# A Matter of Energy

The last discussion highlighted some initial lines of inquiry surrounding
the concept of energy, which classical physics defined in units of mass
[kg], although this idea appears to become increasingly abstracted when
considered in terms of quantum physics. The nature of the inquiry also
touched upon the speculative idea of space being a medium of wave propagation,
where each point is space might be modelled as a *‘not-so-simple harmonic
oscillator’ *. However, within this model, there would only be two
fundamental forms of energy described in terms of the potential and
kinetic energy associated with the displacement of a wave media, i.e.
space, from its ground state as waves propagate through it. Of course,
this description must appear in stark contrast to the many manifestations
of energy normally encountered in most physics textbooks, e.g.

rest mass, thermal, mechanical, electrical, electrostatic, magnetic, electromagnetic, gravitational, chemical, radiation, nuclear, ionization, chromo-dynamic …. etc.

However, any basic line of inquiry might lead to the conclusion that
most of the forms of energy listed above are only a manifestation of
a more fundamental form of energy. For example, we might readily accept
that thermal energy is a statistical aggregation of the kinetic energy
[mv^{2}/2] of individual particles linked to the concept of
mass, which in turn may require further consideration if we question
the nature of physical mass. Of course, if we continue to inquire into
the most fundamental nature of reality, then we might question whether
every action requires a causal explanation rooted in the concept of
energy.

[1]

In [1], we see the classical definition of energy [E] in terms of the SI base units of mass [kg], length [m] and time [s], where the latter two terms might be seen as being representative some form of velocity [v], which might initially be quantified in terms of rest energy or kinetic energy associated with mass [m].

[2]

In the previous discussion, we introduced the idea of the potential energy associated with the model of a simple harmonic oscillator defined as a function of force [F] and a displacement amplitude [A].

[3]

In [3], we have generalised the description of potential energy [Ep]
as a function of force [F] with radial distance [r] as the two primary
examples of potential energy might be described in term of gravitation
and electric charge, which are normally characterised in terms of the
radial distance [r] between two masses [m_{1},m_{2}]
or two charges [q_{1},q_{2}] as follows.

[4]

While we might note the similarity in the form of the two potential
energy equations shown in [4], we shall defer the discussion of these
equations until the discussion entitled ‘*A Matter of
Semantics*’. However,
we might initially inquiry as to whether the idea of gravity or charge
is best described as an attribute of a single ‘ *particle *’ or
rather as a field of potential energy between two concentrations of
energy-density in space. However, for now, we will simply highlight
that the total energy at any point in space might be quantified as follows:

[5]

Having made a general introduction of potential energy, the rest
of this discussion will now focus on the issue of kinetic energy [E_{K}]
and rest energy [E_{R}] and the effects of relativity on the
classical definitions shown in [2]. For the theory of special relativity,
published in 1905, required a qualification of the expressions in [2]
associated with the Lorentz factor [g] that effects mass, and therefore
the energy, in motion with velocity [v].

[6]

In [6], mass [m_{0}] will be described as the rest mass,
when velocity [v=0]. As such, we can now use the formulations to extend
the basic energy equations in [2] to include the idea of a relativistic
mass [m], which is a function of velocity [v].

[7]

Expanding and re-arranging, we can transpose [7] into a relativistic energy equation as follows:

[8**]
**

In the final form of [8], we see the expression for the total energy
[E] being linked to rest energy plus another energy term in the form
[pc], where [p] might represent the momentum of a photon or particle
within the *wave-particle duality* of quantum mechanics. However,
while this expression leads logically to the definition of the Compton
wavelength, shown below, it is less obvious when applied to a particle
having a velocity range [v=0..c]. For if we accept that a photon has
no rest mass [m_{0}] and a velocity [v=c], this immediately
provides a simplification of [8] as follows:

[9]

We can then extend the specific case in [9] using an equivalence
of the Einstein and Planck energy equations, which when re-arranged
in terms of wavelength [λ_{C}] arrives at what is known
as Compton’s wavelength, where [p] corresponds to the momentum of a
photon, which we might initially assume propagates through space as
a wave-particle with velocity [c].

