In the particulate - matter wave dualism of existence of matter, the relationships proposed by Max Planck and Louis de Broglie were

E     is the total energy of the matter wave

h      is Planck's constant of proportionality, ( » 6.62E-27 erg secs).

f       is the frequency of the matter wave.

M     is the momentum of the matter wave particle.

v    is the wave number, (reciprocal of wavelength), of the matter wave.

M    is the spatial-temporal momentum of the particle, (see [1]).

m     is the energy mass of the particle.

m₀     is the rest mass of the particle.

v     is the spatial vector velocity of the particle.

c      is the spatial terminal velocity in D₀, ( » the velocity of light), and is also the magnitude of Existence Velocity in D₀, (see [1]).

j     is a unit vector in the temporal direction.

If the components of (2.1) and (2.2) are equated, it is immediately clear that the momentum component of (2.1) must be expanded to possess both a spatial and a temporal part thus

v_s     is the wave number of a matter wave in a spatial vector direction.

v_t     is the wave number of a matter wave in the temporal direction.

It will be seen later that this type of expanded representation will also be applicable to the energy component of (2.1).

The implications of the equivalence of (2.1), (with (2.3) inserted), and (2.2) are explored in the following three Subsections. Before the general case of a finite spatial velocity is considered, the two extreme cases of | v| = 0 and | v| = c are briefly reviewed.

2.2  A Stationary Particle.

2.2.1  Energy.

Equating the energy components of (2.1) and (2.2) for a stationary particle, | v| = 0 gives,

m0 c 2 = hf0

(2.4)

where

f₀     is the matter wave frequency of a stationary particle.

2.2.2  Momentum.

Equating the momentum components of (2.2) and (2.3), noting that v_s must be zero, gives

j m0 c =  j hvt0

(2.5)

where

v_t0     is the temporal wave number of a stationary particle.

Clearly from (2.4) and (2.5)

j f0 l0 =  j c

(2.6)

This shows that for a stationary particle, the associated matter wave propagates along the temporal axis with the velocity c. Its existence within the spatial dimension is solely corpuscular with zero spatial velocity. Its existence within D₀, in both forms therefore conforms to the criterion of existence within that Domain, (see [1]).

2.3  A Terminal Velocity Particle.

2.3.1  Energy.

Equating the energy components of (2.1) and (2.2) for such a particle, | v | = c, yields

hfc = m0 c 2 æ è 1 -  v 2 c 2 ö ø 1 /2   ê ê ê ê v® c

(2.7)

where

f_c     is the frequency of the matter wave for a terminal velocity particle.

If the limit in (2.7) is explored the RHS becomes infinite, but a simple re-arrangement gives

m0 = hfc c 2 æ è 1 -  v 2 c 2 ö ø 1/2   ê ê v® c

(2.8)

and this shows that the only way for matter energy to achieve a velocity of c, the spatial terminal velocity of D₀, is for it to possess zero rest mass. The energy of such a particle must therefore all be kinetic in nature. This suggests that if such a particle were brought to rest, all of its energy would be transferred to the arresting body and the particle itself cease to exist. For instance, consider the Compton effect. By the law of conservation of energy, from [2], Appendix X,

hfc + me0 c 2 = hfc" + me c 2

(2.9)

where

f_c"     is the frequency of deflected light after the collision.

m_e0, m_e     is the mass of the electron before and after the collision.

If the collision is such that f_c" is zero, i.e. a "head on" collision, then the mass of the electron after the collision is, from (2.9)

me = me0 +   hfc c 2         =   me0 æ è 1 -  v 2 c 2 ö ø 1/2

(2.10)

Thus the energy given up in this exchange is as per (2.8). This energy is transferred to the electron as kinetic energy and the electron acquires a velocity of, from (2.10)

v = c ì í î 1 - 1 æ è 1 +   hfc me0 c 2 ö ø 2   ü ý þ 1/2

(2.11)

It will be shown later that this kinetic energy is transferred to the electron as an increase in its own matter wave frequency.

2.3.2   Momentum.

Equating the momentum components of (2.2) and (2.3) under this condition gives

hvc = m0 c æ è 1 -   v 2 c 2 ö ø 1/2   ê ê ê ê v® c

(2.12)

where

v_c     is the spatial wave number of a particle when |v|® c, (note that v_t in (2.3) is zero in this case).

so that

lc = h æ è 1 -   v 2 c 2 ö ø 1/2   m0 c ê ê ê ê v® c

(2.13)

and therefore from (2.7) and (2.13)

fc lc = c

(2.14)

The matter wave for such a particle therefore propagates entirely along the spatial axes at the terminal velocity in D₀ , i.e. c. For this particle, time is stationary and it possesses no temporal existence at all. Both f_c and l_c are indeterminate from the above analysis and depend upon the characteristics of the emitting medium.

2.4   A Particle With a Finite Spatial Velocity.

This is the general case.

