Example of observables for the EPR paradox – using 2 free non-interacting particles
EPR Argument with Two Pairs of Observables
The EPR argument suggests that if we can measure two different pairs of observables and determine the values of one observable from each pair without disturbing the system, then those values must have been pre-determined. This challenges the quantum mechanical principle that observables do not have definite values before measurement.
Two Pairs of Observables
- Pair A and B: Total momentum (Ptotal) and individual particle momentum (p₁).
- Pair C and D: Relative position (xrel = x₁ – x₂) and individual particle position (x₁).
1. First Pair: Ptotal and p₁
Observable A: Total Momentum
If we measure Ptotal, we determine that:
Observable B: Individual Particle Momentum
Measuring p₁ gives us p₂ immediately since we already know Ptotal.
EPR’s Argument: If measuring p₁ does not disturb particle 2, then p₂ must have been pre-determined before measurement.
2. Second Pair: xrel and x₁
Observable C: Relative Position
If we measure xrel, we obtain a relation between the two positions.
Observable D: Individual Particle Position
Measuring x₁ immediately gives us:
EPR’s Argument: If measuring x₁ does not disturb particle 2, then x₂ must have been pre-determined.
Conclusion: Why This Challenges Quantum Mechanics
- In quantum mechanics, momentum and position are conjugate variables—they cannot be simultaneously well-defined (Heisenberg uncertainty principle).
- But EPR argued that since we can measure p₁ and immediately determine p₂, and similarly for x₁ and x₂, both properties (momentum and position) of particle 2 must have been pre-determined—contradicting quantum mechanics.
- This led EPR to conclude that quantum mechanics is incomplete and must be supplemented with hidden variables.
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