- The EPR Problem and Bell’s Inequality. - When a quantum system possesses more than one degree of freedom, the asso- ciated Hilbert space is a tensor product of the spaces associated to each degree of freedom. - 11.1 The Electron Spin. - Consider a unit vector u ϕ in the (z, x) plane: u ϕ = cos ϕ u z + sin ϕ u x , where u z and u x are unit vectors along the z and x axes. - u ϕ the component of the electron spin along the u ϕ axis.. - ϕ which, in the limit ϕ = 0, reduce respectively to the eigenvectors | e. - Assume the electron is emitted in the state | e. - One measures the component ˆ S eα of the spin along the direction u α = cos α u z + sin α u x . - What is the probability P + (α) of finding the electron in the state | e. - α ? What is the expectation value S ˆ eα in the spin state | e. - 100 11 The EPR Problem and Bell’s Inequality. - 11.2 Correlations Between the Two Spins. - We first assume that, after the dissociation, the electron–proton system is in the factorized spin state | e. - What is the probability P + (α) of finding. - h/2 when measuring the component ˆ S eα of the electron spin in this state?. - Having found this value, what is the state of the system after the measure- ment?. - Is the proton spin state affected by the measurement of the electron spin?. - Calculate the expectation values S ˆ eα and S ˆ pβ of the components of the electron and the proton spins along axes defined respectively by u α and u β = cos β u z + sin β u x. - Calculate E(α, β) in the state under consideration.. - 11.3 Correlations in the Singlet State. - We now assume that, after the dissociation, the two particles are in the singlet spin state:. - One measures the component ˆ S eα of the electron spin along the di- rection u α . - Later on, one mea- sures the component ˆ S pβ of the proton spin along the direction u β . - Would one have the same probabilities if the proton spin had been measured before the electron spin?. - Why was this result shocking for Einstein who claimed that “the real states of two spatially separated objects must be independent of one another”?. - Calculate the expectation values S ˆ eα and S ˆ pβ of the electron and the proton spin components if the system is in the singlet state (11.2).. - Calculate E(α, β) in the singlet state.. - 11.4 A Simple Hidden Variable Model. - For Einstein and several other physicists, the solution to the “paradox” uncov- ered in the previous section could come from the fact that the states of quan- tum mechanics, in particular the singlet state (11.2), provide an incomplete description of reality. - In the case of interest, a very simplified example of such a theory is the following. - In this case, ϕ is the hidden variable. - Owing to this ignorance of the value of ϕ, the expectation value of an observable ˆ A is now defined to be:. - Using the definition (11.1) for E(α, β) and the new definition (11.3) for expectation values, calculate E(α, β) in this new theory. - Compare the result with the one found using “orthodox” quantum mechanics in Sect. - The first precise experimental tests of hidden variable descriptions vs. - 11.1 presents experimental results obtained by A. - It gives the variation of E(α, β) as a function of the difference α − β, which is found to be the only experimentally relevant. - 102 11 The EPR Problem and Bell’s Inequality quantity.. - Which theory, quantum mechanics or the simple hidden variable model de- veloped above, gives a good account of the experimental data?. - 11.5 Bell’s Theorem and Experimental Results. - As proved by Bell in 1965, the disagreement between the predictions of quan- tum mechanics and those of hidden variable theories is actually very general when one considers correlation measurements on entangled states. - We now show that the correlation results for hidden variable theories are constrained by what is known as Bell’s inequality, which, however, can be violated by quantum mechanics in specific experimental configurations.. - Consider a hidden variable theory, whose result consists in two functions A(λ, u α ) and B (λ, u β ) giving respectively the results of the electron and pro- ton spin measurements. - It depends on the value of the hidden variable λ for the con- sidered electron–proton pair. - The nature of this hidden variable need not be further specified for the proof of Bell’s theorem. - The result A of course de- pends on the axis u α chosen for the measurement of the electron spin, but it does not depend on the axis u β . - Give the correlation coefficient E(α, β) for a hidden variable theory in terms of the functions A and B and the (unknown) distribution law P(λ) for the hidden variable λ.. - Derive Bell’s inequality. - Consider the particular case α − β = β − α = α − β = π/4, and compare the predictions of quantum mechanics with the constraint imposed by Bell’s inequality.. - 0.04) for α − β = π/4 and E(α, β. - Is a description of these experimental results by a local hidden variable theory possible?. - Are these results compatible with quantum mechanics?. - 11.6 Solutions. - Section 11.1: The Electron Spin. - In the eigenbasis | e. - 104 11 The EPR Problem and Bell’s Inequality. - Section 11.2: Correlations Between the Two Spins. - I ˆ p , where ˆ I p is the identity operator on the proton states. - Section 11.3: Correlations in the Singlet State 11.3.1. - This result is a consequence of the rotational invariance of the singlet state.. - The state after the measurement of the electron spin, yielding the result. - This simple result is also a consequence of the rotational invariance of the singlet state, which can be written as. - Now the two possible results for the measurement of the proton spin. - The fact that a measurement on the electron affects the probabilities for the results of a measurement on the proton, although the two particles are spa- tially separated, is in contradiction with Einstein’s assertion, or belief. - This is the starting point of the Einstein–Podolsky–Rosen paradox. - From a single measurement of the proton spin, one cannot determine whether the electron spin has been previously measured.. - It is only when, for a series of experiments, the results of the measurements on the electron and the proton are later compared, that one can find this non-local character of quantum mechanics.. - Section 11.4: A Simple Hidden Variable Model 11.4.1. - cos(ϕ − α) dϕ 2π = 0 and similarly S ˆ pβ = 0. - 106 11 The EPR Problem and Bell’s Inequality. - Therefore, in this simple hidden variable model, E(α, β. - The experimental points agree with the predictions of quantum me- chanics, and undoubtedly disagree with the results of the particular hidden variable model we have considered. - We must however point out that the data given in the text is not the actual measured data. - 11.2, where the error bars correspond only to statistical errors. - quantum mechanics) is due to systematic errors, mainly the acceptance of the detectors.. - Section 11.5: Bell’s Theorem and Experimental Results. - In the framework of a hidden variable theory, the correlation coeffi- cient is. - Note that we assume here that the hidden variable theory reproduces the one-operator averages found for the singlet state:. - If this was not the case, such a hidden variable theory should clearly be re- jected since it would not reproduce a well established experimental result.. - We multiply the result (11.4) by P (λ) and integrate over λ. - Bell’s inequality follows immediately.. - The shaded areas correspond to results which cannot be explained by hidden variable theories.. - 2, which clearly violates Bell’s inequality. - This system constitutes therefore a test of the predictions of quantum mechanics vs. - any local hidden variable theory.. - The numbers given in the text lead to | 3E(π/4. - 0.15) in excellent agreement with quantum mechanics (2. - 2) but in- compatible with hidden variable theories.. - As in the previous question, the actual measurements were in fact E(π/4)
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