EPR Paradox for Two Non-Interacting Particles

The EPR paradox (named after Einstein, Podolsky, and Rosen) arises when considering quantum entanglement and the nature of reality. It challenges the completeness of quantum mechanics by demonstrating an apparent contradiction between quantum mechanics and locality.

1. Consider an Entangled State

For two particles (e.g., electrons) in an entangled state of momentum or position, we can write the wave function as:

Ψ(x₁, x₂) = ∫-∞∞ dp ψ̃(p) ei p x₁ / ħ e-i p x₂ / ħ

This describes a situation where the total momentum is well-defined (a delta function constraint), but individual momenta are uncertain.

2. The EPR Thought Experiment

  • Suppose we measure the momentum of particle 1 and find p₁.
  • Because of momentum conservation, we can instantly predict that the momentum of particle 2 is p₂ = -p₁, no matter how far apart they are.
  • Alternatively, if we measure the position of particle 1, then we instantly know the position of particle 2, even though they are non-interacting and distant.

This suggests that particle 2 “knows” about the measurement of particle 1 instantaneously, which violates locality (i.e., no information should travel faster than light).

3. The Paradox

Einstein, Podolsky, and Rosen (1935) argued:

  • If quantum mechanics is complete, then measuring one particle instantaneously determines the other’s state.
  • If information cannot travel faster than light, then the particle must have had a pre-existing state (hidden variables).
  • But quantum mechanics states that these values did not exist prior to measurement.

Thus, EPR concluded that quantum mechanics is incomplete and must be supplemented with hidden variables.

4. Resolution: Bell’s Theorem

John Bell later derived Bell’s inequalities, showing that any hidden variable theory consistent with locality would obey certain statistical constraints. Experiments showed violations of Bell’s inequalities, confirming quantum nonlocality—suggesting that entanglement is real and fundamentally quantum.

Conclusion

For two non-interacting particles, the EPR paradox shows that:

  • Quantum mechanics predicts correlations that seem to allow instant knowledge transfer.
  • Local realism (classical intuition) fails—measurement affects the system nonlocally.
  • Hidden variables (pre-existing states) are unlikely, based on experimental violations of Bell’s inequalities.