2.4: Beyond Bohr

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Eventually, as they considered the problems with the Bohr model, scientists came back to the idea of the wave–particle duality as exemplified by the photon. If light (electromagnetic radiation), which was classically considered to be a wave, could have the properties of a particle, then perhaps matter, classically considered as composed of particles, could have the properties of waves, at least under conditions such as those that exist within an atom. Louis De Broglie (1892–1987) considered this totally counterintuitive idea in his Ph.D. thesis. De Broglie used Planck’s relationship between energy and frequency (E = hν), the relationship between frequency and wavelength (c = λν), and Einstein’s relationship between energy and mass (E = mc²) to derive a relationship between the mass and wavelength for any particle (including photons).⁴³ You can do this yourself by substituting into these equations, to come up with λ = h/mv, where mv is the momentum of a particle with mass m and velocity v. In the case of photons, v = c, the velocity of light.

Although the math involved in deriving the relationship between momentum (mv) of a particle and its wavelength λ is simple, the ideas behind it are most certainly not. It is even more difficult to conceptualize the idea that matter, such as ourselves, can behave like waves, and yet this is consistent with a broad range of observations. We never notice the wavelike properties of matter because on the macroscopic scale, the wavelength associated with a particular object is so small that it is negligible. For example, the wavelength of a baseball moving at 100 m/s is much smaller than the baseball itself. It is worth thinking about what you would need to know to calculate it. At the atomic scale, however, the wavelengths associated with particles are similar to their size, meaning that the wave nature of particles such as electrons cannot be ignored; their behavior cannot be described accurately by models and equations that treat them as simple particles. The fact that a beam of electrons can undergo diffraction, a wave-like behavior provides evidence of this idea.

References

⁴³ Although the resting mass of a photon is zero, a moving photon does has an effective mass because it has energy.