Determination of the differential equation for
.
We wish to minimize
subject to
.
This is called the isoparametric problem. Introduce a single Lagrange
multiplier
,
such that;
Now show that terms in which
is varied are just the complex conjugate of the terms in which
are varied, thus the variation in
,
alone insures the minimization is satisfied.
Proof: Let H = T + V, where
Note
Also
if
(V is real)
Now the kinetic energy part,
.
Using Green's Theorem,
The surface is out at
,
because the volume V is all space
and
at
.
Thus the surface integral is zero. (evaluate this by partial integration.)
Then,
Now perform the variation with respect to
only subject to
.
the constraint
does not have to be diagonal but one can perform a unitary transformation on
to make the constraint diagonal.
Then
for an arbitrary variation in
,
the integrand [ ] must vanish. This implies that we now have a PDE
for
,
or rewriting the last term,
This is an eigenvalue equation of the form,
is the same single particle Hamiltonian, this implies that
.
This is the energy required to remove
the
particle from the system, which is known as Koopman`s Theorem.