# 13: Variational Methods

- Page ID
- 15812

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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.

## Contributors and Attributions

Richard Fitzpatrick (Professor of Physics, The University of Texas at Austin)

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