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Zero Temperature Limit
In the preceding section, it was shown that for the hydrogen atom the
Gaussian variational density matrix, as a function of converges
at low temperature to the Gaussian ground state wave function given by
the RayleighRitz variational principle. It is generally true that the
RayleighRitz ground state
corresponds to the zero temperature limit of the VDM as we now show.
The RayleighRitz principle states that for any real parameterized
wave function
the variational energy

(156) 
is greater than or equal to the true ground state energy even at the minimum determined
by

(157) 
For the VDM ansatz, an amplitude parameter is assumed such that

(158) 
As in the one particle example, it is expected that at low
temperature,
, the other
while
constant. From this assumption, Eq. 3.19 implies
that as



(159) 
for all variational parameters, where we have defined
and
. Since
and
,
Eq. 3.41 for
implies
. So Eq. 3.41 may be rewritten as



(160) 
at the
fixed point. With the correspondence

(161) 
this is equivalent to Eq. 3.39 and thus the RayleighRitz ground
state corresponds to a zero temperature fixed point in the dynamics of
the parameters.
is a function of and , which is calculated by
integrating from with Eq. 3.23 as initial conditions.
The zero temperature limit of is a constant, , which
means in the low temperature limit can written as

(162) 
The function can be rewritten as,

(163) 
where the function
is introduced to describe the
variational error in the solution of the Bloch equation. It is
identical to zero if the variational ansatz includes the exact
solution. It leads to loss of symmetry in and , which will
discussed in the next section. Eq. 3.43 now reads,

(164) 
For certain potentials, several fixed points of the dynamics can
exist. From Eq. 3.46, it follows that only the lowest energy
state contributes to physical observables calculated from
Eq. 2.4. This completes the argument that the zero
temperature limit of the VDM corresponds to the RayleighRitz ground
state.
In the case of an antisymmetrized ansatz for the density matrix,
it can be shown that the fixed point of the dynamics in imaginary time
corresponds to the RayleighRitz ground state for an antisymmetrized
wave function.
Next: Loss of Symmetry
Up: Variational Density Matrix Properties
Previous: Variational Density Matrix Properties
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Burkhard Militzer
20030115