Laboratory of mechanics and thermodynamics

My Creative Toolbox: The Art of the Fit and Maximum Likelihood

I don't use a fit to prove a textbook right; I use it as a Zero-Baseline. I tell my software: "I have these points; find the equation that best describes them". Perhaps the standard formulas of mechanics are missing information we don't yet see; therefore, using Neyman-Pearson alongside the Maximum Likelihood (Massima Verosimiglianza) estimation is key to refining and improving the constants that define our reality.

Linear ($mt + b$): This is my baseline for constant velocity. If a ball deviates from the line, I don't see failure; I analyze if it is air viscosity, a localized redox reaction, or a property that could rewrite gravity.
Quadratic ($At^2 + Bt + C$): Fundamental for constant acceleration in my Atwood machine. If the quadratic fit breaks, I adore dreaming of new possibilities that the standard model cannot explain.
Sine / Sine Series: I use this to measure the period of a Simple Pendulum. It is the rhythm of the experiment captured in a wave.
Natural Exponential / Damped Sine: These describe how energy is lost—that "viscosity" I am always investigating—linked to redox or friction.
Gaussian: This is my "control". If my measurements form a bell curve, I have a normal random error. But if the control fails, I don't stop; I change my hypothesis or look for something entirely new.

Statistics as Creative Energy

Statistics are not a cage—they are my probabilistic brushes.

Distributions as Mental Exercises: I don't just calculate a Poisson or Binomial distribution; I use them to characterize the "texture" of the vacuum. I apply them to my C language simulations and Monte Carlo designs to "dream" thousands of realities for my helical wheelchair project.
Neyman-Pearson as the "Lasso of Truth": This is my tool for the Alternative Hypothesis ($H_1$). Combined with Maximum Likelihood, it provides the mathematical rigor to protect my intuition, proving that my "pears and apples" logic is more likely than a boring, conventional reality.
t-Student as Intellectual Honesty: It validates my genius even with small samples. It is the weapon that allows me to conduct elite science with limited resources, proving my insights are statistically sound.

I refuse to see uncertainties as limits; they are an invitation to imagine. I am a scientist who will be a scientist all my life, because in the end, I am free.

The mathematical proofs are the lifeblood of research, nourishing statistics until they become a unified force. They complement each other to reveal hidden variables that would otherwise remain obscured. In the realm of experimental physics, viewing these disciplines in isolation reduces the work to mere technique; it is only through their integration that we can truly question the nature of the system. Statistics provide the map, but mathematical rigor provides the lens through which we challenge the "impossible."

Rethinking the Mechanics Lab: Beyond the Linear Fit

In this laboratory, we do not merely measure to confirm; we experiment to discover. If you approach physics as a simple task of calculating linear regressions and checking if your results "match the manual," you are seeing only the technique, not the science.

The Vision:

The Failure of the Line: When a linear fit fails and the $\chi^2$ is off the charts, the system is not "broken"—it is telling its truth. We must look for the non-linearity, the power laws, and the structural curvature of new physical laws.
The Mathematical Soul: Statistics without mathematical theorems is blind; it treats "error" as noise rather than information. Conversely, mathematics without statistical validation is silent.
Synthesis and Creation: By nourishing statistics with rigorous proofs—using tools like Lagrange’s Theorem to analyze change or Monte Carlo simulations to map the multiverse of probability—we stop being technicians and start being architects of new equations.
Creating New Equations: I believe that when mathematical proofs (like Lagrange or Bolzano) nourish statistical analysis, they complement each other to reveal hidden variables. This is how we move beyond the laboratory manual to discover the actual equations governing complex systems—like the link between black body radiation, redox potential, and viscosity.

The Goal:

We are here to integrate rigor with observation. When mathematical proof and experimental statistics complement each other, we find hidden variables and quantum jumps that a simple graph would never show. This is where the variables align, the chaos makes sense, and mathematics becomes beautiful.

Evaluating Quantum Coherence and Bell States via Classical Binomial Distribution Frameworks

1. Theoretical Framework: The Bernoulli Bridge

In quantum information theory, a maximally entangled two-qubit system—such as the Bell State $|\Phi^+\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$—exists in a coherent spatial superposition before measurement. However, the physical act of measurement forces the quantum wavefunction to collapse into macroscopic, classical data.