[10]

However, this simplification is not immediately available to matter
particles, which have rest mass [m_{0}] plus a particle velocity
range [v=0..c]. Therefore, we might initially consider two solutions
of [11] corresponding to the extremes of the particle velocity range:

[11]

In the first case, where [v=0], we see a solution that appears very
similar in form to the Compton wavelength in [7], although there is
no obvious reason why this stationary particle-wave would have any momentum
[p], such that we might question the inference of [c]. In contrast,
a particle with velocity [v→c] would have an increased mass [m≫m_{0}]
that allows the simplification shown in the second case and leads to
a wavelength [λ_{D}] known as the deBroglie wavelength.
While the structure of any matter-wave may be far from clear at this
point, we might make an initial assumption that any wave that can remain
stationary in space, i.e. when [v=0], might suggest a
*standing wave
superposition* of some form created by two or more waves propagating
at velocity [c]. Likewise, a waveform capable of moving through space
in the velocity range [v=0..c] might correspond to some form of
*beat
waveform* , where the beat wavelength corresponds to the deBroglie wavelength.
As such, we might test these assumptions in terms of the energy associated
with non-dispersive wave propagation relationship [v=fλ], which
will also need to be clarified later in the discussion.

[12]

In part, the equations in [12] appear to make sense in that a stationary matter-wave has energy corresponding to its rest energy, while the matter-wave with velocity [v] appears to have an energy closely resembling the classical equation for kinetic energy, i.e. linked to velocity [v], albeit missing a factor of 2. Given this discrepancy, we might take a slightly different approach and assume that the deBroglie matter-wave is associated with its classical kinetic energy propagating with velocity [v], not [c]:

[13]

However, while [13] started with the classical equation for kinetic energy, the inclusion of the factor (2) now causes a discrepancy with the accepted form of the deBroglie wavelength, which does not include the factor (2). While this issue will be taken up later in the discussion, the following equation is tabled for initial consideration:

[14**]
**

In [14], we see the classical equation for kinetic energy, which
is known to only be an approximation at non-relativistic velocities,
i.e. [v≪c]. However, we might derive an expression for the kinetic
energy in terms of the known total energy [E] minus the known rest energy
[E_{0}]

[15**]
**

However, equations [12], [13], [14] and [15] all possibly suggest
that further review of the nature of kinetic energy is required, especially
in the context of a wave model, where the distinction of a classical
particle with mass [m_{0}] has to be replaced by a concept of
a wave energy-density confined within some unit volume of space-time.
Therefore, we shall return to the wave-like definitions associated with
the Compton [λ_{C}] and deBroglie [λ_{D}]
wavelengths, which were also derived based on the relativistic energy
equation plus some generalised assumptions about kinetic energy.

[16]

While the structure of any matter wave is still highly speculative,
it was suggested that the existence of both the Compton [λ_{C}]
and deBroglie [λ_{D}] wavelengths might indicate some form
of composite wave structure, such as a stationary standing wave, when
the particle velocity [v] equals zero, which then becomes some form
of beat waveform when the particle velocity [v] is greater than zero.
While this idea was partially supported by the following energy equations,
it was highlighted that the equation relating to the kinetic energy
of a particle-wave in motion had an erroneous factor of 2.

[17]

However, the idea of kinetic energy [E_{k}] can also be mathematically
derived in terms of the work done by a force [F] acting on a mass [m]
over a distance [0..r], which can then be more accurately calculated
by integration.

[18]

However, the integration form above reflects a force that can also be expressed as the rate of change of momentum as a function of time, i.e. dp/dt, where momentum [p=mv]:

[19]

Relativity then extends the basic idea of mass to include the idea
of a rest mass [m_{0}] and a relativistic adjusted mass [m],
such that [19] has to be further modified:

[20]

The form of [20] has to be resolved using the method of integration by parts, i.e.

