Approximate solutions to the Schrödinger equation

There two common methods for obtaining approximate solutions to the Schrödinger equation.

The variational method
Perturbation theory

The expectation value (or average) energy is given by

The denominator áf|fñis a normalization term. The expression shown is written in this manner because it is valid even if the wave function is not normalized. Now, I may know the energy for a particular wave f. The variational theorem states that if I try another wave function f_try that satisfies the same boundary conditions (e.g. goes to zero at infinity) then the energy I calculate from the expectation value

will always be greater than the true energy E₀. The better the wave function the closer the energy will be to E₀, but it will never be less than E₀. The way to proceed is to formulate a wave function that has a variable in it and then minimize the energy with respect to that variable. A common example in text books is the exponent in a radial wave function. For example, if we wish to improve the energy for an atom that cannot be solved exactly using the Schrödinger equation we could use a hydrogen-like solution and then minimize the energy by varying the exponent. Suppose f_try = e^-ar, then the minimum energy will be found by taking the derivative with respect to a and setting that equal to zero.

The value of a that minimizes the energy is the best value that can be chosen for this particular form of the wave function.

In linear variation theory, the variables are coefficients so

f_try = c₁f₁ + c₂f₂ + ….+ c_nf_n

This is the LCAO-MO approach to the solution of the Schrödinger equation for molecules. To get the lowest possible energy for a given set of functions f_i (the basis set), we set ¶áEñ/¶c_i = 0 for each of the n coefficients. This leads to a set of simultaneous equations:

where

A solution exists if the determinant of the matrix of the H_ij - ES_ij is zero

Example: solution to the particle in a box using a variational method.

We use a trial function that satisfies the boundary conditions that Y = 0 at x = 0 and x = a.

Y = x(a - x) for 0 £ x £ a and Y = 0 outside the box.

Normalization requires

Thus,

Recall that the exact solution for the particle in a box is

Thus, the percent error of our variational energy is:

Perturbation Theory

A second method of approximate solution of the Schrödinger equation begins with a solution for an exact hamiltonian H⁽⁰⁾ and then treats a small perturbation to that exact solution using power series expansions of the wave function and energy. For example, the interaction of electromagnetic radiation with a molecule can be treated by starting with the energy levels and wavefunctions of the molecule and adding to these to account for the effect of radiation. The hamiltonian is

where l is just a number that "turns on" the perturbation. When l = 0 the perturbation (i.e. radiation field) is off and when l = 1 it is on. The term H⁽⁰⁾ is called the zero-order hamiltonian and the term H' is the perturbation. The zero-order Schrödinger equation is:

The (0) superscript (zero-order) refers to the exact solution. We now expand the wavefunction:

and energy:

The terms in (1), (2), etc. are referred to as first-order, second-order, etc., respectively. The successive correction terms must decrease in magnitude in order for the series to converge. To find the different orders we substitutie the expression for the hamiltonian, the wavefunctions, and the energy into the Schrödinger equation. If the zero order states are not degenerate:

It is clear that the first order correction can only be found if the zero order terms are known. The first order wavefunction consists of a linear combination of zero order wavefunctions. The magnitude of the coefficients depends on the strength of the coupling H_ji' and the magnitude of the energy denominator E_i⁽⁰⁾ - E_j⁽⁰⁾.