**1. Introduction**

338 Advanced Topics in Measurements

[16] Shande Shen, power system parameters identification, Beijing. Hydro and Power

[17] Zhenyu Huang, Dmitry Kosterev and Ross Guttromson, etc. Model Validation with

Hybrid Dynamic Simulation, IEEE 2006 Power Engineering Society General Meeting, Print ISBN: 1-4244-0493-2, Montreal, Canada, 18-22, Jun,2006,pp.1-9,

Press, 1993

#### **1.1 Two types of double slit experiments**

Quantum statistics play a key role in Quantum Mechanics QM [Feynman et al. (1965,1989); Penrose (2004)]. Two types of Double Slit Experiment are used to explore the core mysteries of quantum interactive behaviors. These are standard Double Slit Experiments with correlated signals and Single Photon Experiments that use ultra low intensity and lengthy exposures to demonstrate quanta self-interference patterns. The key significance is that intrinsic wave properties are observed in both environments [Barrow et al. (2004); Hawkingand & Mlodinow (2010)].

### **1.2 Two types of probabilities**

Multivariate probabilities acting on multinomial distributions occupy a central role in classical probability theory and its applications. This mechanism has been explored from the early days in the study of modern probability theories [Ash & Doléans-Dade (2000); Durret (2005)]. Conditional probability is a powerful methodology at the heart of classical Bayesian statistics. In the history of probability and statistical developments, there have been long-running debates and a persistent lack of agreement in differentiating between prior distributions and posterior distributions [Ash & Doléans-Dade (2000); Durret (2005)]. It is worthy of note that the uniform distributions or normal distributions of conditional probability are always linked to a relatively large number of probability distributions in non-normal conditions. This points to practical problems with random distributions.

3. analysis of key issues of QM 4. conditional construction proposed

6. analysis of visual distributions

**2.1 Heisenberg uncertain principle**

7. using the variant solution to resolve longstanding puzzles

The Heisenberg Uncertainty Principle HUP was established in 1927 [Heisenberg (1930)]. The HUP represented a milestone in the early development of quantum theory [Jammer (1974)]. It implies that it is impossible to simultaneously measure the present position of a particle while also determining the future motion of a particle or any system small enough to require a Quantum mechanical treatment. From a mathematical viewpoint, the HUP arises from an equation following the methodology of Fourier analysis for the motion [*Q*, *P*] = *QP* − *PQ* =

<sup>341</sup> From Conditional Probability Measurements to Global Matrix Representations on Variant

Construction – A Particle Model of Intrinsic Quantum Waves for Double Path Experiments

This equation shows that the non-commutativity implies that the HUP provides a physical

The HUP provided Bohr with a new insight into quantum behaviors [Bohr (1958)]. Bohr established the BCP to extend the idea of complementary variables for the HUP to energy and time, and also to particle and wave behaviors. One must choose between a particle model, with localized positions, trajectories and quanta or a wave model, with spreading

Under the BCP, complementary descriptions e.g. wave or particle are mutually exclusive within the same mathematical framework because each model excludes the other. However, a conceptual construction allowed the HUP, the BCP and wave functions together with observed results to be integrated to form the Copenhagen Interpretation of QM. In the context of double slit experiments, the BCP dictates that the observation of an interference pattern for waves and the acquisition of directional information for particles are mutually exclusive.

Bohr and Einstein remained lifelong friends despite their differences in opinion regarding QM [Bohr (1949; 1958)]. In 1926 Born proposed a probability theory for QM without any causal explanation. Einstein's reaction is well known from his letter to Born [Born (1971)] in which

Then in 1927 at the Solvay Conference, Heisenberg and Bohr announced that the QM revolution was over with nothing further being required. Einstein was dismayed [Bohr (1949); Bolles (2004)] for he believed that the underlying effects were not yet properly understood.

**2. Wave and particle debates in QM developments**

*ih*¯. The later form of HUP is expressed as �*p* · �*q* ≈ *h*.

wave functions, delocalization and interferences [Jammer (1974)].

he said "I, at any rate, am convinced that HE [God] does not throw dice."

**2.3 Bohr-Einstein debates on wave and particle issues**

interpretation for the non-commutativity.

**2.2 Bohr complementarity principle**

5. exemplar results

8. main results 9. final conclusions

#### **1.3 Advanced single photon experiments**

#### **1.3.1 Applying the bohr complementarity principle**

The Bohr Complementarity Principle BCP, established back in the 1920s brought us the foundations of QM [Bohr (1949)]. In Bohr's statement:"... we are presented with a choice of either tracing the path of the particle; or observing interference effects ... we have here to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements." It is significant that BCP provided a powerful intellectual basis for Bohr in key debates in the history of QM and especially in his debates with Einstein [Jammer (1974)].