[21]

Using an online integration calculator to resolve the second integral, we get

[22]

The expression in [22] can then be rationalised as follows:

[23]

In the final form of [23], which matches the form of [15], we see
that the kinetic energy [E_{k}] equals the total energy [ g
m_{0}c^{2}] minus the rest energy [m_{0}c^{2}],
which must now replace the classical form [mv^{2}/2], although
it is still unclear how this expression would lead to the required form
of the deBroglie wavelength without the factor [ ϒ-1] appearing.
However, we might pursue this issue using a second form of the derivation
above, starting with a re-statement of equation [22].

[24]

By moving the rest energy [m_{0}c^{2}] back over
to the left-hand side, we are left with 2 expressions on the right-hand
side, which might be interpreted in a different way.

[25]

Clearly, this segregation of the various energy expressions in [25] might be questioned, but let us first provide some form of description for each expression as follows.

[26]

The summary in [26] may initially appear little more than mathematical manipulation and a somewhat strange interpretation in the context of classical physics or even relativistic physics, especially when viewed in particle terms, where the rest energy is usually assumed to be constant. However, it may be more valid when viewed from the perspective of a composite waveform in motion, where the total energy of a perceived particle is not necessarily fixed within the division of an absolute rest mass versus a variable kinetic mass. Again, this might be a questionable interpretation, but if the concept of mass is only a manifestation of a wave energy-density, then the idea of rest mass may be no more real than the perceived increase in kinetic mass, especially if the division of energy is only considered in terms of the potential and kinetic energy associated with a waveform in motion. For example, we might consider the following comparative table showing the energy before and after a conceptual elastic collision in terms of the 3 representations of energy discussed.

The elastic collision being described takes place between 2 conceptual
‘*particles*’, i.e. [A] and [B], where the concept of rest mass
energy might be assumed to have been replaced by the idea of a potential
energy [E_{P}]. In the ‘*before*’ state, [A] also has a
kinetic velocity [v=0.5c] with an associated Lorentz factor [ϒ=1.155],
while [B] is assume to be at rest, although this may depend on the scope
of the *reference frame* in question. Based on the conservation of energy,
we require the total energy [E_{T}] before and after the elastic
collision to be identical. In the first Newtonian case, kinetic energy
is based on the classical approximation [mv^{2}/2], which is
known to be inaccurate at the relativistic velocity [v=0.5c], but might
be seen as a comparative benchmark for the two remaining cases. In the
relativistic case, the kinetic energy is calculated based on the revised
formulation [(ϒ-1)m_{0}c^{2}], as derived in [15]
and [23]. As expected, the total energy in the relativistic case is
higher than the Newtonian case as relativistic energy has now been taken
into account. The final case, in the row labelled WSE, is based on the
equations in [26], which shows the same conserved total energy [E_{T}]
before and after the collision, as per the relativistic case. As such,
the discussion is primarily related to the percentage breakdown of this
total energy described in terms of potential energy [E_{P}]
or kinetic energy [E_{K}], which in the WSE model might be a
somewhat arbitrary division. However, consideration of the interpretation
in [26] may help to resolve the propagation velocity [v] associated
with the deBroglie wavelength in terms of its kinetic energy, i.e.

[27]

Although, some of this interpretation is only conjecture, the validity
of the deBroglie wavelength formula appears to have been experimentally
verified via the
*Davisson–Germer experiment*, first conducted between
1923–1927. At this stage, this line of inquiry is only suggesting that
the idea of kinetic energy [E_{k}] may require some further
clarification in the context of any WSE model. However, another line
of inquiry might now question the relationship between atomic line spectra
and the atomic orbital structure of electrons, which are required to
be integer multiples of the electron deBroglie wavelength, although
often explained in terms of the quantization of angular momentum, e.g.

[28]

However, a possibly more logical rearrangement of [28] might be given in the form, which provides a causal mechanism in terms of integer wavelengths, such that we are no longer making reference to orbiting electron particles.

[29]

In many respects, we might start to perceive an issue of descriptive semantics that has its roots in classical Newtonian physics, which has been maintain through to quantum physics even though some of the concepts no longer really apply. This issue is now taken up in the next discussion.