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Physics · Ch 14 — Dual Nature of Radiation and Matter

Wave-Particle Duality of Matter

14.7

Wave-Particle Duality of Matter

If every moving material object has an associated de Broglie wavelength, a natural question arises: why do we never notice a 'wavelength' for a running child, or a moving car, in everyday life? The answer lies entirely in the SIZE of the wavelength involved compared to the size of the object and its surroundings.

Example 14.5 makes this concrete: a 45 kg student running at 8 km/hr has a de Broglie wavelength of only about 6.63×10−366.63\times10^{-36} m, and a 1200 kg car moving at 60 km/hr has a wavelength of only about 3.32×10−383.32\times10^{-38} m -- both utterly negligible not just compared to the size of the objects themselves, but compared even to the width of the paths (a few metres) they are moving on. Such fantastically small wavelengths can never be measured by any instrument, and play no observable role whatsoever in the objects' motion -- which is exactly why ordinary Newtonian (classical) mechanics, with no reference to waves at all, works perfectly well for describing everyday macroscopic motion.

The picture changes completely, however, for MICROSCOPIC particles like electrons. Example 14.6 shows that a 100 eV electron has a de Broglie wavelength of about 1.23×10−101.23\times10^{-10} m = 1.228 Å -- a length scale comparable to the spacing between atoms in a solid, or to the size of a small aperture like a 2 Å hole. Whenever a particle's de Broglie wavelength becomes comparable to the size of an obstacle, slit, or confining structure it encounters, its wave nature becomes directly relevant to how it behaves -- exactly the situation exploited in the Davisson-Germer experiment, where electron wavelengths of a few tenths of a nanometre were comparable to the atomic spacing in a nickel crystal.

This is the general rule that emerges: for both electromagnetic radiation and material particles alike, PARTICLE nature dominates during their direct INTERACTION with matter (as in the photoelectric effect or a particle collision), while WAVE nature dominates during PROPAGATION through space, especially when the relevant confining dimensions (slit widths, atomic spacings, aperture sizes) are of the same order as the associated wavelength. Wave-particle duality, in other words, is not an either/or choice but a genuine dual character of both light and matter -- with which aspect is 'visible' in a given situation depending entirely on the physical scale of that situation. …

Misc Ex.14.5de Broglie wavelengths of a running student and a moving car -- both utterly negligible

Worked out. For a 45 kg student running at 8 km/hr (momentum p1=100p_1=100 kg m/s), the de Broglie wavelength λ1=h/p1=6.63×10−36\lambda_1=h/p_1=6.63\times10^{-36} m; for a 1200 kg car moving at 60 km/hr (momentum p2=20000p_2=20000 kg m/s), λ2=h/p2=3.32×10−38\lambda_2=h/p_2=3.32\times10^{-38} m. Both wavelengths are described as negligible compared not only to the size of the moving objects themselves but also to the widths of the paths (2 m footpath, 20 m road) they are moving along -- this pair of results is the worked-example proof of the section's opening question: why we never notice a 'wavelength' associated with everyday moving objects. Their momenta being enormous compared to a photon's or an electron's is exactly why their matter-wave wavelengths are correspondingly minu …

Misc Ex.14.6de Broglie wavelength of a 100 eV electron passing through a 2 Å hole -- now significant

Worked out. For an electron with kinetic energy 100 eV, its speed is found from 12mv2=100 eV\frac{1}{2}mv^2=100\text{ eV} to be about 5.93×1075.93\times10^{7} m/s, giving momentum p≈5.40×10−24p\approx5.40\times10^{-24} kg m/s and hence a de Broglie wavelength λ=h/p≈1.23×10−10\lambda=h/p\approx1.23\times10^{-10} m =1.228=1.228 Å. This wavelength is explicitly compared to the diameter of the circular hole the electron is passing through (2 Å) -- of the SAME order of magnitude, in fact comparable to the size of a helium atom and more than double that of a hydrogen atom -- making the electron's wave nature directly relevant to how it interacts with the aperture, in sharp contrast to Example 14.5's macroscopic objects. This is the contrast the whole section is built to establish: matter's wave nature only becomes apparent when the associated wavelength is comparable …

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