“Know how to solve every problem that has been solved.” “What I cannot create, I do not understand.” — Richard Feynman

The Variational Principle by Hand — When the trial contains the exact answer

Exercises

Quantum Chemistry I › Unit 2 · The variational game › The Variational Principle by Hand › all problems

Practice Problem 2 of 8

When the trial contains the exact answer

Problem

Do the same for the hydrogen atom with trial

e^{- ζ r}

, whose energy is

E (ζ) = \frac{1}{2} ζ^{2} - ζ

hartree. Find

ζ^{⋆}

and

E (ζ^{⋆})

, and compare with the exact

- 0.5

Solution

Predict before reading on. Helium landed strictly above exact. This one will land on it — what is special about $e^{- ζ r}$ for hydrogen that helium’s trial lacked?

\frac{d E}{d ζ} = ζ - 1 = 0 \Rightarrow ζ^{⋆} = 1

E (ζ^{⋆}) = \frac{1}{2} (1)^{2} - 1 = - 0.5 hartree (exact)

The trial form $e^{- ζ r}$ is the exact 1s eigenfunction, so at $ζ = 1$ the trial equals the true ground state and the bound is saturated. The variational answer is exact precisely when your trial space contains the truth.