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When the Hamiltonian of a quantum mechanical problem
is independent of time,
the time evolution of the quantum state (wavefunction)
can always be written as the
sum of components with harmonic timedependence.
Each component, or stationary state, ,
is a solution of the timeindependent Schrodinger equation
(TISE)

(5.1) 
The (hermitian) operator is linear, so we have a linear eigenproblem,
which only has solutions for (real) energies .
If we ignore all discrete quantum numbers (spin)
then is a function of continuous variables only
which can be written generally as a vector .
The dynamical variable may represent the location of a particle, or
many particles, or more general coordinates such as relative distances,
angles, etc.
One can define position states
which are the eigenstates of
the position operator
, and which form a complete basis
for the quantum problem.
is the position representation of the wavefunction,
and
is the position representation of .
If is local, as is the case in a huge variety of physical
situations,
then
is a function
only of the local value and spatial derivatives of the wavefunction.
The function is commonly known as the potential.
For instance a very common form of Hamiltonian arising in the case of
isotropic mass and quadratic dispersion relation is

(5.2) 
written as an operator in the position representation.
The potential is
.
However, when does not simply indicate location of a single particle
in a conservative potential,
then (5.2) is not sufficiently general.
If the Hamiltonian is infinite (or much larger than the energies of interest)
outside a finite region, then
is effectively confined to this region and
the eigenenergies are discrete.
This is also true if is not strictly confined, but if the energy
is below the continuum transition[128], giving bound states.
In this latter case can take nonzero values over an infinite
region of ; however outside
a certain
(classicallyallowed) region
the decay is exponential so the effective region of confinement is finite.
I will be interested in solving for these resulting discrete states.
The problem is to find the eigenvalues ,
or equivalently the eigenfunctions
, for which (5.1)
is obeyed.
If an is known then it is easy to get the corresponding
,
and vice versa.
If there is no analytical solution, then a numerical approach
is requiredthe subject of this and following chapters.
The issue of boundary conditions (BCs) now enters; notice that
it is not actually present in the eigenproblem (5.1).
BCs arise because of the computational need to consider only
a finite region.
The finite region is chosen such that
is negligible outside the
region.
I call the region and its boundary (see Fig. 5.1).
If
dies smoothly to zero (as is true for a smoothpotential
or softwalled problem), then the choice of is determined
by the required accuracy.
The smallest region is found
such that
to a sufficient approximation.
In this case, the exact nature of BCs imposed at is irrelevant
for the problemthe only important BC is then the
asymptotic restriction
with increasing distance outside the finite region.
Another class of problems results if the potential suddenly changes (for instance
if it jumps to infinity corresponding to a hard wall) at the edge of the
problem region.
It is then natural to locate at this edge (often this
is essential since it avoids the difficulty of representing
singularities which occur at the edge).
Therefore , and the BCs at , are exactly specified.
To summarize,
the BCs result from truncation of the region of space to
.
BCs can generally be written

(5.3) 
where is a linear operator which acts on the function space of
, and is sensitive only to its values, gradients,
and higher derivatives, at the boundary .
The result of the operator is the smaller function space
of functions of the boundary position coordinate .
Generally is local on , meaning it measures only
properties of
local to a point
.
For instance, most BCs are described by special cases of the operator
which returns a
(positiondependent) linear combination of the local value and normal gradient
(this is known as `mixed BCs').
BCs may be specified for other reasons.
They may enter in the context of solving for resonance states in the continuum,
where they can model the effect of an infinite region of space
whose propagator is known analytically.
These ``radiative BCs'' may be nonlocal on , and the resulting
will be complex.
The calculation of eigenstates in the continuum (which are usually
highly degenerate) is a scattering problem, and
I shall not consider it further.
Also the eigenstates in a region with certain BCs can be needed for use in
other methods, for instance Rmatrix theory[131] for
scattering problems.
Figure:
dimensional domain and
dimensional boundary of a quantum solution region.
The local normal
is also shown.

Table 5.1:
Two general classes of quantum eigenproblem solution strategies using
basis sets.
The desired solutions are the eigenenergies and the states
(written in terms of basis coefficients)
.
For other physical wave eigenproblems, the TISE (5.1)
should be replaced by
the appropriate timeindependent
wave equation obeyed in the domain (e.g. Helmholtz equation).
In the methods of this chapter a diagonalization will replace the SVD
(singular value decomposition).
The scaling method of Vergini and Saraceno (not listed) combines
the best elements of both
classes.

(A) `basis diagonalization' 
(B) `Green's function matching' 



basis 
Independent of . 
Depend on . 
funcs. 
Obey BCs. 
Don't obey BCs. 

Don't obey TISE. 
Are solutions of TISE. 



basis size 
Scales like volume of 
Scales like surface area
of 



technique 
Write TISE in terms of basis 
Write BC matching equations 

coefficients. 
in terms of basis coefficients 


and boundary locations. 



resulting 


equation 






Diagonalization of matrix 
Hunt in for
, needs 

finds many
at once. 
many SVDs per
found. 



examples 
Finite Element Method 
Boundary Integral (or Element) 

(FEM)[19]. 
Method (BIM or BEM)[121]. 

DiscreteVariable 
Plane Wave Decomposition 

Representation (DVR)[17]. 
Method (PWDM) of Heller
[91]. 

Discretization on a lattice[161]. 
Bogomolny's method[34]. 

Robnik conformal mapping[171]. 
LupuSax tmatrix inversion[140]. 


Methods of this chapter. 




Next: Categories of existing numerical
Up: Introduction and history of
Previous: Introduction and history of
Alex Barnett
20011003