Exploring quantum entanglement through Bell theorem
To complete my analysis on Bell’s theorem (here and here), I designed an experiment to directly observe the effects of quantum entanglement and correlation as outlined in quantum mechanics theory. This experiment, based on principles derived from Bell’s theorem, aimed to test the quantum mechanical predictions through empirical evidence, gathered using a meticulously designed setup involving entangled particles.
Experimental setup
The core of the experimental apparatus consisted of three parts: two detectors positioned at a considerable distance from each other and a source located midway between them. The placement of the source was crucial for ensuring the emitted particles had an equal probability of being detected by either unit, facilitating the examination of entanglement and quantum correlation effects.
Each detector was equipped with a switch that could be set to one of three positions. Upon detecting a particle, the detectors illuminated a light—red or blue—reflecting the particle’s spin orientation as it interacted with the magnetic field within the detector.
Procedure and observations
During the experiments, each detector’s switch was independently and randomly set to one of the three positions. The source then emitted particles, and the detectors responded by displaying a light color corresponding to the detected spin state. This process was repeated multiple times to gather a substantial dataset.
Two key observations emerged from the experimental runs:
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Consistency in Identical Settings: When the switches on both detectors were set to the same position, the detectors invariably flashed the same color, indicating a strong correlation under uniform conditions.
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Randomness in Varied Settings: In runs with different switch settings, the pattern of the lights was entirely random—flashing the same colors 50% of the time and different colors 50% of the time.
These findings are significant as they demonstrate the influence of the switch settings on the experimental outcomes, corroborating the predictions of quantum mechanics.
Mathematical analysis
Using the principles of quantum mechanics, particularly the properties of particles in a singlet state, I could calculate the probabilities consistent with these observations. In a singlet state, the total spin of the system is zero, which means the spins of the individual particles are entangled and always opposite in orientation. The mathematical formula governing the probabilities is:
P = \left(\cos(120^\circ)\right)^2 = \frac{1}{4}
Combining these probabilities for various detector settings:
P_t = \frac{1}{3} \times 1 + \frac{2}{3} \times \frac{1}{4} = \frac{1}{3} + \frac{1}{6} = \frac{1}{2}
This calculation showed that, overall, the probability that the detectors would flash the same color was 50%, exactly as observed.
Quantum mechanic description
The source in the setup emitted pairs of spin-½ particles in a singlet state. The wavefunction characterizing this state is:
\psi = \frac{1}{\sqrt 2}(| ud \rangle - | du \rangle)
When measurements are taken with the detectors set at an angle of 120° relative to each other, the theoretical prediction for the probability amplitude of the correlated outcomes is:
E_{\mathbf a \mathbf b} = -\cos( \theta_{\mathbf a \mathbf b})
Selecting an angle of ±120°:
E_{\mathbf a \mathbf b} = - \cos\left(\pm \frac{2}{3}\pi\right) = \frac{1}{2}
So, the probability of correlated outcomes is:
P = \left(E_{\mathbf a \mathbf b}\right)^2 = \frac{1}{4}
Conclusion
The results of this experiment not only validate the quantum mechanical model of particle entanglement and spin correlation but also enrich our understanding of quantum correlations in a controlled experimental setting. The data conclusively demonstrated the inherent randomness and deep-seated correlations dictated by the quantum mechanical framework.
For more insights into this topic, you can find the details here.