Wave function models

According to the variation principle, we can obtain the ground state wave function if the variation of the energy functional with respect to the wave function is zero: \[\begin{equation} \require{physics} \delta_{\Psi} E[\Psi] \leq 0 \thinspace , \end{equation}\] in which \(\delta_{\Psi}\) means a variation over all possible wave function forms and \(E[\Psi]\) is the energy functional, which for a normalized wave function is given by the expectation value of the Hamiltonian of the system: \[\begin{equation} E[\Psi] = \ev{ \hat{\mathcal{H}} }{\Psi} \thinspace . \end{equation}\] Since it is hard to describe variations over a wave function space, what is usually done is to choose a certain wave function model \[\begin{equation} \ket{\Psi(\vb{p})} \thinspace , \end{equation}\] which functionally depends on some parameters \(\vb{p}\). Such a wave function model is often called a wave function , which is German for approach or attempt.

In order to determine the parameters \(\vb{p}^\star\) that are optimal in some sense, every method13 also introduces a way of solving a set of equations such that its optimal parameters \(\vb{p}^\star\) can be somehow determined. These equations are most generally written as \[\begin{equation} \vb{F}(\vb{p}^\star) = \vb{0} \thinspace , \end{equation}\] in which \(\vb{p}^\star\) defines a solution. The wave function \(\ket{\Psi(\vb{p}^\star)}\) that is obtained by filling in a solution \(\vb{p}^\star\) is then called a wave function that belongs to/is associated with the corresponding method. The effective kind of equations that are solved in order to obtain such a solution is very essential, leading to some methods that are automatically variational and others have to be made variational.

In some sense, we can regard the wave function model as defining a whole space of possible wave functions \(\ket{\Psi(\vb{p})}\), but only those of interest are characterized by the solutions \(\vb{p}^\star\).

  1. that confusingly and often receives the same name as its wave function model↩︎