$$\require{cancel}$$

13: Variational Methods

We have seen, in Sect. 8.3, that we can solve Schrödinger's equation exactly to find the stationary eigenstates of a hydrogen atom. Unfortunately, it is not possible to find exact solutions of Schrödinger's equation for atoms more complicated than hydrogen, or for molecules. In such systems, the best that we can do is to find approximate solutions. Most of the methods which have been developed for finding such solutions employ the so-called variational principle discussed below.

• 13.1: Variational Principle
The variational principle states, quite simply, that the ground-state energy is always less than or equal to the expectation value of H calculated with the trial wavefunction
• 13.2: Helium Atom
A helium atom consists of a nucleus of charge +2e surrounded by two electrons. Let us attempt to estimate its ground-state energy.
• 13.3: Hydrogen Molecule Ion
The hydrogen molecule ion consists of an electron orbiting about two protons, and is the simplest imaginable molecule. Let us investigate whether or not this molecule possesses a bound state: that is, whether or not it possesses a ground-state whose energy is less than that of a hydrogen atom and a free proton.

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