# A Matter of Perspective

Science has offered up many descriptions of energy, which classical
physics might have described in terms of the kinetics of a mass particle
in motion or the potential offset in a gravitation or electrostatic
field or even the propagation of mechanical waves through a physical
media like water or air. However, science has also provided a description
of radiated energy dissociated from any rest mass in the form of electromagnetic
waves, which are described as self-propagating without any obvious requirement
or interaction with any form of physical media. Later, the description
of all energy interactions was updated in terms of photons, which developed
into an interpretation of the
*quantum wave function* that may only have a
‘*probability existence’* within one of many mathematically abstract
*quantum fields*. While all of these descriptions have had success within
some restricted domains, we might still inquire into one fundamental
issue:

Does cause and effect take place within some tangible reality?

Today, such questions may only be considered as having philosophical
scope and debated by those who simply do not understand the mathematical
framework that now underpins all scientific method. Of course, the counter-argument
is that scientific certainty is always the position up until somebody
proves an alternative model may have more validity. However, while this
discussion is not an attempt to forward an alternative model, it does
question whether any sort of wave model is possible and, as such, assumes
that all potential energy may be linked to a displacement of the wave
propagation media from a point of equilibrium. For if a WSE model underpins
the structure of everything, i.e. all sub-atomic particles, then the
idea of kinetic energy attributed to any ‘*particle*’ in motion
is just another manifestation of wave energy in motion. However, as
a starting point, we need a basic description of how a ‘*displaced
point’* in space might be modelled, possibly starting with a
*simple
harmonic oscillator*. As such, we might initially anchor this description
to the physics of Hooke’s Law, where the force [F] associated with a
displacement [A] is a function of some elasticity constant [k].

[1]

The negative sign reflects that the force [F] always acts in the opposite direction of the displacement [A]. However, a force [F] can also be defined in terms of a rate of change of energy [E] with displacement [A], i.e. F= dE/dA, such that the input energy of this system can be determined by integrating the force over its displacement range [0..A]:

[2]

This equation would represent the initial potential energy [E_{P}]
of the oscillator in terms of some maximum displacement [A_{0}].
We might then proceed by assuming that the initial displacement [A_{0}]
can be modelled as a sine wave function of time [t], where the idea
of angular frequency [ω] replaces frequency [f] primarily for mathematical
convenience.

[3]

While [3] expresses the displacement [A] as a function of the angular
velocity [w] and time [t], we have not provided any physical explanation
of what might determine the angular velocity [w] of our model oscillator,
which in classical terms might be achieved by combining Hooke’s law
and Newton’s 2^{nd} law of motion

[4]

This is essentially a classical description of simple harmonic motion, where [4] describes a unit mass [m] undergoing acceleration [a], which is subject to a rate of change of displacement [A] from its equilibrium position.

Note: For the moment, we will continue to use the idea of [m], while highlighting that any WSE model must ultimately redefine mass as an energy-density.

However, in order to further develop the oscillator model, we need to equate the acceleration of the wave motion, as defined by [3], by recognising that acceleration [a] is a second order differential of displacement [A].

[5]

Based on the result in [5], we can now substitute for acceleration [a] in [4].

[6]

In [6], we have quantified an expression for the elasticity [k] that can be used in the oscillator model, when substituted back in [2].

[7]

Note: The purpose of highlighting the proportionality between energy
[E] to both frequency [f^{2}] and amplitude [A^{2}]
will be discussed further in the context of Planck’s energy equation
[E=hf]. However, given that Planck’s equation is often considered important
within the quantization of energy, it seems we will need to understand
why amplitude [A] disappears in Planck’s energy equation and why this
equation is only proportional to frequency [f] not [f^{2}].