#### **1.3.2 Testing bell inequality**

To help decide between Bohr and Einstein on their approaches to wave and particle issues, Bell proposed a set of Bell-Inequations in the 1960s [Bell (1964)]. In 1969, CHSH proposed a spin measurement approach [Clauser et al. (1969)] and experiments by Aspect in 1982 did not support local realism [Aspect et al. (1982)].

#### **1.3.3 Afshar's measurements**

In 2001 Afshar set up an experiment to test the BCP [Afshar (2005)]. This experiment generated strong evidence contradicting the BCP, since both particle and wave distributions can be observed simultaneously. In Afshar's experiments, there are four measurements: *ψ*<sup>1</sup> - signals via left path, *ψ*<sup>2</sup> - signals via right path, *σ*<sup>1</sup> - interactive measurements of {*ψ*1, *ψ*2} on the distance of *f* , and *σ*<sup>2</sup> - separate measurements of {*ψ*1, *ψ*2} on the distance of *f* + *d* respectively. In this experiment, a measurement quaternion is

$$
\langle \psi\_1, \psi\_2, \sigma\_1, \sigma\_2 \rangle. \tag{1}
$$

#### **1.4 Current situation**

From the 1920s through to the start of the 21st century, there was no significant experimental evidence to show that there were problems with the BCP. However, Afshar's 2001 experimental results are clearly not consistent with the BCP and further experimental results have provided solid evidence against the BCP. [Afshar (2005; 2006); Afshar et al. (2007)]. It is interesting to see that neither local realism nor the BCP are validated by the results of modern advanced single photon experiments [Afshar et al. (2007); Aspect (2007)]. It will be a major challenge in this century to redefine the principles on which the quantum approach may now be safely founded.

#### **1.5 Chapter organization**

Following on from multivariate probability models, this chapter focusses on a conditional approach to illustrate special properties found in conditional probability measurements via global matrix representations on the variant construction. This chapter is organized into nine sections addressing as follows:


2 Measurement Systems

The Bohr Complementarity Principle BCP, established back in the 1920s brought us the foundations of QM [Bohr (1949)]. In Bohr's statement:"... we are presented with a choice of either tracing the path of the particle; or observing interference effects ... we have here to do with a typical example of how the complementary phenomena appear under mutually exclusive experimental arrangements." It is significant that BCP provided a powerful intellectual basis for Bohr in key debates in the history of QM and especially in his debates

To help decide between Bohr and Einstein on their approaches to wave and particle issues, Bell proposed a set of Bell-Inequations in the 1960s [Bell (1964)]. In 1969, CHSH proposed a spin measurement approach [Clauser et al. (1969)] and experiments by Aspect in 1982 did not

In 2001 Afshar set up an experiment to test the BCP [Afshar (2005)]. This experiment generated strong evidence contradicting the BCP, since both particle and wave distributions can be observed simultaneously. In Afshar's experiments, there are four measurements: *ψ*<sup>1</sup> - signals via left path, *ψ*<sup>2</sup> - signals via right path, *σ*<sup>1</sup> - interactive measurements of {*ψ*1, *ψ*2} on the distance of *f* , and *σ*<sup>2</sup> - separate measurements of {*ψ*1, *ψ*2} on the distance of *f* + *d*

From the 1920s through to the start of the 21st century, there was no significant experimental evidence to show that there were problems with the BCP. However, Afshar's 2001 experimental results are clearly not consistent with the BCP and further experimental results have provided solid evidence against the BCP. [Afshar (2005; 2006); Afshar et al. (2007)]. It is interesting to see that neither local realism nor the BCP are validated by the results of modern advanced single photon experiments [Afshar et al. (2007); Aspect (2007)]. It will be a major challenge in this century to redefine the principles on which the quantum approach may now

Following on from multivariate probability models, this chapter focusses on a conditional approach to illustrate special properties found in conditional probability measurements via global matrix representations on the variant construction. This chapter is organized into nine

�*ψ*1, *ψ*2, *σ*1, *σ*2�. (1)

**1.3 Advanced single photon experiments**

support local realism [Aspect et al. (1982)].

respectively. In this experiment, a measurement quaternion is

with Einstein [Jammer (1974)].

**1.3.2 Testing bell inequality**

**1.3.3 Afshar's measurements**

**1.4 Current situation**

be safely founded.

**1.5 Chapter organization**

sections addressing as follows: 1. general introduction (above)

2. key historical debates on the foundations of QM

**1.3.1 Applying the bohr complementarity principle**

9. final conclusions