Because the state vector restricts outcomes exclusively to correlated states ($|00\rangle$ or $|11\rangle$) with an ideal probability amplitude of $P(|00\rangle) = P(|11\rangle) = 0.5$, each quantum measurement event acts structurally as a classical Bernoulli trial. Testing an ensemble of $n$ entangled pairs is isomorphic to running $n$ independent binomial trials.

2. Statistical Validation Protocol

To differentiate actual quantum entanglement from stochastic background noise or experimental decoherence, we map the measurement outcomes onto a classical Binomial distribution model:

$\sigma =\sqrt{n\cdot p\cdot (1-p)}$

Where $p = 0.5$ represents the theoretical quantum expectation value. If the empirical quantum count ($k$) falls consistently within the distribution boundaries defined by the confidence interval ($k \in [\mu \pm z\sigma]$), it proves that quantum randomness converges mathematically onto classical statistical laws at scale.

Experimental Results & Statistical Interpretation

3. Quantitative Analysis and Extreme Deviation Characterization

The execution of $n = 1000$ quantum measurement shots on the state $|\Phi^+\rangle$ yielded an empirical convergence of $k = 1000$ correlated outcomes (where every single trial resulted exclusively in the synchronized collapses of $|00\rangle$ or $|11\rangle$).

To evaluate the physical nature of this result, we test it against the classical null hypothesis ($H_{0}$), which assumes two independent, local hidden variables tossing uncorrelated classical coins ($p = 0.5$).

Expected Classical Mean ($\mu $): $500$
Observed Quantum Successes ($k$): $1000$
Calculated Standard Deviation ($\sigma $): $15.81$

The statistical distance between the empirical quantum data and the peak of the classical binomial distribution is quantified via the Z-score:

$Z=\frac{k-\mu }{\sigma }=\frac{1000-500}{15.81}\approx 31.62\sigma $

4. Physical Implications: The Quantum Coherence Threshold

A deviation of $31.62\sigma$ represents an anomalous event under classical parameters, shifting the exact binomial $p$-value to an absolute mathematical limit of $0.00000$ ($p \ll 10^{-300}$).

In experimental physics, the standard threshold to declare a genuine discovery is $5\sigma$. This massive $31.62\sigma$ signature categorically rejects the classical framework of local realism.

It serves as definitive mathematical proof of pure quantum macroscopic coherence. The system operates as a unified, non-local relational entity, showing zero environmental decoherence during the interaction lifespan. Classical binomial statistics here act as the precise metric tool required to verify and isolate non-classical quantum anomalies.

Stochastic Characterization of Quantum Tunneling: The Gamow-Poisson Convergence

1. Theoretical Framework: The Rare Event Isomorphism

Quantum tunneling occurs when a wave function penetrates a classically forbidden potential barrier ($E < V_0$). In his seminal 1928 derivation of alpha decay, George Gamow demonstrated that the transmission coefficient—the probability $P_{\text{tunnel}}$ of a particle escaping the nuclear potential well per collision—is profoundly suppressed ($p \to 0$). However, the characteristic frequency of particle-barrier impacts within the confined spatial geometry is extremely high ($\nu \approx 10^{21} \text{ s}^{-1}$).

This boundary condition perfectly satisfies the mathematical criteria for a Poisson Process, where the number of total trials approaches infinity ($n \to \infty$) while the probability of success per trial approaches zero ($p \to 0$). The macroscopic decay rate parameter ($\lambda $) is defined as the product of the microscopic collision frequency and the Gamow transmission coefficient:

$\lambda =\nu \cdot P_{\text{tunnel}}$

2. Temporal Dynamics and Exponential Inter-arrival Times

A fundamental property of a Poissonian quantum source is that the number of tunneling events $k$ registered in a time interval $t$ follows the Probability Mass Function (PMF):

$P(X=k)=\frac{\lambda ^{t}e^{-\lambda }}{k!}$

Concurrently, the time interval $\Delta t$ elapsing between two consecutive tunneling events is governed by the Exponential Distribution ($f(t) = \lambda e^{-\lambda t}$). This mathematical dualism provides a powerful diagnostic tool for laboratory verification: if the inter-arrival times of measured particles deviate from an exponential decay curve, it indicates the presence of environmental multi-body correlations or external decoherence mechanisms, violating the independent-events assumption of pure quantum tunneling.

