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 [EP] of the oscillator in terms of some maximum displacement [A0]. We might then proceed by assuming that the initial displacement [A0] 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 2nd 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 [f2] and amplitude [A2] 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 [f2].

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 [ET] as the sum of the potential energy [EP] and the kinetic energy [EK] 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 [ET] 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 [mc2] 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 [m0], 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 [mv2] 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 m0] becomes a relativistic function of velocity [v]. As such, we might consider an alternative interpretation based on kinetic energy [Ek=mv2], 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=mc2, there is a suggestion that each frequency [f] would be associated with a different mass [m=hf/c2]. 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 [mv2] 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 [ψ2], 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.