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From now on we will
consider a quantum mechanical canonical ensemble at temperature .
Such a system is described by a density operator
|
(15) |
denoting the Hamiltonian of the system.
Specifically, we will focus on repeated measurements
of a single particle in a heat bath of temperature
with sufficient time between measurements
to allow the heat bath to reestablish the canonical density operator.
For (non-degenerated)
particle coordinates
the likelihood for
becomes the thermal expectation
|
(16) |
denoting the thermal expectation with probabilities
|
(17) |
and energies and orthonormalized eigenstates
|
(18) |
In particular, we will consider a hermitian Hamiltonian of the
standard form
= ,
with kinetic energy ,
being times the negative Laplacian
for a particle with mass
(setting = ),
and local potential
=
.
Thus, in one dimension
|
(19) |
where the -functional is usually skipped.
For independent position measurements
the likelihood for , and thus for , becomes
(writing now
=
)
|
(20) |
We remark that it is straightforward to allow
to vary between measurements.
Next: Maximum likelihood approximation
Up: The likelihood model of
Previous: Measurements in quantum theory
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Joerg_Lemm
2000-06-06