2.4.1  Energy.

First equating the energy components of (2.1) and (2.2) gives

hfv = m0 c 2 æ è 1 -   v 2 c 2 ö ø 1/2

(2.15)

and thus from (2.4)

fv = f0 æ è 1 -   v 2 c 2 ö ø 1/2

(2.16)

With reference to this result, it was stated in [1] that when a material body is spatially accelerated from rest, the kinetic energy it gains was stored as an increase in mass. From (2.16) it is now clear that the manner in which that storage takes place is by an increase in the matter wave frequency of the accelerated body as mentioned in relation to the Compton effect in Subsection 2.3.1. Consequently, it is also seen from (2.16) that to spatially accelerate any mass to the terminal velocity c would result in its matter wave frequency becoming infinite. A result also evident in (2.7). Therefore from the energy component of (2.1) this would require an infinite amount of accelerating energy, a result that concurs which other analyses in the literature, i.e.[1].

2.4.2  Momentum.

Now consider the momentum components of (2.1) and (2.2), noting that the full spatial-temporal form of (2.1) is now invoked, i.e. (2.3)

h( vsv + jvtv ) = m ì í î v + j æ è 1 -   v 2 c 2 ö ø 1/2   ü ý þ                             =  m0 v æ è 1 -  v 2 c 2 ö ø 1/2    + jm0c

(2.17)

First consider the spatial components of (2.17), these give

lsv = h æ è 1 -   v 2 c 2 ö ø 1/2   m0 v

(2.18)

where v is the magnitude of v.

This is the relativistic version of de Broglie's equation which will be central in the resurrection of the Bohr/Sommerfeld theory of atomic structure in a future series of papers. Combining (2.15) and (2.18) gives

fv lsv =  c 2 v

(2.19)

This is the so called 'phase velocity' of the matter wave and in view of the fact that it is greater than the velocity of light, it has been stated in the literature, [2] to have no physical significance. Such a statement needs clarification which will be provided in the next Section.

Now consider the temporal parts of (2.17), they yield

j ltv =  j h m0 c

(2.20)

This result is identical to (2.5), (for a stationary particle), and confirms that the temporal component of the particle matter wave is unchanged by the spatial motion. This is because the motion has been spatially induced and no energy has been either added or subtracted in the temporal direction. However, from (2.15) and (2.20)

fv ltv = c æ è 1 -  v 2 c 2 ö ø 1/2

(2.21)

This is the 'phase velocity' in the temporal direction and is again seen to be greater than the terminal velocity of D₀. Eq(2.21) along with (2.19) will be discussed in detail in the next Section.

Finally consider the magnitude of (2.17), this yields

lv = h æ è 1 -   v 2 c 2 ö ø 1/2   m0 c

(2.22)

and therefore from (2.15)

fv lv = c

(2.23)

Clearly, from (2.15), (2.17) and (2.23) the matter wave is propagating along the same path as the Existence Velocity Vector of the particle, (see [1]), and at the same velocity. In addition, it is now clear that this is also the case for the stationary particle in Subsection 2.1 and the terminal velocity particle in Subsection 2.2. This now helps the clarification in the next Section of the matter wave 'phase velocities' of (2.19) and (2.21).

2.5   Matter Wave Phase Velocities.

These are purportedly the velocities with which the matter wave propagates through the spatial dimension, (2.19), and the temporal dimension, (2.21). Both of these velocities are greater than the terminal velocity in D₀, and so contravene both the primary criterion of existence in that Domain and Einstein's maximum velocity of spatial propagation. This apparent conflict has arisen because the product of the parameters in (2.19) and (2.21) are in fact invalid, for the following reason. The matter wave frequency f_v is the frequency of the matter wave as it propagates along the same path as the Existence Velocity Vector of the associated particle. The wavelength l_sv is that of the spatial component of this matter wave, and the wavelength l_tv is that of the temporal component. These frequency and wavelength parameters are not coincident and therefore the products of (2.19) and (2.21) are invalid as true velocities. To resolve this it is necessary to associate with l_sv in the spatial dimension a frequency such that

fsv lsv = c

(2.24)

and with l_tv in the temporal dimension a frequency such that

ftv ltv = c

(2.25)

It is then clear from (2.19) and (2.21) that

fsv = fv v c ftv = fv æ è 1 -  v 2 c 2 ö ø 1/2

(2.26)

and it is also clear that f_sv and f_tv are the projection of f_v into the spatial and temporal dimensions respectively.

As a result of this it is now possible to represent the dual corpuscular/wave function existence of matter in D₀ as in Fig. 2.1 below.

Fig. 2.1 Pictorial Representation of the Matter Wave Associated with a Particle in D₀

It is emphasised that Fig. 2.1 is not a literal representation, only a pictorial one.

Note that the associations of (2.24), (2.25) and (2.26) now require that energy as well as momentum has both spatial and temporal components. This is also evident from (2.2) and (2.6), and is further analysed in the next Section

From (2.26) it is a simple matter to determine the relationship between the wavelength components of the wave.

P1 Version 1.0.0
Ó P.G.Bass, October 2006