It should also be highlighted that the oscillator model developed,
so far, does not account for there being some minimum energy as required
by quantum mechanics in the form of the uncertainty principle. We might
also recognise that [7] only tells us the potential energy stored within
a classical oscillator and not the total energy. Again, from a classical
perspective, we might quantify the total energy [E_{T}] as the
sum of the potential energy [E_{P}] and the kinetic energy [E_{K}]
at any point in time [t] based on the following components.

[8]

Based on [8], we can combine the two energy components to create
an expression of the total energy [E_{T}] at any point in the
oscillation cycle. However, this single oscillation model is cycling
between the maxima of potential and kinetic energy. At the top of the
cycle, all energy is potential as the mass [m] is stationary. Similarly,
as the mass [m] passes the centre-point, all energy is kinetic and velocity
is at a maximum. However, it needs to be highlighted that the only thing
in motion in this model is a conceptual point in space, which at the
quantum scale only exists as a scalar energy quantity associated with
the positional amplitude [A] of the oscillator.

Note: When subsequent aspects of the derivations to follow make reference to a distance [x], they are implicitly orientated towards the amplitude [A] of the oscillation offset from its equilibrium point, which in some contexts might be seen as a vertical [y] offset.

[9]

In [9], we see the combination of both potential and kinetic energy,
although we will possibly need to question the accuracy of the classical
kinetic energy component used. Also, before proceeding, we need to establish
some of the basic assumptions of quantum mechanics anchored to the derivation
of the *Compton wavelength* and the
*Planck energy equation* *
,* which is underpinned by the
*uncertainty principle*. We will start
with a basic derivation of Compton’s wavelength linked to the energy
equations of Einstein [mc^{2}] and Planck [hf].

[10]

In [10], we see an inverse relationship between wavelength [λ]and
momentum [p]. However, the use of [c] rather than [v] suggests that
the equation applies to a photon rather than a particle with mass. However,
an analogous form of this equation exists in the form of the
*deBroglie
wavelength*, such that the relationship [h/p] will be consider generic
at this stage. Of course, if a photon has no rest mass [m_{0}],
the kinematics of momentum being assigned to the mass [m] would have
to be explained in terms of a wave energy density. However, based on
[10], we might proceed to defined the uncertainty in momentum [p] and
distance [x=A] as follows.

[11]

In [11], we see the basic form assumed to support the uncertainty
principle, although at this stage, it is simply showing that an inaccuracy
in measuring either momentum [p] or distance [x] will result in a corresponding
inaccuracy in the other quantity. While [11] is most commonly used form
of the uncertainty principle, it may be useful to also highlight that
this implied ‘*uncertainty*’ extends to energy [E] and time [t]
based on Planck’s energy equation.

[12]

Before commenting further on the scope of equations [10], [11] and [12] in terms of the uncertainty principle, we shall develop the total energy oscillator model in [9] based on the uncertainty of momentum [Δp] and distance [Δx].

[13]

In [13], we have translated the amplitude displacement [A] into a positional uncertainty [Δx] using [11] to map the uncertainty in momentum [Δp] into a positional uncertainty [Δx], such that we can take the derivative with respect to the positional uncertainty [Δx] and equate the result to zero:

[14]

Solving [14] in terms of the positional uncertainty [Δx] gives:

[15]

Finally, we can substitute the result in [15] back into [13]

[16]

First, we possibly need to highlight the significance of [16] in
that it suggests that a ‘*quantum oscillator model*’ cannot have zero
energy, if the uncertainty principle of quantum mechanics is adopted.
However, it also leads us down the path of quantum logic based on a
*time dependent solution* of
*Schrodinger’s wave equation* that then
deviates from classical physics. While the previous links attempt to
discuss the development of quantum mechanics in a little more detail,
this discussion wants to highlight an assumption, based on the formulation
of the *deBroglie wavelength* that leads to idea of a quantum
*wave-function
collapse* and its
dispersive nature. While this discussion will
not reproduce the detail in previous discussions, it will reproduce
some of the logic used to derive the deBroglie wavelength based on the
Newtonian formulation of kinetic energy.

