Interpretation of Quantum Mechanics by olliegoblue30


									Interpretation of Quantum Mechanics

    Lectures Quantum Mechanics - January 2009

                                                Lectures-QM-08/09 – p.
  Critique of Einstein, Podolski and Rosen

1935 paper by Einstein, Podolski and Rosen
Can Quantum-Mechanical Description of Physical Reality Be Considered Complete ?

Question posed:
Is the description given by quantum mechanics complete ?
What is meant by complete ?

Herefore two propositions:

      A. Every element of physical reality must have a counterpart in the physical theory

      B. If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal
      to unity) the value of a physical quantity, then there exists an element of physical reality
      corresponding to this physical quantity

                                                                                                         Lectures-QM-08/09 – p.
  EPR critique
EPR invented Gedankenexperiments, each of which can determine the eigenvalues of one
member of a group of incompatible observables without disturbing the system to which
the observable pertains. This they did by by considering a system composed of two
parts, I and II, which interacted in the past, but which subsequently separate to an extent
where a measurement performed on one part is no longer able to produce an influence
on the other.


      Two-particle decay of a system
      Spin singlet state decays into two spin   2

                          |0 >= √ (|+ >1 |− >2 −|− >1 |+ >2 )                            (1)

      - Perfect correlations between the two spin projections

Classical physics of decaying particles would also exhibit correlations, but no influence of
the measurement of one particle on the other if they are spatially separated.

                                                                                     Lectures-QM-08/09 – p.
  EPR critique
What causality requires is that there be no influence on an object due to any action taken
in a region that is at a space-like position with respect to the object. It is the combination
of this requirement with the existence of incompatible observables that creates the
enigmatic situations that EPR pointed to.

The paradox:
      Let A be an observable for system I, with eigenvalues aα and eigenfunctions
      uα (x1 ) and the total state is
                                 Ψ(x1 , x2 ) =       ψα (x2 )uα (x1 )                       (2)

      B is another observable belonging to I, which does not commute with A, with the
      eigenvalues bβ and eigenfunctions vβ (x1 ) and
                                 Ψ(x1 , x2 ) =       φβ (x2 )vβ (x1 )                       (3)

                                                                                        Lectures-QM-08/09 – p.
  EPR critique

       I and II are spacelike separated: No real change in I or II due to an action in II or I,
       From a measurement of the observable A in I with outcome uν we know from the
       wave function that the distant state of system II is ψν . Alternatively, by measuring
       B, and finding I to be in the state vλ , we know from the wave function that II is in
       the state φλ
       Suppose further: ψα (x2 ) are the eigenfunctions of an observable P belonging to
       system II, with eigenvalues pα , while the φβ (x2 ) are the eigenfunctions of another
       observable Q of system II with eigenvalues qβ , and that P and Q do not commute.
       As no interaction between I and II is possible when they have a space-like
       separation this has the following consequences:

C. By measuring either A or B we are in a position to predict with certainty, and without in any way disturbing
the second system, either the value of the quantity P (that is pα ) or the value of the quantity Q (that is qβ ).
In accordance with our criterion of reality, in the first case we must consider the quantity P as being an
element of reality, while in the second case the quantity Q is an element of reality.

                                                                                                           Lectures-QM-08/09 – p.
  EPR critique

However, quantum mechanics tells us

D. When two operators corresponding to two physical quantities do not commute the two quantities cannot
have simultaneous physical reality.

and furthermore that any attempt to determine Q experimentally will alter the state of the
system in such a way as to destroy the knowledge of P.

C. and D. contradict each other !

Since the Gedankenexperiment satisfies proposition B, EPR concluded that quantum
mechanics is a theory that fails to comply with proposation A i.e. the quantum mechanical
description of physical reality given by the wave function is not complete !

=⇒ EPR: Not every element of physical reality has a counterpart in quantum

Bohr argued subsequently: The EPR conception of reality is in conflict with how factual
statements must be made in the quantum domain.

                                                                                                  Lectures-QM-08/09 – p.
  EPR critique

Bohr further argues:

     Distinct experiments measure mutually incompatible observables of one particle
     (Spin projections along different axes as being examples for the two observable A
     and B)
     EPR were mislead by their definition of elements of physical reality
     The procedure of measurement has an essential influence on which the very definition of the
     physical quantities rests

Nowadays typical attitude: EPR does not pose a paradox, because it is impossible to
devise any experiment that simultaneously measures precise values of incompatible

Arguments for EPR paradox typical employ counterfactual arguments: Contradictions
between an actual experiment that does measure P, and another that could, were it done
instead, measure the incompatible observable Q.

                                                                                                 Lectures-QM-08/09 – p.
  EPR critique

Nevertheless, entangled states have properties that appear to be due to non-local
interventions, as is made strikingly clear by the spin singlet state. For if a long string of
measurements of teh distant spins σ1 , σ2 are made along the same direction, then in
each and every individual case an observer who knows the outcomes v1 = ± at end I
can, without fail, tell the distant observer at II the outcome v2 = ∓, and vice versa. Thsi
does not allow superluminal signaling between I and II because observer I cannot
control which of the outcomes ± will appear in any one measurement at I, nor for taht
reason the correlated outcomes at II. The data have to be brought together before the
’prediction’ can be confirmed.

Nevertheless, the correlations provoke the thought that a conspiracy is afoot, with the
members of each pair carrying instructions that tell them what to do when they are

                                                                                         Lectures-QM-08/09 – p.
  Hidden variables and Bell theorem

The EPR argument raises the question:

Is there a description of nature that is more refined than that of quantum mechanics, and
which explains the the spooky action at a distance that quantum mechanics appears to imply

Request for a ’local realistic’ theory satisfying EPRs definitions of realism and
completeness, while leaving the established consequences of quantum mechanics intact

=⇒ Introduce additional ’hidden variables’ that provide a ’complete’ description of
individual systems by assigning values to all the observables of such an individual prior
to measurement, with an average over the hidden variables giving the quantum state as
a description of ensembles of individuals. (Note: Quantum mechanics assigns no value
to the observable prior to measurement)

Bell-Kochen-Specker (BKS) theorem:
It is not possible to consistently assign values to all the observables in any Hilbert space
larger than H2 . Quantum mechanics is not a statistical rendition of an underlying hidden
variable theory.

                                                                                       Lectures-QM-08/09 – p.
  Hidden variables and Bell theorem

Bell’s theorem and its generalizations establish that hidden variable theories can only
reproduce the statistical predictions of quantum mechanics if the hidden variables in one
region can be affected by measurements carried out in a space-like separated region (as
they are in the de Broglie-Bohm theory).

Such theories therefore violate the relativistic causality principle, surely an ironic
outcome from the perspective of EPR !

The experiments motivated by Bell’s theorem give results in excellent agreement with
quantum mechanics, hence only non-local hidden variable theories are viable options,
which, one could say, offers a cure that is worse than the malaise .....

                                                                                         Lectures-QM-08/09 – p.1

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