# Wave Function Collapse

We have already touched on the subject of the wave function collapse;
first in the terms of the double-slit experiment and then in terms of
the dispersion of the wave function. In the case of the ‘*double-slit
experiment’ *, it was suggested that some aspect of a single photon,
or electron matter wave, must pass through both slits in order to produce
an interference pattern; subject to the following caveats. While ‘*passively*’
observing the process, the wave function continues to represent a probability
distribution, which is consistent with a wave-like interference pattern.
However, if any attempt is made to determine which slit, e.g. A or B,
the ‘*wave-**particle*’ actually passes through, the interference
pattern disappears, i.e. the wave-like function collapses.

In the case of the ‘*time evolution of the wave function’ *we
modelled the particle-wave packet as a superposition of plane waves,
where each wave had to propagate with a different velocity. As a result,
the probability density of the wave packet, which was initially localised
in space, becomes increasingly dispersed as a function of time. Of course,
if we reverse this logic, we might reasonably ask:

*How does the wave packet probability ever get localised as a particle?
*

Unfortunately, in order to address this question, we have to also
consider the larger issue of the wave function collapse, which we might
introduce as a process where the quantum probability density collapses
back into classical certainty. However, the nature and cause of the
wave function collapse continues, to this day, to be an issue of intense
debate in terms of both its scientific and philosophical implications.
As such, the scope of following discussion will expand somewhat beyond
the ‘*Pre-War Years’*, so as to consider some of the wider implications
that have subsequently arisen. However, we will begin by questioning
the scope of the quantum model:

*What exactly is quantum theory telling us about the nature of
existence? *

The field of quantum theory has developed a number of competing interpretations, not yet discussed, which all apparently agree with experimental data, even though they must differ in some details. We might also question what these quantum models are actually telling about the underlying nature of physical reality. However, for the purposes of this discussion, we shall try to rationalise the scope of the debate into one of two philosophical positions:

**The Ontological Position:**- The wave function has some form of physical existence
- It exists independent of any observer
- It aligns to Schrödinger’s
wave description

**The Epistemological Position:**- The wave function
is not
*‘real*’ and has no physical existence - It is a just mathematical construct for determining probability
- It aligns to Born’s probability density description

- The wave function
is not

In the current context, we might define the ontological position
in terms of the nature of physical existence, while the epistemological
position relates more to the acquisition of knowledge, which might then
be abstracted from physical existence. Initially, within the ‘*Pre-War
Years’ *, the accepted position came to be defined in terms of the
‘*Copenhagen Interpretation’*, primarily based on Neils Bohr’s
weight of authority, which we shall align more to the epistemological
position based on the physical ambiguity of the following aspects of
quantum mechanics:

- The wave probability amplitude collapses upon measurement.
- Physical particles are always detected at localized positions
- There is no obvious wave-like dispersion of matter.

However, in recent years, the opinion within this debate has possibly
become more circumspect, such that it might be useful to consider the
ontological position in a little more detail, as it possibly holds out
more *'hope*' for some better insight to the nature of quantum reality. In the jargon
of quantum physics, we might characterise a quantum system in terms
of its state vector |Ψ>, which in-turn might be said to consist
of an infinite number of ‘*basis vectors |ψ _{i}>
*’. Each basis vector can then be characterised as representing just
one possible outcome of measuring the system given the caveat that each
possible outcome is different. For example, when the system is subject
to no measurement, the system continues to evolve as described by

*Schrodinger’s wave equation*as a function of time. However, when the system is subject to measurement, it is always found to be in a state described by just one of the basis vectors. Within the limitation of this description, we might try to characterise an inconsistency; for while it might be assumed that any measurement would be representative of the total state vector, as predicted by Schrodinger’s wave equation, it turns out to only represent a single basis vector. For example, the double slit experiment seems to suggest that the system, immediately prior to measurement, reflects the totality of all quantum basis states, while immediately after measurement, only one basis state is realised. In this context, the reduction of the state vector is referred to as the wave-function collapse, which we might try to quantify in terms of the following question:

*What is the role of observation and measurement in determining
the outcome of physical reality? *

Embedded in this question is the suggestion that the observer’s participation in the measurement process might play an active role in the outcome. Although, it has to be said that, in practice, there can be some considerable ambiguity in defining where the observer and measurement systems ends and where the quantum system itself starts. Therefore, let rephrase the question above in a different way:

*How may we characterise the state of the system before and after
measurement? *

We have assumed that prior to some measurement of the quantum system,
it exists as a superposition of basis states, which then immediately
collapses after measurements to just one of many potential basis state. As such, we might
characterise this state change in terms of many ‘*wave*’ states
collapsing into one ‘*particle*’ state. If we follow this line
of argument, the concept of any reality in quantum mechanics is represented
by ‘*wave*’ states rather than *‘particle’* states of existence.
While this statement makes no change to the mathematics, it could be
argued that it does help clarify the scope of *‘reality’* before
and after measurement. For example,
*in a previous discussion*,
we estimated the dispersion times for an electron and a 1g marble:

*Electron Dispersion Time: 2.75*10 ^{-16 }seconds
Marble Dispersion Time: 3*10^{23} seconds *

If we accept the description of quantum mechanics, the implication of quantum dispersion seems to suggest that fundamental particles, like electrons, exist in a wave state that represents the sum of all basis vectors or superposition waves, depending on your preferred description. In contrast, the 1g marble maintains its particle state, representative of just one basis vector, even though its composite structure might still be described in terms of a superposition of quantum waves. However, while we intuitively accept our perception of physical reality in the case of the marble, the notion of a wave state, as described by quantum mechanics is far more difficult to reconcile.