[17]

In [17], we see similar logic for both the Compton and deBroglie
wavelengths, but where the latter substitutes for kinetic energy in
the form [mv^{2}] and assumes a wave velocity [v] not [c]. While
the missing factor of (2) in the expression for kinetic energy will
be discussed further in the next discussion, it is normally assumed
that the energy associated with the deBroglie wavelength leads to a
relationship between [k] and [ω] that appears to be non-linear,
i.e. dispersive.

[18]

However, we might inquire as to whether there is an ambiguity in
the form of [18], if we take into consideration that [h] itself is quantified
in terms of energy, i.e. [h=joules.seconds], while the concept of mass
[m= g m_{0}] becomes a relativistic function of velocity [v].
As such, we might consider an alternative interpretation based on kinetic
energy [E_{k}=mv^{2}], to be discussed next, which then
leads to the suggestion that matter-waves might be non-dispersive.

[19]

Note: The inference in [19] with respect to [18] is wrong because
it is only showing the wave velocity relationship
v=fλ=ω/κ, while [18] is inferring that this velocity
with be a function of frequency [ω=2πf]. However, the dispersion
model might still be questioned on the grounds that the implied
velocity [v] in [18] is dependent on the mass [m] of the wave-matter
particle as a whole, on the basis of
E=hf=mc^{2}, there is a suggestion that each
frequency [f] would be associated with a different mass [m=hf/c^{2}]. Based on [18], a matter-wave
particle with zero velocity [v=0] would have an infinite wavelength
[κ=2π/λ] or zero frequency [ω=2πf]. As such, it might be
assumed that this wave structure might conform to some form of 3D standing
wave. If so, we would also need to understand how this
standing wave structure might propagate in space-time with velocity [v=0..c]
- see
Beat Waves. In this wave model, the underlying travelling waves
that form the composite beat wave might all propagate with velocity
[c] in free-space, such that this wave is not really a wave, but
rather an interference effect with an associated energy distribution
at some point in space. As such,
we possibly still need to question what it physically means for a
wave-matter particle
to 'disperse' in space as a function of time as described in the
wave function collapse.

It might also be suggested that the
kinetic energy of matter-wave is always [mv^{2}] within a WSE
model and may therefore not necessarily be dispersive. If this is the
case, it might then question some aspects of the following quantum description
that usually accompanies the idea of the quantum oscillator, i.e. the
wave function collapse and the scope of the uncertainty principle.

The diagram above shows the solution of the Schrödinger equation
for the first four energy states [n=0..3] in terms of a normalized wave-function.
In this context, it shows the difference in the solutions of the harmonic
oscillator model in both the quantum and classical domains in terms
of the ‘*probability*’ of finding the implied mass [m], i.e. energy,
at a given amplitude [A=x] from the equilibrium position. In the classical
case, shown by the blue curve, the probability is highest at maximum
amplitude as the conceptual mass approaches the maximum potential energy
and minimum kinetic energy, i.e. the velocity approaches zero, such
that the implied probability is the inverse of velocity [v]. However,
in the quantum case, the probability is now defined in terms of the
square of the quantum wave-function amplitude [ψ^{2}],
which has no obvious physical interpretation, and in the ground state
[n=0] appears to completely contradict the classical interpretation.
However, in the progression towards higher energy states shown right,
we start to see an approximation of the quantum probabilities towards
the classical blue curve. It is also highlighted that the classical
probability is confined within the maximum amplitude [+A,-A], while
the quantum red curve extend beyond this limit. However, it is usually
assumed that the oscillations within the quantum probability model only
have consequences if observed, which then causes a collapse of the wave-function.
It is then argued that the
*uncertainty principle* will prevent any practical
resolution of the distance [x] without any measurement altering the
physical state of the oscillator. This process is also described in
terms of the *wave-function collapse* when any measurement is attempted.

