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Physics Special Topics

P5_7 The Wavelength of a Cow

Hannah Clear, Daniel Evan, George McGilvray, Edward Raymond Turner

Published Dec 13, 2018• 0 citations
OpenAlex

Abstract

In this paper, we explore the de Broglie wavelength of an average cow, and investigate if it is possible for a cow to also act as a wave. It is perfectly viable to calculate the de Broglie wavelength of a macroscopic object, such as in this case a cow, if we consider each atom of the cow to be coherent. The de Broglie wavelength of an average cow was found to be 1.3 × 10^−37 m.

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In this paper, we explore the de Broglie wavelength of an average cow, and investigate if it is possible for a cow to also act as a wave. It is perfectly viable to calculate the de Broglie wavelength of a macroscopic object, such as in this case a cow, if we consider each atom of the cow to be coherent. The de Broglie wavelength of an average cow was found to be 1.3 × 10^−37 m.
In quantum optics an ensemble of photons is treated as a Bose condensate which has a de Broglie wavelength given by ${\ensuremath{\lambda}}_{0}/N$ where ${\ensuremath{\lambda}}_{0}$ and NNN are the wavelength and average number of constituent photons, respectively. We describe an interferometer which is capable of measuring this de Broglie wavelength. We show specifically that a coherent state input has a de Broglie coherence length proportional to 1/N1/\sqrt{N}1/N​. Finally we show that the interferometer is Heisenberg limited in sensitivity ( $\ensuremath{\Delta}{\ensuremath{\varphi}}_{\mathrm{min}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1/N$) and thus forms a new class of interferometers operating at the fundamental quantum limit.
The undergraduate courses on modern physics generally consider the particle interpretation of Compton effect. Motivated by a student's solution in an undergraduate examination on modern physics here we consider the wave characteristics of recoiled electrons in the Compton effect. The de Broglie wavelength, wave and clock frequency of the scattered electrons are expressed in terms of the wavelength and the frequency of the incident and the scattered photons respectively using the familiar particle interpretation of the Compton effect, where initially the electron is at rest and its spin is ignored. Both non-relativistic and relativistic cases are considered. Numerical values of de Broglie wavelength, wave and clock frequency of the scattered electron are calculated for an incident photon energy that was used in the original experiment of Compton as a function of the scattering angle of the recoiled electron. Considering the relativistic effects which are however insignificant for the de Broglie wavelength of the recoiled electron under these conditions, the minimum value obtained is in the range of X-rays. The non-relativistic de Broglie wave frequency obtained by neglecting the rest mass of the electron leads to an underestimation of its value. The implications of de Broglie wavelength and clock frequency for Compton scattering experiments are briefly discussed and possible extensions of the obtained mathematical formulations are indicated. The results are useful for understanding the wave-particle duality of the recoiled electron in the context of the Compton effect.
In modern physics, the de Broglie wavelength is considered to be the matter wave. However, the de Broglie wave has a series of strange properties. It is not mathematically defined for a rest-mass particle, when v=0. However, one can claim a particle never stands still and that the de Broglie wavelength only converges towards infinite when v converges to zero. An infinite matter wavelength would also be strange. We have good reasons to think that the de Broglie wavelength only is a mathematical derivative of the true matter wavelength, which we believe is the Compton wavelength. Although noted briefly here, this has already been described by Haug \cite{Hau20UnifiedA} and is a topic for another article. What we will focus on here is that the length of the de Broglie wavelength, if we use an observational window of one second and the minimum observable velocity, is the Planck length per second. Then the de Broglie wavelength for a proton actually has a length very close to the assumed radius of the observable universe. We think most likely this is a coincidence, particularly since one second is an arbitrarily chosen time unit, and not a fundamental time unit such as the Compton time, or the Planck time, for example. Still, we think this finding is worth mentioning and could be the basis for further discussion.
Macroscopic objects do not exhibit quantum interference, even though the de Broglie relation applies universally. A common argument is that their momentum is too large for observable wavelengths. However, this does not explain why an object at rest does not display an infinite wavelength. This work develops a thermal-ensemble-based explanation: the internal degrees of freedom of any object at finite temperature possess a Maxwell--Boltzmann distribution of momenta, each associated with its own de Broglie wavelength. By integrating over this thermal momentum distribution, we show that the internal wave contributions cancel due to isotropy and rapid phase oscillations. This yields an effectively vanishing internal de Broglie wavelength and provides an intuitive mechanism for intrinsic decoherence in macroscopic matter.
A fourth-order Young interference experiment was done to demonstrate a practical way to measure the de Broglie wavelength of a two-photon wave packet. A two-photon collinear beam is generated by type-II spontaneous parametric down-conversion. By modifying the transverse field profile of the pump laser beam that generates the two-photon beam we demonstrate that it is possible to measure the de Broglie wavelength of the single-photon constituents of the two-photon wave packet, the de Broglie wavelength of the two-photon wave packet as a whole and an ill defined intermediate de Broglie wavelength between the two cases.