Understanding Bell's Inequality and Its Violation in Quantum Mechanics

Bell's Inequality is a cornerstone of the theory of quantum mechanics, providing a profound insight into the nature of reality at the quantum level. This article explores the concept of Bell's Inequality, its mathematical formulation, and its violation in experiments involving entangled particles.

Introduction to Bell's Inequality

Bell's Inequality can be beautifully explained using principles from quantum mechanics, particularly in the context of entangled particles. In quantum mechanics, it is believed that when electrons pass through a magnetic field, half of them get deflected to the right while the other half gets deflected to the left. This behavior can be intuitively understood in a setting where students take a yes-no test and are cheating off a third student.

Theoretical Basis of Bell's Inequality

Consider three students, A, B, and C, taking a yes-no test. If Student A and Student C are both cheating off Student B, and Student A is 99% correlated with B, and C is 99% correlated with B, then we expect Student A and C to be highly correlated, at least 98%. This correlation is based on the simple observation that if there are 1000 questions, with 10 different between A and B, and 10 different between B and C, then at most 20 different questions might exist between A and C.

Mathematical Formulation

The general statement regarding pre-determined answers to yes-no questions can be mathematically expressed as:

1 - CAC 1 - CAB - 1 - CBC

Where CXY is the degree of correlation between X and Y, a real number between 0 and 1. This is the essence of Bell's Inequality. Despite its apparent simplicity, Bell's Inequality has profound implications when applied to quantum mechanics.

Quantum Mechanics and Bell's Inequality

When you measure the spin of entangled particles far apart and you measure in the same direction, you always get the same answer (correlation). This is in stark contrast to the classical scenario, where you would expect the correlation to be lower. In a typical setup, if angles A, B, and C correspond to the directions of measurement, particle 1 and particle 2 have the same answer on their internal crib sheets.

Experimental Setup

If you measure the spin of particle 1 along A and particle 2 along B, and the angles A and B are close, you can make A and B 99% correlated. Similarly, if you measure the spin of particle 1 along A and particle 2 along B, you can also make B and C 99% correlated. Thus, if you measure A on particle 1 and B on particle 2, you will get the same answer 99% of the time. However, if you measure A on particle 1 and C on particle 2, you will find they are only 96% correlated.

Violations of Bell's Inequality

This situation is impossible to arrange with crib sheets, indicating a deeper physical reality. The probable cause is that the probability amplitude is maximum at equal angles and is a smooth function. Therefore, the probability for getting a different answer is quadratic in the small angle:

Mathematical Explanation

To understand why this occurs, consider the probability amplitude for A and B. The amplitude is given by:

Amplitude 1 - e

Thus, the corresponding probability is:

Probability 1 - 2e

When you quadruple the angle between A and C, the amplitude becomes:

Amplitude 1 - 4e

And the corresponding probability becomes:

Probability 1 - 8e

This violation of Bell's Inequality implies that either there are no crib sheets, as in standard interpretations of quantum mechanics, meaning the information is produced through a strange irreducible interaction of the measuring device and the spinning particle at the point of measurement. Alternatively, the crib sheets can be modified at the point of measurement faster than light, as in Bohm's variation on quantum mechanics, leading to nonlocal links between measurements at distant points.

Conclusion

The intricate nature of quantum mechanics and Bell's Inequality challenges our classical understanding of reality. The violation of this inequality is not merely a theoretical curiosity but a profound statement about the nature of physical reality. It highlights the need for a more comprehensive theoretical framework that can accommodate these seemingly paradoxical observations.