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wave function


What is Quantum Reality?

-:Copenhagen Interpretation:-

This is the dominant view within the Physics community, and for good reason, however it is very difficult to actually find a definitive statement of exactly what the Copenhagen Interpretation is. In fact, there are multiple subtle versions of it. The main variant historically is presented below.....
..........The Copenhagen Interpretation is a Positivist interpretation. The quantum world can only be examined by constructing machines that perform measurements. Quantum Mechanics is a probability calculus which can be applied to a specific combination of measuring devices and quantum system. The waveform is not "real"; it is a mathematical construction that represents the observer's knowledge of the system. No deeper level of understanding is possible......
-:Measurement & the Measuring Devices:-
This is a really tough question. We shall go with the following definition, although it is obviously incomplete and is revised later in the document: A measuring device is a thing which produces a real valued outcome to an experiment. i.e Measuring devices measure "quantities" such as position, momentum, energy, charge etc whose values are real numbers. Measurements never return complex numbers. To avoid confusion with other uses of the word "real", we shall use to indicate the set of real numbers and the term -valued to indicate a quantity which takes a value which is a real number.
..............How can we tell what we are measuring? Bohr's view was that measuring devices are "essentially classical" in the sense that (1) they deal with classical concepts such as position, momentum etc. and (2) the examination of their macroscopic structure determines what they measure. There are problems with this view. (See below) Whatever it's shortcomings, this is the standard view repeated ad nauseum in text books. We shall have more to say on this topic later....................
-:Born's Probability Interpretation:-
What interpretation can be attached to wave function? The square of the amplitude of the waveform yields the probability density function. This is consistent with experiment (e.g. Diffraction experiments where single photons can be detected) and is not disputed by any of the interpretations (Hidden Variable Theories view the probability description as incomplete). The notion is generalized to the norm of the state vector in a Hilbert space.

-:Heisenberg's Uncertainty Principle (HUP):-
The Copenhagen Interpretation objects to statements such as "It is not possible to measure the momentum of an electron without disturbing it's position". HUP makes no statement about "disturbing" anything, only that simultaneous measurements are not possible. In fact, as we shall see, the position of an electron may not even be well defined. HUP is a consequence of the mathematical structure of Quantum Mechanics and not directly challenged by any interpretation.
--:Observer Created Reality:--
Copenhagen does NOT say that the observer creates reality, whatever that means. It does not say that the world is a purely mental construct (Idealism).

Copenhagen does say that a property of a system however may not be defined until it is measured. For example, if an electrons waveform is smeared out over space, it's position is not just unknown but undefined. The electrons position is only defined once it is measured. Copenhagen treats the experiment and measuring apparatus as a gestalt (whole). The interested reader is referred to the later section on Philosophy.

Bohr's Complementary Principle or Wave-particle Duality:-
Quantum particles are said to exhibit a wave-particle duality. Bohr's Complementary Principle says that it is possible to design an experiment to show the particle nature of matter, or show the wave nature of matter, even though each picture of matter is mutually exclusive.
..............For example, Double Slit Diffraction using a single photon at a time demonstrates the wave-nature of light as the light moves from the source through the diffraction grating and builds up a diffraction pattern on the final screen. The particle nature of light is demonstrated by arrival of discrete packets of energy (photons) at the final screen.

Bohr’s Complementary Principle is at the heart of the Copenhagen Interpretation. Matter is neither a wave nor a particle; each picture is complementary not contradictory. It is the nature of the experiment which "chooses" which picture (or aspect) of matter is demonstrated.

The Principle is much more subtle and deep than it first appears; it is best understood by reference to Kantian or Positivist Philosophy. We know the world through our senses, or experimental devices. What we know about the world is as much a product of the devices we use to interact with the "real world" as the "real" world itself, which ultimately lies beyond our senses and essentially unknowable.

The author recommends that the reader re-visit this topic after reading the sections on Philosophy in the rest of this document.

Is the Waveform real?
The standard version of Copenhagen is clear on this issue. Consider the following statement by Bohr.

