When rest mass m = 0, the energy of this wave follows Planck’s law for energy E and follows de Broglie’s hypothesis for momentum p:

E = h ν = ℏ ω

and

p = h λ = ℏ k (3)

where h is Planck’s constant, and where ν and λ are frequency and wavelength respectively. In wave mechanics, as in relativity, the products:

ν λ = ω k = c (4)

and, since, in the present case, E = pc, then the derivative of hω = hkc gives:

d ω d k = c (5)

which is the beat velocity of an unbalanced tuning fork and, by integration, the group velocity of the symmetric wave function [2] . Massless particles, in free space, have phase and group velocities that are both equal to c. They progress in phase, at their respective centers.

The wave function ϕ = e^X has unit amplitude for all x and t. However, by definition, in a mono-atomic crystal having conditions where all atoms scatter an electron beam in phase, the structure factor is the sum of all the atomic scattering factors in the unit cell. However, when half of the atoms scatter in anti-phase, the structure factor is zero, and the corresponding diffraction line is forbidden. The wave function ϕ is therefore genuine with typical properties. The energy of the quantum is proportional to the area under then envelope.

When m > 0, it is well known that the relativistic mass m’ of a particle increases with its velocity [7] :

E = m ′ c 2 (6)

Since

E 2 = p 2 c 2 + m 2 c 4 (7)

and, by definition of the Lorentz factor γ:

m ′ = γ m = m 1 − β 2 (8)

where β = v_g/c. Consider the transmission electron microscope. We need to work with the most convenient reference frame which might include one of the following: near the field emission tip maintained at accelerating voltage V; or at the laboratory specimen chamber; or within the specimen, etc. The last option is needed in fine measurements, but it is specific to specimen composition, precise orientation, and to the probe energy. The appropriate adjustment is comparatively small [8] , a few electron volts in energy, so we will calculate parameters in the laboratory frame as a general approximation. This becomes more valid for probes of energy pc > 100 eV.

Where does the accelerating energy Ve (voltage x electronic charge) of the probe enter into Equation (1) for mass? Though a magnetic field would be velocity dependent, that is not the case with electrostatic interaction, so it is included as a scalar like the coulomb potential in Schrödinger’s equation. The accelerating energy is added to the rest mass energy, i.e. independent of momentum. The frequency of the probe is therefore given by:

ν = E h = m c 2 + V e h (9)

The wavelength of the probe depends on momentum through de Broglie’s hypothesis, so that when applying Equation (7) (Figure 2) and the Pythagoras theorem: p 2 c 2 = E 2 − m 2 c 4 ; p = ± E 2 − m 2 c 4 / c and:

λ = h p = h c ( E 2 − m 2 c 4 ) 1 / 2 = h c ( ( V e ) 2 + 2 ( V e ) m c 2 ) 1 / 2 (10)

From which the phase velocity v_p = νλ is calculated (Table 1). Meanwhile, following Equation (8), γ = E / m c 2 = 1 / 1 − v g 2 / c 2 and the group velocity:

v g = c 1 − γ − 2 (11)

excepting low energies when E ® mc², the product of the group and phase velocities tends increasingly to c² as the probe becomes more relativistic (Figure 3), so that when pc?mc², v_p ≈ v_g ≈ c. More generally, the Pythagorean relationships illustrated in Figure 2 underpin the complex nature of ϕ : since mass displacement varies with time while momentum displacement varies in space, complex ϕ combines them. Notice that in the free electron probe the potential energy Ve is positive; whereas in Schrodinger’s solutions for electrons in the bound hydrogen atom, the potential energy eV and eigenvalues are negative.

This occurs because the positive kinetic binding energy of a bound state is less than the Coulomb potential energy. | 〈 p c 2 / ( 2 m ) 〉 | < | 〈 V e 〉 | (For a free anti-particle, such as the positron, the relationships between m, p and E would need adjustment [3] ).

To understand Figure 3, return to Equation (1). Differentiating with respect to k gives the important result:

ω k ⋅ d ω d k = c 2 = v p ⋅ v g (12)

where the group and phase velocities are inversely related. The more familiar group velocity is used in the Lorentz factor (the inverted bracket in Equation (8)) in special relativity. At an acceleration of 10 eV, the free electron has a phase velocity over 4 orders of magnitude greater than its group velocity.

Anti-particles behave a little differently (Appendix 1).

The evidence in electron microscopy and in creation and annihilation experiments is so copious and longstanding that the claim that “Nothing travels faster than light” should now be considered unscientific, because phase velocity is not “nothing”. Obviously, Einstein meant only that v_g < c [7] ; the range of v_p is limited to space within the corresponding wavefunction. Nevertheless, his views were realistic [9] : he accepted hidden variables with their explanatory power [10] .

However, we should wonder why, for the free electron, since λ and ν were certainly calculated many decades ago, it passed unnoticed that their product is faster than light. The fact is not found in common textbooks, and it may never have been discussed in journals. Other characteristic features of dispersion dynamics are also neglected. If we are to understand what, in general, we do as scientists, we are obliged to speculate on the multiple reasons that have produced this outcome.

Obviously therefore, the method described here cannot be called mainstream. However, our logic [2] does adhere to falsifiability [11] [12] ; but weakly to Bohr’s phenomenology [13] and yet less to Dirac’s preference for beauty and simplicity [14] over tested truth. Our method is realistic: the quantum is a property; not an entity that has been multiplied without necessity. The property is not a substantial mathematical axiom: but is a consequence of harmonious interaction of matter with regularities in spatial harmonics (as in Schrödinger’s solutions), or with crystalline material (as with momentum quanta in diffraction patterns), including logarithmically periodic quasicrystals [15] , etc.

Our dispersion dynamic is relativistic, unlike Schrödinger’s equation that is in principle classical. Approximate relativistic corrections can however be applied [16] ; the greatest difference is that our states are for free electrons in an electron microscope, whereas Schrödinger’s states are bound atomic electrons where the total energy E < mc². This comparison is illustrated in Appendix 2.

A further difference between the two methods is our separation of Equation (1) into a conservative part (green) and a responsive part (pink) [2] . The former provides the condition for particulate properties, energy, momentum, charge, mass, spin etc.; the response part provides conditions for interference, annihilation, creation, entanglement etc.:

m 0 2 = ω 2 − k 2 = ( ω + k ) ( ω − k ) (15)

These two operations remain separable in the corresponding wave function for the free particle:

ϕ ( x , t ) = A ⋅ exp ( X 2 2 σ 2 + X )

with

X = i ( ω ¯ t − k ¯ x ) (16)

and elastic uniform response intensity: ( e X ) * ⋅ e X = 1 for all x and t inside the normal envelope. These formulae are a direct expression of wave-particle duality.

A significant reason for the general ignorance of the phase properties in dispersion dynamics is Dirac’s opinion that the wavepacket is unstable [1] . This claim is falsified by two facts: one is that many photons are currently observed by cosmologists and others, even though the microwaves are many billions of years old; the second fact is that unstable photons would break Newton’s first law of motion. These facts suggest that the high mobility of the phase is constituent in its stability.

His influence is widespread: his relativistic theory by rank 8 matrices is precise and widely used, particularly with Heisenberg’s representation of quantum mechanics by non-commuting canonical variables. Dirac showed that the latter representation provides similar results to Schrödinger’s alternative, which however is more commonly used for practical reasons, especially in chemical applications.