizes the
attention of intellectuals.** While quantum mechanics did seem a
profound break with classical physics, in the case of the uncertainty
principle most of the argument can be done classically. Indeed the
only quantum mechanics needed is the connection, pointed out in 1900 by
Planck, that the energy of light is quantized -- that is, the energy of
a photon *E* is connected to its angular frequency *ω* by the
small constant ℏ (the Planck constant ℎ divided by 2π):
*E*
= ℏω

**
The classical part of the argument [3] is rooted in the question: is it
possible classically to localize a variable, such as an electric field,
in both frequency and time?** To start, consider an electric field
proportional to cos(ω*t*).
While this field is localized in
frequency space (namely, at ω) it is unlimited in time. To
produce a field more localized in time, it is necessary to give up some
of the localization in frequency. Consider adding two fields with
frequencies on both sides of ω, specifically, fields with half
the strength of central field with frequencies: ω - δω
and ω + δω. The figure shows the resulting electric
field both in terms of its component frequencies and as a function of
time.

**
;rom the two parts of the figure it is possible to extract the
localization in frequency and in time.** The left hand part of figure
shows that the delocalization in frequency is of order δω.
The right hand part of the figure show the the oscillation at angular
frequency ω is modulated, having zeros at &delta *t* = +
π/(δω). In other words the localization in time is
of order π/(δω). Combining these two results gives
an estimate of the classical uncertainty δω~δ *
t* =π.
Improving this simple model [3] yields a lower limit on the best that can be
achieved. ω δ *t* ≤ ½.
This classical form of the uncertainty principle
sets the limit on how localized any physical classical variable can
`simultaneously' be in both time and angular frequency.

The step to the quantum uncertainty principle is now straight forward.
As foreshadowed in the second paragraph, it is only necessary to invoke
*E*
= ℏ ω to get the quantum uncertainty principle δ
*E* δ *t* ≥ (ℏ/2). This argument was not unknown in late
1920's, [4] but it seems to have played no major role in the discussions
that have continued to this day. [5,6]

[1] Oxford English Dictionary (Oxford University Press, 1989). In that definition were several quotations showing usage, the weirdest of which was from the 1977 March 14th New York Times: "Even in the age of the Uncertainty Principle and culture fracture, Warren has not lost his sense of life as a sustained drama." [2] For example, in philosophy it prompted a reconsideration of the principle of determinism -- the theory that all events, including moral choices, are completely determined. [3] J. D. Jackson,ClassicalElectrodynamics, 2nd Ed. (Wiley, New York, 1962), p. 301. [4] N. Bohr,Nature121, 580 (1928). In his article reviewing the central features of quantum mechanics for a general audience, Bohr cites the theory of optical instruments as setting the localization of classical field "under the most favorable conditions" as $\delta\omega~\delta t = 2\pi.$ Clearly he and his physics contemporaries were unaware the lower limit was an order of magnitude smaller (namely, $1/2$). Then no one contemplated an experiment to look at this limit, now we do and can. [5]Theuncertaintyprincipleandfoundationsofquantummechanics:afiftyyears'survey, Eds. William C. Price and Seymour S. Chissick (Wiley, New York, 1977). This essay collection included a reminiscence of Heisenberg written shortly before his death in 1976. [6]Sixty-twoyearsofuncertainty:historical,philosophical,andphysicalinquiriesintothefoundationsofquantummechanics, Ed. Arthur I. Miller (Plenum, New York, 1990). The title indicates any excuse suffices for intellectuals to twitch about quantum mechanics.

Your comments and suggestions are appreciated.

Originally posted by wilkins@mps.ohio-state.edu [September 1997]