Ritz method

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In physics, the Ritz method is a variational method named after Walter Ritz.

It can be applied in quantum mechanical problems to provide an upper-bound on the ground state energy.

As with other variational methods, a trial wave function, $Ψ$ , 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, $E 0$ , satisfies an inequality:

$E_0 \le \int \Psi^* \hat{H} \Psi \, d\tau$

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 $\left\lbrace\Psi_i\right\rbrace$ , parametrized by unknown coefficients:

$\Psi = \sum_{i=1}^N c_i \Psi_i.$

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

$\varepsilon = \frac{\left\langle \sum_{i=1}^N c_i\Psi_i|\hat{H}|\sum_{i=1}^Nc_i\Psi_i \right\rangle}{\left\langle \sum_{i=1}^N c_i\Psi_i|\sum_{i=1}^Nc_i\Psi_i \right\rangle} = \frac{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jH_{ij}}{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jS_{ij}} \equiv \frac{A}{B}$ .

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

$\frac{\partial\varepsilon}{\partial c_k^*} = \frac{\sum_{j=1}^Nc_j(H_{kj}-\varepsilon S_{kj})}{B} = 0$ ,

which leads to a set of N secular equations:

$\sum_{j=1}^N c_j \left( H_{kj} - \varepsilon S_{kj} \right) = 0 \;\;\;\;\;\;\;\; \mbox{for} \;\;\; k = 1,2,...,N$ .

In the above equations, energy $\varepsilon$ and the coefficients $\left\lbrace c_j \right\rbrace$ 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:

$\det \left( H_{kj} - \varepsilon S_{kj} \right) = 0$ ,

which in turn is true only for N values of $\varepsilon$ . Furthermore, since the hamiltonian is a hermitian operator, the H matrix is also hermitian and the values of $\varepsilon_i$ will be real. The lowest value among $\varepsilon_i$ (i=1,2,..,N), $\varepsilon_0$ , 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 $\left\lbrace c_j \right\rbrace$ from the corresponding secular equation.

1 Papers
2 Books
3 External links
4 See also

[edit] Papers

W. Ritz, "Ueber eine neue Methode zur Lösung gewisser Variationsprobleme der mathematischen Physik" J. Reine Angew. Math. 135 (1908 or 1909) 1
J.K. MacDonald, "Successive Approximations by the Rayleigh-Ritz Variation Method", Phys. Rev. 43 (1933) 830