*So how might we fit a photon within this description?*

At one level, it would seem that the idea of a photon is coupled
to the wave-particle debate as much as an electron, although we might
recognise that there are some key differences. As discussed, it was
the Einstein’s work into the *photoelectric effect*, in 1905, which
renewed the classical debate as to whether light had a wave-like or
particle-like nature. However, in many ways, the idea of photon never
fitted the classical description of a particle, as it was essentially
a definition of a quantum of energy, although its localisation in space
appears to remain an ambiguous issue to this day. If we now introduce the concept
of relativity into this discussion, it might be argued that within the
frame of reference of a photon, elapsed time does not exist:

[1]

So, according to relativity, a photon in vacuum must move at the speed of light [c], with respect to any observer, and therefore its proper time is always zero. If we reverse this description, it means that irrespective of the elapsed time in any observer’s spacetime, the elapsed time of the photon within its own frame of reference is zero, which might lead us to another question:

*Does a photon really exists in spacetime?*

We might also consider whether any attempt to measure a photon only
result in the collapse of any implied wave function, although without
rest mass, it would be unable to exist in a ‘*particle*’ state
and would simply ceases to exist. This said, some consideration of its
energy must be accounted for within the observed system as a whole.
However, based on this logic, we might come to question whether any
object, in its quantum ‘*wave*’ state, fully exists in the spacetime
of any given observer. If so, the transition from unobserved ‘*wave*’
state to observed *‘particle’ *state, as linked to the wave function
collapse, might also be described in terms of a wave function collapse
into the observer’s spacetime. However, based on the dispersion times
given above for an electron and a marble, it would seem that larger
composite quantum systems remain anchored in normal spacetime, while
fundamental particles tend to revert back into a ‘*wave*’ state,
which are possibly subject to quantum rules that may operate outside
normal spacetime. Of course, while the terminology introduced above
may help visualise the possible nature of the wave-duality duality and
the role of the wave function collapse, it is unclear that it addresses
the more fundamental question:

*Is quantum mechanics a description of physical reality or just
a useful mathematical construct? *

As a personal perspective, it seems that quantum mechanics, as discussed
so far, fails to provide sufficient insight to the true nature of any
physical existence on the quantum scale, which is possibly why science
has continued to search for alternative paradigms, such as *
string
theory*. Of course, some might question whether there can be any
tangible existence, which might be described as ‘*physical*’ at
the quantum level. However, it would seem that this is really an issue
of ‘*substance*’ rather than necessarily suggesting that everything
is an illusion, because if the latter were true, the concept of science might cease
to have any real meaning.

*So what can be inferred as physical or tangible substance?*

Based on the building blocking blocks of quantum theory, Planck associated
a wave with energy and Einstein then associated energy with matter,
although we must eventually question the ‘*substance*’ of any fundamental
particle like the electron:

[2]

The common denominator in both these equations in energy [E], which is a concept with its own set of ambiguities. First of all, energy is a scalar quantity that comes with no explicit description of how it moves from one place to another in spacetime. Of course, if we revert to the most fundamental definition of energy, i.e. potential and kinetic, we might described the movement of energy associated with the movement of mass or by its transport within a wave. If we accept that the concept of a particle is questionable at the quantum level, then some form of wave appears to be the only known mechanism by which energy can be transported in spacetime. However, if a fundamental particle, such as an electron, is known to have rest mass energy, kinetic energy and potential energy within some gravitational field, it is unclear whether quantum mechanics can describe how this energy is distributed across the quantum superposition model. If so, we need table one final question within this discussion:

*If quantum mechanics is only a logical
mathematical construct, can it ever be forwarded as a description of
any form of underlying
physical reality? *

In the context of this question, we might now realise why the debate
between the ontological and epistemological positions has continued
for so long. The epistemological position, as associated with original
*Copenhagen interpretation*, may simply reflect a recognition that while
quantum mechanics can produce very accurate predictions, it appears
to fail to
fully explain these predictions; although we should possibly note that
some people believe that any deeper understanding beyond quantum mechanics
is impossible. In contrast, the ontological position philosophically
seeks an understanding of the nature of being, i.e. existence, and therefore
rejects the assumption that our present understanding, based on quantum
mechanics, must forever defined the limits of our understanding of any
deeper reality, assuming that it does exist.