**What it shows:**

A pulse-modulated electromagnetic signal is simultaneously displayed in the time domain (on an oscilloscope) and in the frequency domain (on a spectrum analyzer). Using ∆n for the frequency spread (uncertainty in frequency) and ∆t for the duration of the pulse (uncertainty in the time domain), the frequency-time uncertainty relation is given by ^{1}

∆n ∆t ≥ ^{1}/_{4π}

By progressively shortening the length of time that the carrier signal is on, the inverse relation between pulse length and spectral-energy density is made evident, i.e.

∆E ∆t ≥ ^{h}/_{4π}

if one identifies the electromagnetic frequency, n, with photon energy, E, via the Planck-Einstein relation E=hν, where h is Planck's constant.

**How it works:**

A function generator ^{2} serves as the source of electromagnetic signal -- a sinusoidal wave which can either be continuously on, or pulsed on for an adjustable length of time. To demonstrate the reciprocity relation, the signal is observed (measured) simultaneously in the two complementary domains: on an oscilloscope for the time domain and on a frequency spectrum analyzer for the frequency (energy) domain.

Typically one starts with a continuous wave (CW) in which case there are no surprises -- a continuous wave is displayed on the oscilloscope and the frequency display shows a single spike corresponding to the frequency of the signal (say, 35 kHz). The significance of this result is that one needs to have the signal continuously on to achieve little or no uncertainty (within the limits of the instrument's resolution) in determining its frequency. The analog of this situation on the atomic scale is that, in order to have a well-defined energy, a physical state must last a long time -- we cannot precisely know a particle's energy and the exact time it has that energy. The fun begins when the signal is no longer continuously on and we try to minimize the uncertainty in the time domain.

The function generator allows one to pulse the sinusoidal signal "on" in the BURST mode -- the burst length being continuously adjustable down to about 30 µsec, which corresponds to a single oscillation of the wave (at 35 kHz). To see a continuous display of this burst on the oscilloscope, it is necessary to repeatedly burst (at a repetition rate somewhere between 10 and 20 Hz) and use the appropriate time base and synchronization on the scope to see one burst on the screen. The spectrum analyzer is an analog device that literally scans through the selected frequency range with band-pass filters and displays the resultant signal strength vs. frequency on a storage CRT. Modern microprocessor-based instruments store the signal digitally and display the calculated Fourier transform. The end result is, of course, the same. As a reminder to the reader, let us first consider the fourier transform of the pulsed sinusoidal wave shown below.

The relevant parameters are (a) signal frequency = ν_{s}, (b) burst length = τ (signal "on" time), c) burst repetition rate = ν_{r}, and (d) the burst repetition period = T = 1/ν_{r}. The Fourier transform of this signal is illustrated below. ^{3}

The experimental results differ from the above in two significant features. First, the polarity of all spectral components is positive which is simply a consequence of the electronic detection scheme. Secondly, the spectrum analyzer's resolution is usually adjusted to 30 Hz or more so that it doesn't take too long to analyze and display the frequency spectrum. ^{4} Consequently the individual spectral components spaced at integral multiples of the burst repetition frequency (10-20 Hz) do not show up -- only the overall envelope of these "spectral lines" is displayed. However, the virtue of this limitation is that the time domain display features only *one* burst of the signal and so the fine structure of the frequency spectrum (the details of which depend on something not obvious to the audience) would only add unnecessary confusing details. ^{5} With only the overall shape (envelope) of the frequency spectrum displayed, the salient features become immediately obvious: as the uncertainty is reduced in determining the "on" time of the signal (reducing τ), the frequency spectrum, or spectral-energy density, is seen to spread out dramatically with accompanying proportional decrease in intensity (the area under the curve represents the total energy). Three representative frequency spectra are illustrated below with pulse durations of 10, 5, and 1 msec.

**Setting it up:**

Many combinations of function generators and spectrum analyzers will do the job with the right selection of frequency. Less sophisticated function generators (without burst mode) might need to be "doubled up," i.e. the pulse output of one is used to gate the other to produce the pulse-modulated signal. All the instrumentation can fit on one cart. Two video cameras will be needed with whatever projection scheme that seems most appropriate. The following setup is suggested for getting you started quickly.

function generator |
spectrum analyzer |

function: 35 kHz SINE wave | resolution bandwidth: 30 Hz |

mode: BURST at 10 - 20 Hz | sweep mode: single |

burst width: 20 ms to 30 µs | freq span/div: 2 kHz/div (± 10 kHz) |

frequncy: center (adj. for 35 kHz) | |

amplitude mode: linear |

**Comments:**

Heisenberg derived his law on quite general and abstract grounds. However, its significance becomes apparent only when one shows how it enters into any specific measuring process. The law works differently in every imaginable measuring process and its application is remarkably general. This demonstration clearly shows its presence when performing measurements on electromagnetic waves.

The uncanny similarity between the shape of the frequency spectrum and the intensity distribution of the single-slit diffraction pattern is remarkable. The similarity is not entirely an accident since the spreading out of the diffraction pattern (while narrowing the slit-width) is, of course, another example of the ubiquitous uncertainty principle rearing its head and is described in the Optical Analog of Uncertainty Principle demo.

As Nunn and Figueroa point out, the ideas and techniques in this demonstration have many practical applications beyond that of demonstrating uncertainty. In communication theory discussions, it can be used to illustrate the origin of upper and lower sidebands in radio transmission, or to show how high data rates (with accompanying narrow pulse widths and high pulse repetition rates) require very wide communication bandwidths, to give but two examples.

1 J.D. Jackson, *Classical Electrodynamics*, 2nd ed (Wiley, NY, 1975), pp 299-306.

2 20 MHz Pulse/Function Generator (Wavetek model 191)

3 W.M. Nunn, Jr. and M. Figueroa, Am J Phys **51**, 239-245 (1983). "An uncertainty demonstration with electromagnetic waves" Our present demonstration is an adaptation of the microwave experiment described by the authors in this paper. They present the underlying mathematics as well as many additional references.

4 This in itself demonstrates the limitations of the instrument as dictated by the uncertainty principle!

5 The analyzer *can* be adjusted so that all the details of the fourier spectrum are displayed, but this would only be appropriate for an advanced and mathematically sophisticated audience. For such an audience, the demonstration can be made as quantitative as the lecturer desires (or has time for).