General definitions

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 time-dependence. Each component, or stationary state, $\psi_\mu$, is a solution of the time-independent Schrodinger equation (TISE)

$$\hat{H} \psi \; = \; E \psi .$$ (5.1)

The (hermitian) operator $\hat{H}$ is linear, so we have a linear eigenproblem, which only has solutions for (real) energies $E_\mu$. If we ignore all discrete quantum numbers (spin) then $\psi$ is a function of continuous variables only which can be written generally as a vector ${\mathbf r}$. The dynamical variable ${\mathbf r}$ 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 $\left\vert{\mathbf r}\right\rangle$ which are the eigenstates of the position operator $\hat{{\mathbf r}}$, and which form a complete basis for the quantum problem. $\psi{({\mathbf r})}$ is the position representation of the wavefunction, and $H({\mathbf r},{\mathbf r}')$ is the position representation of $\hat{H}$. If $\hat{H}$ is local, as is the case in a huge variety of physical situations, then $H({\mathbf r},{\mathbf r}') = \delta({\mathbf r}-{\mathbf r}') [h_0({\mathbf r}') + h_1({\mathbf \nabla}_{{\mathbf r}'}) + \cdots]$ is a function only of the local value and spatial derivatives of the wavefunction. The function $h_0$ is commonly known as the potential. For instance a very common form of Hamiltonian arising in the case of isotropic mass $m$ and quadratic dispersion relation is

$$H \; = \; -\frac{\hbar^2}{2m} \nabla^2 + U{({\mathbf r})},$$ (5.2)

written as an operator in the position representation. The potential is $U{({\mathbf r})}$. However, when ${\mathbf r}$ 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 ${\mathbf r}$ is effectively confined to this region and the eigenenergies $E_\mu$ are discrete. This is also true if ${\mathbf r}$ is not strictly confined, but if the energy is below the continuum transition[128], giving bound states. In this latter case $\psi$ can take non-zero values over an infinite region of ${\mathbf r}$; however outside a certain (classically-allowed) 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 $E_\mu$, or equivalently the eigenfunctions $\psi_{\mu}{({\mathbf r})}$, for which (5.1) is obeyed. If an $E_\mu$ is known then it is easy to get the corresponding $\psi_{\mu}{({\mathbf r})}$, and vice versa. If there is no analytical solution, then a numerical approach is required--the 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 $\psi{({\mathbf r})}$ is negligible outside the region. I call the region $\mathcal{D}$ and its boundary $\Gamma$ (see Fig. 5.1). If $\psi{({\mathbf r})}$ dies smoothly to zero (as is true for a smooth-potential or soft-walled problem), then the choice of $\mathcal{D}$ is determined by the required accuracy. The smallest region $\mathcal{D}$ is found such that $\psi({\mathbf r} \in \Gamma) \approx 0$ to a sufficient approximation. In this case, the exact nature of BCs imposed at $\Gamma$ is irrelevant for the problem--the only important BC is then the asymptotic restriction $\psi{({\mathbf r})}\rightarrow0$ 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 $\Gamma$ at this edge (often this is essential since it avoids the difficulty of representing singularities which occur at the edge). Therefore $\mathcal{D}$, and the BCs at $\Gamma$, are exactly specified. To summarize, the BCs result from truncation of the region of space to ${\mathbf r} \in \mathcal{D}$.

BCs can generally be written

$$(\mathcal{L} \psi){({\mathbf s})}\; = \; 0 , \hspace{1in} \mbox{for all ${\mathbf s} \in \Gamma$} ,$$ (5.3)

where $\mathcal{L}$ is a linear operator which acts on the function space of $\psi$, and is sensitive only to its values, gradients, and higher derivatives, at the boundary $\Gamma$. The result of the operator is the smaller function space of functions of the boundary position coordinate ${\mathbf s}$. Generally $\mathcal{L}$ is local on $\Gamma$, meaning it measures only properties of $\psi{({\mathbf r})}$ local to a point ${\mathbf s}{({\mathbf r})}$. For instance, most BCs are described by special cases of the operator $\mathcal{L}$ which returns a (position-dependent) 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 non-local on $\Gamma$, and the resulting $E_\mu$ 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 R-matrix theory[131] for scattering problems.

Figure: $d$ dimensional domain $\mathcal{D}$ and $d{-}1$ dimensional boundary $\Gamma$ of a quantum solution region. The local normal ${\mathbf n}{({\mathbf s})}$ is also shown.

Table 5.1: Two general classes of quantum eigenproblem solution strategies using basis sets. The desired solutions are the eigenenergies $E_\mu$ and the states (written in terms of basis coefficients) ${\mathbf x}_\mu$. For other physical wave eigenproblems, the TISE (5.1) should be replaced by the appropriate time-independent 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 $E$.	Depend on $E$.
funcs.	Obey BCs.	Don't obey BCs.
	Don't obey TISE.	Are solutions of TISE.
basis size	Scales like volume of $\mathcal{D}$	Scales like surface area of $\Gamma$
technique	Write TISE in terms of basis	Write BC matching equations
	coefficients.	in terms of basis coefficients
		and boundary locations.
resulting
equation	$H{\mathbf x}\; = \;E{\mathbf x}$	$A(E) {\mathbf x}\; = \;0$
	Diagonalization of $H$ matrix	Hunt in $E$ for $\det [A(E)] = 0$, needs
	finds many $(E_\mu,{\mathbf x}_\mu)$ at once.	many SVDs per $(E_\mu,{\mathbf x}_\mu)$ found.
examples	Finite Element Method	Boundary Integral (or Element)
	(FEM)[19].	Method (BIM or BEM)[121].
	Discrete-Variable	Plane Wave Decomposition
	Representation (DVR)[17].	Method (PWDM) of Heller [91].
	Discretization on a lattice[161].	Bogomolny's method[34].
	Robnik conformal mapping[171].	Lupu-Sax t-matrix inversion[140].
		Methods of this chapter.