Ritz Method - Discussion

Discussion

As with other variational methods, a trial wave function, is tested on the system. This trial function is selected to meet boundary conditions (and any other physical constraints). The exact function is not known; the trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration.

It can be shown that the ground state energy, satisfies an inequality:

That is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than the ground-energy (or at least, equal to it).

If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state.

The Ritz ansatz function is a linear combination of N known basis functions, parametrized by unknown coefficients:

With a known Hamiltonian, we can write its expected value as

The basis functions are usually not orthogonal, so that the overlap matrix S has nonzero nondiagonal elements. Either or (the conjugation of the first) can be used to minimize the expectation value. For instance, by making the partial derivatives of over zero, the following equality is obtained for every k = 1, 2, ..., N:

which leads to a set of N secular equations:

In the above equations, energy and the coefficients are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero:

which in turn is true only for N values of . Furthermore, since the Hamiltonian is a hermitian operator, the H matrix is also hermitian and the values of will be real. The lowest value among (i=1,2,..,N), will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients from the corresponding secular equation.