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American Journal of Physics

An uncertainty demonstration with electromagnetic waves

W. M. Nunn, M. Figueroa

Published Mar 1, 1983• 0 citations
Publisher

Abstract

An experiment is described in which a pulse-modulated electromagnetic signal is simultaneously displayed in the time domain by means of an oscilloscope and in the frequency domain with a spectrum analyzer. The inverse relation between pulse length and spectral-energy density can readily be observed, and an interpretation of the experiment as a demonstration of an uncertainty principle is presented.

DOI
10.1119/1.13301

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An experiment is described in which a pulse-modulated electromagnetic signal is simultaneously displayed in the time domain by means of an oscilloscope and in the frequency domain with a spectrum analyzer. The inverse relation between pulse length and spectral-energy density can readily be observed, and an interpretation of the experiment as a demonstration of an uncertainty principle is presented.
It is considered constraints imposed by the quantum mechanics on the measurement of the density of the electromagnetic energy. First, the energy of the electromagnetic wave and the volume (time) are bound with the Heisenberg uncertainty principle. It reduces from double to ordinary the relativistic effect for the density of the electromagnetic energy. Second, the frequency of photons and the number of photons are bound with the Heisenberg uncertainty principle. Then relativistic effects for the frequency of photons and for the number of photons are incompatible.
Abstract Two complementary properties, such as frequency and duration of a pulse, cannot be precisely measured simultaneously. The lower bound on the time-bandwidth product, known also as uncertainty relation, plays a very important role in quantum theory. In this work, we consider single-photon pulses with arbitrary temporal and spectral profiles, and derive a lower bound for the single-photon time-bandwidth product. We investigate how this bound depends on the modal purity, the single-mode fidelity, and the effective number of modes. Our results can be used to solve the inverse problem, namely, to estimate characteristics of the single-photon pulse based on the time-bandwidth product obtained from temporal and spectral measurements of the pulse.
We confirmed that the uncertainty relation between time and energy is established in fluctuations in the space and time domains that appear when coherent optical pulses propagate through a disordered static medium. The widths of temporal and spectral correlation functions of these fluctuations satisfy the relation $\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}\ensuremath{\omega}\ensuremath{\sim}1$, that is, $\ensuremath{\delta}\ensuremath{\tau}\ensuremath{\delta}E\ensuremath{\sim}\ensuremath{\hbar}$. The number of fluctuations is discussed on the basis of the uncertainty relation.
The problem of measuring the energy of a quantum system, the process of emission of an electromagnetic field quantum, and their connection with the provisions of the Heisenberg uncertainty principle are considered. It is shown that the processes of measuring the energy of a quantum system and the time of transition between levels are performed not only simultaneously, but in a single act of measurement, and cannot be fulfilled in another way. This contradicts the physical ideas embedded in the uncertainty principle. As an alternative, the concept of "determinacy relations" is proposed.
We report an observation of a change in the energy spectrum of a matter wave due to fast modulation in time. The experiment, carried out with slow neutrons, achieved pulse widths of 33 n, and has been analyzed using time-dependent optics based on the nonrelativistic Schr\"odinger equation. A good fit to the data is obtained. The modulation of a matter wave in time leads to a characteristic energy spread. These phenomena can be seen as a consequence of the time-energy uncertainty relation. In our case, the amplitude modulation is done by a rapid periodic opening and closing of a slit. In the time domain this causes a so-called ``diffraction in time.'' In this work we demonstrate the operation of a time-of-flight spectrometer (FOTOF), and achieve an energy resolution of 20 neV, which to our knowledge represents a new limit for a TOF with slow neutrons.