Axiomatic approach to the formulation of Quantum Mechanics

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Axiomatic approach to the formulation of Quantum
Mechanics

• We assume there exists a function
• Which contains all the information about a system
  at time,t. We shall say that is the
  state of the system

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Axiom

• It is impossible to measure all the properties of
  a physical system simultaneously
• Example and

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The Principle of Superposition of states

• Suppose we start with a physical systems and we
  want to measure physical quantities A,B,C,
• We describe the system in terms of abstract
  quantities called kets

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Postulate

• Every Physical state corresponds to a ray in a
  Hilbert space over C.
• The addition of Hilbert space elements
  corresponds to the
• Principle of Superposition of States

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PostulateEvery physical dynamical variable
will be represented by a linear self-adjoint
operator whose eigenvectors span H
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PostulateEvery physical dynamical variable
will be represented by a linear self-adjoint
operator whose eigenvectors span H

• We need the operator to be self adjoint since we
  want real eigenvalues
• Measuring devices give us real numbers, measuring
  real and imaginary parts separately wont work,
  violate uncertainty principle

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PostulateEvery physical dynamical variable
will be represented by a linear self-adjoint
operator whose eigenvectors span H
Since operator is s.a, eigenvectors are
orthogonal or can be orthogonalized
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PostulateEvery physical dynamical variable
will be represented by a linear self-adjoint
operator whose eigenvectors span H
Any result of measurement of a real dynamical
variable is one of its eigenvalues, after the
measurement the system must be in an eigenstate
of the operator, prior to measurement it must be
in a linear combination of the set of all
eigenstates into which it may jump. The
eigenvectors must span the space since they
represent all possible results of measurements
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• The average, or expectation value of the
  observable, A in the state ?gt is defined to be

If we make exactly the same measurement on
identically prepared systems then the average
value will be After a sufficiently large
number of measurements
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• With loss of generality we may assume

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• Now we expect ai to be some how related to the
  probability of getting the value qi

So we may interpret
As the probability of getting the result qi when
we measure A, on ?gt
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• Note

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Stern-Gerlach again!
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Matrix Representation
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• When we subjected the Sxgtbeam to a SG
  measurement in the z direction
• The beam was split into two components of equal
  intensity

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Over all phase of is irrelevant , w.l.g take
coefficient ofSzgt to be real,ve
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• Now the observable Sx can be written

A similar argument leads to the observable Sy
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• It is convenient to take the matrix elements of
  Szreal by taking

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• Easy to see

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• Matrix Representation

Easy to check have all the properties required
of the x, y components of spin
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• We can orientate our SG detector at will

Follows from our matrix representatiom
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• So clearly if the electron is known to be in the

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• To arrive at this result I have only used
• The oberved results of Stern Gerlach Measurement
• Our probability concepts
• The principle of superposition of states

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2 obervers very far apart

• Einstein-Podolsky-Rosen (EPR)

Observer 1 measures the spin in the Z direction
at time t0, he gets the value h/4?
Observer 2 measures the spin in the Z direction
at time t0, he gets the value -h/4?
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Observers can chose to rotate their SGs at
random
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• If 1 measures Sz
• 2 measurers Sx then there is a
  completely random correlation between the 2
  measurements
• If 1 measures Sz
• 2 measurers Sz there is a
  100(oppositite sign) correllation
• Between the 2 measurements

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• If 1 choses to make no measurement and 2 measures
  Sz, this measurement shows no correllation
• The outcome of 2s measurement appears to depend
  on what kind of measurement 1 decides to perform
• But 1 and 2 can be light years apart

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But on one supposition we should, in my
opinion, hold fast the real factual situation of
the system S2 is independent of what is done to
S1which is spatially independent
A .
Einstein(1935) Einstein would argue that the
most simple explanation was that the electron
was in a given spin state
before the
measurement but we just didnt know it!
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• The quantum mechanical predictions for
• Sz performed on the z-direction
• Spin up states are reproduced provided only that
  there are as many particles
• of type

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Photodouble ionization of Helium
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• Einsteins argument is essentially that
• For an arbitary beam of electrons
• Each electron has a definite spin value in the
  directions
• Determined by some unknown physical law,
• He would accept that the act of measuring Sz
  would effect the values of the spins in the

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Follow Einsteins argument

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• The only assumptions we have made are
• The electron spins have a reality independent of
  the experiment
• The Stern-Gerlach resultsi.e the result of the
  component of spin in a given direction is
• Signals cannot travel faster than light

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Let us return to our Quantum Mechanical formalism
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There is thus a measurable difference between
assuming reality and causality

• And starting from the superposition of states

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• The only assumptions we have made are
• The electron spins have a reality independent of
  the experiment
• The Stern-Gerlach resultsi.e the result of the
  component of spin in a given direction is
• Signals cannot travel faster than light

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FAlSe For example qp/4