Note: While the uncertainty principle is usually attributed to Heisenberg, he actually preferred the word ‘indeterminacy’ rather than ‘uncertainty’. In this context, Heisenberg argued that all system observables, at the quantum level, simply have no definite value and, as such, there is not uncertainty, just an inability to assign an exact value of position [x] and momentum [p], when approaching the quantum scale.

Based on the note above, Heisenberg’s initial position might be described
in terms of a
*practical measurement approach* versus a
*conceptual
wave approach* . In terms of the conceptual approach, it is often assumed
that a matter wave is more analogous to a short pulse in that it is
confined to within in a small region [Δx] of space, which is not
related to the amplitude [A] in the oscillator model. However, Fourier
analysis allows this short pulse to be constructed as a superposition
of sine waves of many different wavelengths quantified in terms of a
wave number [k=2π/λ]. This superposition of waves when confined
to a small region [Δx], must contain a range of different spatial
wave numbers [Δk] that are described using the following relationship.

[20]

This equation is based on calculus and has nothing to do with the quantum behaviour being outlined, i.e. it applies to mechanical waves. However, we might restate the deBroglie wavelength in terms of momentum [p] and formulated in terms a small delta [Δ] change to both variables.

[21]

Now let us multiply [20] by [h-bar] and substitute for [Δk] as per [21]

[22]

At one level, [22] appears to replicate the uncertainty principle
used in the previous energy derivation. However, it is again highlighted
that the result in [22] appears to be predicated on the assumption that
the wave-particle being modelled is a superposition of multiple wavelengths,
which would ‘*propagate’* through a dispersive media, at least
mathematically, such that the wave-function conforms to the
*time dependent
solution* of
*Schrodinger’s wave equation* that results in the
*wave-function collapse* when any measurement is attempted. At this point,
the introduction of some of the basic assumptions of quantum mechanics
may appear to have created a degree of ambiguity surrounding the idea
of energy, at least, in terms of a definition predicated on the mass
[kg] of a particle.

So what other lines of inquiry might be pursued?

Well, the typical mainstream response might be to study the range
of different mathematical formulation used in the early development
of *quantum mechanics* , e.g.
*Lagrange mechanics**, *
*Hamilton
mechanics**, *
*quantum notation**, *
*wave mechanics*
and quantized wave operators *. *Further recommended
study might include the formulation of quantum mechanics in terms of
*Heisenberg’s matrix mechanics*,
*Schrodinger’s wave mechanics*
and *Dirac’s equation* covering the additional consideration
of *relativistic energy and momentum**. *Having assimilated sufficient
mathematical understanding of this framework of topics, it might then
be suggested that some appreciation of the
*philosophical implications*
of quantum mechanics is necessary, only later to be told that many of
these earlier ideas would be superseded by
*post-war quantum theory* .
Unfortunately, this second phase of the quantum learning curve will
also require further detailed study of the
*semantics* of
*particles
and fields* plus
*forces and interactions* plus further
*mathematical
abstractions*, e.g.

natural units, quantum postulates, quantum operators, bra-kets and matrices, quantum states, probability and quantum states, probability and bra-ket notation, abstract vector space, coherence states.

Persistence in the process of inquiry may eventually lead you towards
the modern overarching
*quantum field theory (QFT)* inclusive of*
*quantum electro-dynamics (QED), quantum chromo-dynamics (QCD) and
electro-weak theory (EWT), which possibly also requires an update of
*quantum philosophy* along with an extension of various subsequent
*quantum interpretations*. Finally, at the end of this long learning
curve, you may return to the original question.

Does cause and effect take place within some tangible physical reality?

Whether you believe quantum theory provides a satisfactory answer to this question may be a matter of personal preference as it is far from clear that the quantum debate regarding the epistemological versus ontological nature of reality has actually been resolved. For if quantum theory only provides a mathematical description, based on probability, rather than a physical model of cause and effect, debate and inquiry may naturally continue.