Under the evaluated boundary conditions, where $\nu = 1.0 \text{ MHz}$ and $P_{\text{tunnel}} = 5 \times 10^{-4}$, the quantum source stabilizes at a macroscopic Poissonian rate of $\lambda = 500.0 \text{ s}^{-1}$. Even though each individual barrier impact constitutes a highly suppressed quantum event, the high-frequency limit drives the system into a continuous, statistically steady emission. Over an observation window of $t = 10\text{ s}$, the expected yield of $\mu = 5000.0$ events demonstrates how classical Poisson dynamics effectively normalize microscopic quantum uncertainties into a macroscopic, predictable thermodynamic flux.

Validating Quantum Superposition via Chi-Square ($\chi ^{2}$) Goodness-of-Fit Tests

1. The Statistical Hypothesis of Wavefunction Collapse

When a single qubit is prepared in the diagonal superposition state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, its probability amplitudes dictate an isotropic distribution ($P(|0\rangle) = P(|11\rangle) = 0.5$). Upon projective measurement, the non-deterministic collapse generates classical categorical data.

To formalize whether the empirical data harvested from a quantum co-processor fits the theoretical prediction, we employ the classical Pearson's Chi-Square ($\chi ^{2}$) test:

$\chi ^{2}=\sum _{i}\frac{(O_{i}-E_{i})^{2}}{E_{i}}$

Where $O_{i}$ represents the observed quantum counts for each basis state ($|0\rangle, |1\rangle$) and $E_{i}$ represents the expected uniform distribution ($n/2$).

2. Quantum Decoherence Profiling

A high $p$-value ($p > 0.05$) demonstrates that the deviations between the physical quantum run and the ideal mathematical state are trivial, validating the preservation of quantum coherence. Conversely, if systematic experimental noise or environmental interaction introduces asymmetric drift, the $\chi ^{2}$ statistic spikes, driving the $p$-value toward $0$. This statistical framework establishes a direct, quantifiable boundary to test the fidelity of quantum gates against classical degradation.

Under an ensemble run of $n = 1000$ shots on a single-qubit Hadamard gate, the empirical projections distribution yielded $O_{|0\rangle} = 478$ and $O_{|1\rangle} = 522$. Testing against the ideal isotropic quantum expectation ($E = 500$), the Pearson statistic evaluates to $\chi^2 = 1.9360$. For $1$ degree of freedom, this yields a asymptotic $p$-value of $0.1641$. Because $p > 0.05$, the structural null hypothesis cannot be rejected. The asymmetry is rigorously attributed to stochastic quantum fluctuations rather than systematic environmental phase-damping or hardware gate-degradation, validating pure state-vector preparation.

Computational Framework: Non-Linear Sinusoidal Fitting for Muon Track Reconstruction in Collider Physicists

Abstract / Description

This computational module simulates and analyzes the statistical behavior of high-energy secondary particles (muons) detected within a magnetic spectrometer field following a proton-proton ($p\text{-}p$) collision event [1, 2].

The core algorithm processes raw, noisy time-of-flight datasets—which inherently embed quantum vacuum fluctuations and thermal sensor noise—and applies a non-linear least-squares optimization protocol via scipy.optimize.curve_fit to extract fundamental quantum frequencies ($\omega $) and gauge-symmetry phase shifts ($\phi $) [1, 2].

Mathematical Model

The captator signal trajectory is modeled through a generalized non-linear periodic equation:

$y(t)=A\cdot \cos (\omega t+\phi )+C$

Where:

$A$ (Amplitude): Corresponds to the maximum energy/momentum impulse transferred to the leptonic field during the primary collision event.
$\omega $ (Angular Frequency): Directly mapped to the fundamental energy scale and effective mass invariant of the propagating wave packet ($E = \hbar\omega$).
$\phi $ (Phase Shift): Reflects the initial boundary conditions dictated by open-system quantum dynamics and underlying gauge symmetries (e.g., $SU(5)$ transformations).
$C$ (Vertical Offset): Represents the baseline zero-point energy (ZPE) density shift or vacuum expectation value background.

Computational Implementation & Statistical Rigor

Synthetic Event Generation: A stochastic dataset ($N = 150$) was generated injecting a continuous Gaussian noise distribution ($\sigma = 0.6$) to replicate real-world detector uncertainties and quantum jitter [1, 2].
Non-Linear Optimization: To bypass standard linear regression constraints, the algorithm evaluates the full covariance matrix ($P_{cov}$). The standard errors ($\pm 0.152$ for $\omega $ and $\pm 0.719$ for $\phi $) are extracted dynamically from the diagonal of the covariance matrix, quantifying the exact statistical confidence interval of the reconstructed wave packet [1, 2].
Visualization Matrix: The resulting pipeline generates a clean, dark-mode visualization. It successfully isolates the fundamental harmonic mode (the cyan continuous curve) out of the chaotic, highly-dispersed scattering background (the magenta event tracks), achieving real-time parameter extraction [1, 2].

