Matter Waves

1Equation [6] of a previous discussion, which attempted to introduce the basic premise underpinning the idea of the deBroglie wavelength, also raised some doubts concerning the central relationship between wavelength [λ] and momentum [p]. One key problem seems to stem from the generalisation of the propagation velocity [v=fλ], which then altered the derivation anchored in equating the Planck and Einstein expressions for energy, e.g.

[1]  1

However, one of the issues implicit in [1] is whether the energy of a matter wave is associated with the concept of total or kinetic energy. However, let us initially examine the suggestion that any wave associated with a particle must somehow remain collocated with the particle in both time and space, such that both have the same velocity [v]. In order to satisfy this requirement, it was suggested that any matter wave associated with a particle, like the electron, might have a structure that was more aligned to the description of a superposition wave, which introduces the concept of both a phase [vp] and group [vg] velocity. Again, at this point, we might introduce some basic wave terminology by way of reference:

[2]      2

As a basic visualization, the following animation is trying to represent two sine waves propagating in space and time. However, these waves are also shown combined in superposition to produce the standing wave pattern below.

1

As indicated, standing waves can also have a propagation velocity depending on the attributes of the underlying waves. However, the length of a superposition wave packet can be finite based on the sum total of all the underlying waves at each point in space. As such, the next animation might possibly be a better initial visualization of a ‘conceptual’ matter wave moving through space, at least, if constrained to one dimension.

2

It is in the nature of these waves that the propagation of the standing wave is associated with the group velocity [vg], while the phase velocity [vp] is associated with the underlying travelling waves.

  • Phase Velocity [vp]:
    This might initially be visualized as the propagation velocity of a single frequency wave travelling with respect to space [x] and time [t]. This velocity results from the fundamental geometry of the ‘travelling wave’ and is defined by the expression [vp=ω/k] that may be reduced to the form [vp=fλ].

  • Group Velocity [vg]:
    This is the speed of a superposition wave,  which is sometimes referred to as a ‘standing wave’. The group velocity is defined by the expression [vg=∂ω/∂k] and may take the range [0<vg<c] on the assumption that no physical travelling wave can exceed the speed of light [c].

While [2] provides a basic definition of the phase velocity [vp] , we also need to try to develop some better insight regarding the relationship between energy [E] and momentum [p], which are both important quantities associated with a wave.

[3]      3

While there are aspects of de Broglie’s derivation of the particle wavelength [λ=h/p], which requires further scrutiny, experiments have apparently confirmed this relationship. It is also known that this relationship is unambiguous in terms of the Compton wavelength for photons. So, from [3], we might now try to quantify the phase velocity of a photon and a particle by substituting for the energy [E] in [3], while making a few assumptions about each ‘particle’ in question:

[4]      4

In the case of [4], we appear to arrived at a definition of the phase velocity [vp] for a photon and a matter particle based on the assumption that the photon has no rest mass [m0] and the particle being constrained to a non-relativistic velocity, i.e. [v<<c]. At this point, we shall simply highlight the inference that the phase velocity of a component particle wave would appear to be greater than the speed of light [c]. However, from [2], we might also extend the arguments in [3] to the group velocity [vg]:

[5]      5

In the context of [5], we are now making reference to the change in energy, which might be interpreted as the kinetic energy. Again, we might now try to quantify the nature of the group velocity [vg] for a photon and a particle:

[6]      6

So based on the assumption that the matter wave might be better described in terms of a superposition wave with a group velocity [vg] aligned to the kinetic velocity [v] of the particle, we might consider revising the definition of energy in [1] to kinetic energy:

[7]      7

In the context of [7], we have now assumed that the energy associated with the group velocity corresponds to the kinetic energy of the particle, not the total energy. However, while the form appears closer to that of the deBroglie wavelength, there is still an additional factor of 2, which would have to be reflected in either the wavelength [λ] or the velocity [v].

Note: It will simply be highlighted at this point that the formulation of kinetic energy in [7] is only an approximation. See 'A Matter of Energy' for a wider discussion and other derivations.

However, if we are to map the particle-like description, of say an electron, into a wave-like description, we also have to consider the issue of localizing the particle and the wave in the same region of space. So let us assume that an electron is moving from A to B through a vacuum. At some intermediate point, between A and B, we might reasonably assume that the wave amplitude, whatever that implies, must be non-zero within this localized region of space, but zero at both A and B. However, if we assume that the electron has kinetic energy [E], it must also have a defined momentum [p], which implies that the wave has a defined wavelength [λ]. Based on these assumptions, we might initially start by considering a general wave function of the form:

[8]      8

However, a sine wave that fits this description would conceptually extend to infinity in both spatial directions and could not therefore represent a particle, whose wave function is assumed to only be non-zero within a small region of space. Again, at this point, we might return to the animation above, which tries to represent a localized waveform moving through space with a group velocity [v=vg] within the range [0<v<c]. As such, we might initially try to formalize a simple waveform in terms of a superposition of just two waves with slightly different wavelengths:

[9]      9

Now, the first term, sin(κxt) is reflective of the average frequency of the two waves being used in this example, which is then modulated by the second term, which creates a standing wave that extends over a distance defined by [π/Δk]. In defining [Δx=π/Δk], we are labeling the distance over which the waves are initially in phase and then go out of phase. However, in order to describe a single electron moving through a localized region of space, we need a wave function that leads to a single wave packet, which can be achieved by assuming a continuous distribution of wavelengths within the wave packet.

2

In this case, the sum of all the waves will still be out of phase after a distance in the order of [Δx=π/Δk] , but because of the continuous range of different wavelengths, the superposition or standing wave never gets back in phase. So, based on previous arguments, we might model an approximate wave packet by estimating the spread in wavelength localized in a region of space defined by [Δx], which can be constructed from waves having spread of [k] over a range [Δk] such that:

[10]    10

In the context of  [10], we see that there is an implication that the precise position of the wave-particle might have a degree of uncertainty due to the distribution of wavelengths underpinning the superposition wave packet. As such, we might use this idea as some sort of initial resolution of the apparent discrepancy in [7]:

[11]    11

In the current context [Δx] is a measure of the length of the wave packet in space, which approximates to the half wavelength [Δλ/2]. In this context, the accepted form of the deBroglie wavelength might actually correspond to [Δx], although eventually the ambiguity of a particle’s position [Δx] and momentum [Δp] will have to be discussed under the wider context of ‘Heisenberg’s Uncertainty Principle’. However, there still seems to be an open question regarding the wave packet model discussed so far:

Does this model really reflect physical reality?

While, at this stage, we shall try to avoid any philosophical implications concerning the scope of physical reality, there is still a basic concern regarding the construction of a superposition wave, i.e. the wave packet, from a continuous distribution of waves with different wavelengths propagating with a phase velocity [vp=c2/v], i.e. in excess of [c].