If and
are random variables and
is their joint probability
distribution, the marginal distribution
of
is given by
![]() |
(6.54) |
![]() |
(6.55) |
![]() |
(6.56) |
![]() |
(6.57) |
Classically the definition of the local (marginal) expectation
values
of observables is
not ambiguous due to the commutativity of all observables.
However, quantum mechanically to each way of calculating the expectation
value in 6.59 corresponds a definition for the local expectation
value and in general these definitions give different results
![]() |
(6.59) |
In general the operator is not selfadjoint, so the
definitions using Dirac brackets above are different.
We can rephrase this by using the
explicit expression for the density operator for
a pure state
.
For a single wave function
we get two
definitions for the local expectation
![]() |
(6.60) |
A good way to define the local expectation is to symmetrize the definitions
This definition has the property that
is
again a selfadjoint operator,
hence the local expectation is real. This is the
way in which the current
is conventionally defined.
A special case which is not treated in this way
is the definition of local expectation for
an energy-like operator where
is selfadjoint.
Then we can define
However, in the Wigner formalism one usually defines energy
as the second
-moment in the form
With this definition for the local expectation value the
local energy can become negative, as was observed in
Wigner function simulations.
To calculate the Wigner transformation of
we use a suitable definition
for the star product.
We get
![]() |
(6.64) |
The question of marginalization is also discussed in [Wlo99].
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