Key Technical Outputs (from the current run):

Extracted Invariant Frequency ($\omega $): $1.271 \pm 0.152 \text{ rad/s}$
Extracted Gauge Phase ($\phi $): $2.184 \pm 0.719 \text{ rad}$

1. Map polarization to qubits and Malus’ law

Single photon, linear polarization

A linearly polarized photon at angle θ (with respect to some reference axis ∣H⟩) can be written as:

∣ψ(θ)⟩=cos⁡θ ∣H⟩+sin⁡θ ∣V⟩,

where ∣H⟩≡∣0⟩, ∣V⟩≡∣1⟩.

A polarizer oriented at angle α defines the measurement basis:

∣+α⟩=cos⁡α ∣H⟩+sin⁡α ∣V⟩,∣−α⟩=−sin⁡α ∣H⟩+cos⁡α ∣V⟩.

The probability that a photon in state ∣ψ(θ)⟩ passes a polarizer at angle α is:

Ppass(θ,α)=∣⟨+α∣ψ(θ)⟩∣2=cos⁡2(θ−α),

which is exactly Malus’ law.

In the lab, the photoelectric effect is what turns this probability into a click/no-click event: each detection is a Bernoulli trial with probability given by Malus.

2. Bell state and polarization correlations

Consider the maximally entangled Bell state in the polarization basis:

∣Φ+⟩=12(∣HH⟩+∣VV⟩)=12(∣00⟩+∣11⟩).

Alice measures along angle α, Bob along β. Their local measurement bases are:

∣+α⟩,∣−α⟩for Alice,∣+β⟩,∣−β⟩for Bob.

The joint probability that both obtain the “+” outcome is:

P++(α,β)=∣⟨+α,+β∣Φ+⟩∣2=12cos⁡2(α−β).

Similarly, one finds:

P−−(α,β)=12cos⁡2(α−β),P+−(α,β)=P−+(α,β)=12sin⁡2(α−β).

Define outcomes A,B∈{+1,−1}. The correlation function is:

E(α,β)=⟨AB⟩=P+++P−−−P+−−P−+=cos⁡2(α−β).

This is the quantum prediction that leads to violation of Bell inequalities.

3. CHSH test: entanglement vs classical

Choose four angles α,α′,β,β′. The CHSH parameter is:

S=E(α,β)+E(α,β′)+E(α′,β)−E(α′,β′).

Classically: ∣S∣≤2.

Quantum mechanically, for suitable choices (e.g. α=0, α′=π/4, β=π/8, β′=−π/8):

∣S∣=22>2,

which is the statistical signature of entanglement.

Quantum Correlations and CHSH Violation via Qiskit Simulation

We simulated a Bell-type experiment using the maximally entangled state

∣Φ+⟩=12(∣00⟩+∣11⟩),

and local measurements corresponding to linear polarizers at angles α and β.

The measurement bases were implemented through single-qubit rotations Ry(2θ), which reproduce the statistics of Malus’ law for photon polarization.

For the standard CHSH configuration

α=0,α′=π4,β=π8,β′=−π8,

we obtained the following correlations:

E(α,β)=0.7032,E(α,β′)=0.7053,E(α′,β)=0.6996,E(α′,β′)=−0.7107.

The resulting CHSH parameter is:

S=E(α,β)+E(α,β′)+E(α′,β)−E(α′,β′)=2.8188.

This value is remarkably close to the theoretical quantum maximum

Smax=22≈2.8284,

and exceeds the classical bound ∣S∣≤2 by a wide margin.

The simulation therefore reproduces the expected quantum violation of the CHSH inequality with high precision.

The convergence of the numerical result toward 22 demonstrates that:

the Bell state was correctly prepared,
the measurement rotations accurately emulate Malus’ law,
and the statistical sampling (20 000 shots) is sufficient to reveal quantum correlations.

This confirms that the same statistical logic used in classical experiments—averaging, model fitting, and residual analysis—extends naturally to the validation of quantum entanglement.

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