de Broglie Wavelength

1

By 1924, nearly a quarter of century has passed since Planck had first introduced the idea that energy might be quantized in the context of blackbody radiation. Although Einstein had extended this idea, as early as 1905, with the publication of his paper addressing the photoelectric effect, the full implications of quantum theory were either not realised or not accepted. In 1913, Bohr’s model of the atom renewed the scope of energy quantization in terms of atomic orbits, but the idea of quantized light, i.e. the photon, was still rejected by Bohr himself. In fact, Bohr was still rejecting the notion of the photon, even after the initial publication of Compton’s scattering experiment, in 1923. However, by this time, the idea of quantized light conforming to Planck’s equation [E=hf] had gained some general acceptance. In 1924, the work of Louis de Broglie would reverse the original direction of the debate by proposing that particles, such as electrons, also had a complementary wave nature. Without initially scrutinising the mathematical logic underpinning de Broglie’s idea too much, we might summarise his thinking as follows:

If a light wave can behave as a particle,
is it possible for a particle to behave as a wave?

However, we might based this premise on some sort of mathematical foundation by simply repeating the logic that led to Compton’s equation showing the relationship between momentum [p] and wavelength [λ] based on the ‘duality’ of Planck’s and Einstein’s energy equations:

[1]      1

If we start from the original accepted position of light as a wave, the idea forwarded by Compton, in support of the photon was actually that light had a kinetic mass, which can be then be described in terms of momentum [p], such that we need to arrange [1] as follows:

[2]      2

In [2], the definition of momentum is anchored in the constancy of the velocity [c] of a photon in vacuum and while having no rest mass, exhibiting an effective kinetic mass [mk], which can be linked to the photon’s frequency [f], as defined in [1]. So, in many ways, de Broglie’s solution appears to simply extend the relationship in [2] to a particle, where the momentum [p] is now the product of the particle’s mass [m=γm0] and its velocity [v].

 [3]     3

Before pursuing the premise underpinning [3], it might be useful to first cross-reference deBoglie’s idea with Bohr’s model of the hydrogen atom, which provided an estimate of the electron’s velocity [v] in the ground state of hydrogen. If we combine this non-relativistic velocity with the estimated mass of an electron, we can calculate its wavelength to a first order of approximation:

[4]      4

As another way of explaining the quantization within Bohr’s atomic model, deBroglie postulated that an electron orbiting the nucleus of a hydrogen atom could only occupy discrete orbits due to the wave nature of the electron. Within this extended wave model; the circumference of the electron’s orbit had to equal an integral number of wavelengths, e.g.

2We might carry out a provisional test of this idea by assuming that the result in [4] corresponds to the circumference of the electron orbit in the ground state of hydrogen. If so, we might divide this figure by [2π] and see if the result corresponds to the estimated orbital radius of the ground state:

[5]      5

Within the limits of the approximations being made, this would appear to be a reasonably accurate estimate, which supports deBroglie’s general hypothesis that particles also have a wave-nature complementary to the particle nature of light. As such, the controversy known as the wave-particle duality now appears to have been extended to matter, although we might try to put this ‘duality’ into some initial perspective:

  • An object is a particle when the probe is much bigger than the wavelength.
  • An object is a wave when the probe is much smaller than the wavelength.

In terms of the result in [4], the suggestion is that probing an atom on a scale larger than 10-10 metres will return the perception that the atom, as a whole, is a single particle, even though it is known to have a sub-structure. So, the suggestion is that sub-atomic particles would require very, very small ‘probes’ in order to examine any implied wave structure associated with these particles. However, while the definition of the deBroglie wavelength appears to be giving us some insight into some potential wave structure, the details of this wave structure remained far from clear within the emerging scope of quantum theory at this point in the timeline of developments. So, having established some basic concepts, let us highlight some possible issues that may require further clarification; starting with an alternative derivation of [3]:

[6]      6

The derivation in [6] has the same starting point as [1], but has now generalised the propagation velocity of the particle wave by using [v], not [c], which is specific to the propagation of light. However, this one simple change appears to have now obscured the key relationship between wavelength [λ] and momentum [p]. At face value, the only way [6] can be restored to the form in [3] is to make what appears to be a very strange assumption given the postulate of special relativity concerning the speed of light:

[7]      7

Clearly, we will need to examine the implications resulting from [7] in more detail, but we might see that there is an initial suggestion for some sort of wave with a phase velocity [vp] greater than the speed of light [c]. However, we would need to question this perception, not only because it appears to violates special relativity, but because we need a model of the matter wave that remains collocated with the particle in space. As such, we need to define some sort of wave that has a propagation velocity in the range [0<v<c], i.e. comparable to the particle, which might be better described in terms of a standing wave, when [v=0]. However, in contradiction of its name, a standing wave can propagate through space, when described in terms of the superposition of two or more travelling waves. However, before discussing the potential structure of a particle wave packet, we need to introduce the concept of ‘Wave Dispersion.