# How Quantum is the Uncertainty Principle?}

## Second draft -- October 8.

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."
The uncertainty principle quickly attracted popular attention which has not slacked to this day. Its interpretation has been of concern not only in the sciences from biology to astrophysics but also in fields outside science. [2] Indeed this quote, also from OED, illustrates how widely the idea has taken hold: "A kind of Heisenberg effect which is far more serious than anything in the physical or biological sciences: the very act of observation distorts that which is being observed."

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(ωt). 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]

## References


[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, Classical Electrodynamics, 2nd Ed.
(Wiley, New York, 1962), p. 301.

[4] N. Bohr, Nature 121, 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] The uncertainty principle and foundations of quantum mechanics:
a fifty years' 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-two years of uncertainty: historical, philosophical, and
physical inquiries into the foundations of quantum mechanics,
Ed. Arthur I.  Miller (Plenum, New York, 1990).  The title indicates
any excuse suffices for intellectuals to twitch about quantum mechanics.
`