The EPR Paradox

Einstein attempts to prove Quantum Mechanics is Incomplete

The premise behind EPR - Completeness

Einstein, with the assistance of Boris Podolsky and Nathan Rosen, constructed the argument that Quantum Mechanics was incomplete via the logical connections between two assertions, one of which must hold:
a) The theory of Quantum Mechanics is incomplete
b) Incompatible quantities – what Bohr termed complementary observables – cannot have simultaneous realities


The paper began with the definition of completeness and reality:
“Every element in physical reality must have a counter-part in the physical theory”
“If, without in any way disturbing a system, we can predict with certainty (ie with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity”

(Physical Review Volume 47 March 1935)


Thus, if a real situation could be described such that a pair of complementary observables, such as position and momentum, could be measured exactly and simultaneously, a phenomena Quantum Mechanics could not explain, the conclusion would be that Quantum Mechanics is incomplete.

The Einstein-Podolsky-Rosen Paradox

This formulation of the Einstein – Podolsky – Rosen paradox, was published in Physics Review in March 1935: Can Quantum Mechanical reality be considered to be complete?

The first formulation:

From Heisenberg’s Uncertainty Principle we cannot simultaneously measure to exact precision the position and momentum of a quantum mechanical system. Therefore if the wavefunction is an eigenfunction of the position operator, it cannot be an eigenfunction of the momentum operator and vice versa.

Suppose we have a system consisting of 2 identical particles, A and B. After interaction, they are separated far apart from each other without undergoing any other interactions. And since this is a “thought experiment”, we can let them travel until they become spacelike separated, i.e. anything which happens to one particle cannot interfere with the other (interference would require signals traveling above light speed, which were forbidden by Einstein's concepts of locality and separability).

Suppose we measure the momentum of A exactly, then the wavefunction would be an eigenfunction of momentum for A. From the Law of Conservation of Momentum, the momentum of B would be of equal magnitude but opposite direction.

And now, let us measure the position of B (at the same time as the measurement of momentum of A). The wavefunction would then collapse and we would get an eigenvalue for its position.

This gives rise to a contradiction! From Heisenberg’s Uncertainty Principle, we cannot know the momentum and position of a particle at a certain time to infinite precision. Therefore, the Copenhagen Interpretation of Quantum Mechanics is incomplete!

 

These images obtained from Soshichi Uchii (2004)

 

The EPR(Bohm) Paradox and alternative formulations

As we shall see, Bohr explained the EPR paradox without abandoning his formulation of quantum theory satisfactorily before the end of the 1930's. A more general formulation was created by Bohm using the intrinsic property of all fundamental particles, angular momentum (spin):

Special relativity tells us that nothing can travel faster than the speed of light. By making use of this fact, a quite similar “thought experiment” was devised. However, this assumption of locality will come back to haunt Einstein in Bohr's reply.
A Particle of zero spin decays into particle A with spin (+1/2) and particle B with spin (-1/2) travelling in opposite directions along the x-axis until they are very far apart.

By the conservation of angular momentum, if we measure the spin (or a directional componet of spin) of A then we instantaneously know that the spin of B would be equal in magnitude and opposite in direction even before measuring it.

But the 2 particles are assumed to be light years apart and thus spacelike. This instantaneous collapse of the wavefunction of B requires non-locality or in Einstein’s words: “spooky action at a distance”.
Therefore, the Copenhagen interpretation of Quantum Mechanics violates Special Theory of Relativity, and thus is incomplete.

 

This is a similar contradiction to the one arising out of the EPR paradox well avoiding the 'mechanical disturbance' solution formulated by Bohr.

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