There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.

The Copenhagen Interpretation regards the waveform as not real. The alternative interpretation which accepts the same probabilistic world-view as the Copenhagen Interpretation but regards the waveform as real will be referred to as the State Vector Interpretation.

Waveform Collapse
The Copenhagen Interpretation makes no comment on waveform collapse. The waveform is viewed as having no reality and is only an aid to calculating probabilities. Consider the following experiment:

A coin is to be thrown from the dark onto a table top. The observer cannot see the coin so the best description he can produce is

Y = ( |heads> + |tails> )/Ö2
The coin tumbles onto the table and settles heads up. The best description of the coin is now

At no time did the description effect the outcome of the experiment and so in this sense it has no reality and the "collapse" of the waveform Y holds no mystery.

The coin example can also be misleading. We are quite happy to accept the "non-reality" of the waveform because we "know" the coin's position and momentum are in well defined states that could be measured, and if the appropriate calculations performed, we could determine in advance which way up the coin would land. The Copenhagen Interpretation on the other hand says the waveform is everything that is knowable about the experiment.


1.2 State Vector Interpretation
Synopsis: This interpretation is the interpretation usually implicitly presented in many physics text books. It accepts the Born Probability Interpretation and the belief that the ultimate description of the universe is probabilistic. The main difference from Copenhagen is that quantum systems are viewed as being in a state described by a wave function which lives longer than any specific experiment. The waveform is considered to be an element of reality. Measurement implies the collapse of the wave function. Once a measurement is made, the wave function is no longer smeared over many possible values, instead the range of possibilities collapses to the known value. Unfortunately the nature of the waveform collapse is problematic and the equations of Quantum Mechanics do not cover the collapse itself.


1.3 Consciousness Causes Collapse
Synopsis: It accepts most of the State Vector Interpretation except measuring devices are also regarded as a quantum systems. This has some interesting consequences:
If the measuring device is a quantum system, the measuring device changes state when a measurement is made but it's wavefunction does not collapse. So when does the waveform collapse? See The Measurement Problem below. The collapse of the wave function can be traced back to its interaction with a conscience observer. Seriously scary!!!


1.4 Many-worlds Interpretation
Synopsis: All possible outcomes are regarded as "really happening" and somehow we "select" only one of those realities (universes). Real SF stuff. Also untestable unless someone comes up with a way of moving between universes. First formulated by Everett.

If |a> and |b> are possible outcomes, then so is c|a> + d|b> for any c,d where |c|2 + |d|2 = 1 so the number of alternate universes is infinite. Since "every possible outcome" is viewed as happening, the many world interpretation implies universes where the laws of physics hold but every toss of a coin has resulted in "tails"! Universes "interfere" with each other "smearing" our universe at the quantum level. Making a measurement "selects" one of these universes.

Proponents of this theory suggest that the many-worlds interpretation of Quantum Mechanics is now the dominant view within the physics community. If true, this is undoubtly due to similarities between many-worlds and Feynman's approach (See below).


1.5 Hidden Variable Theories
Synopsis: Usually proponents have philosophically objections to what Einstein would call "God playing dice". It views electrons and other quantum objects as having properties with well-defined values that exist separately from any measuring device. (e.g. position and momentum) Quantum Mechanics is viewed as a high level statistical description of the underlying theory. Deterministic theories imply that the fate of universe was determined at creation and free will is an illusion. This view was favoured by Einstein and others.

The terminology "Hidden Variable Theories" is misleading. Many Quantum theories have "hidden" variables within their formulation. A better term would be "Deterministic theories". However we maintain what has now become a universally accepted convention and will also continue to call such theories Hidden Variable theories.

Hidden Variable Theories may or may not consider the waveform to be an element of reality. I.e. It may be purely a statistical creation, or it may have some phyiscal role (e.g. Bohm).


1.6 Transactional Interpretation
Synopsis: The statistical nature of the waveform is accepted but the waveform is broken up into an "offer" wave and "confirmation" wave, which both are "real". Quantum particles interact with ...
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