130_notes.dvi

We wish to show thatE′errors are second order inδψ

⇒

∂E

∂ψ

= 0

at eigenenergies.

To do this, we will add a variable amount of an arbitrary functionφto the energy eigenstate.

E′=

〈ψE+αφ|H|ψE+αφ〉

〈ψE+αφ|ψE+αφ〉

Assumeαis real since we do this for any arbitrary functionφ. Now we differentiate with respect to
αand evaluate at zero.

dE′

dα

∣

∣

∣

∣

α=0

=

〈ψE|ψE〉(〈φ|H|ψE〉+〈ψE|H|φ〉)−〈ψE|H|ψE〉(〈φ|ψE〉+〈ψE|φ〉)

〈ψE|ψE〉^2

=E〈φ|ψE〉+E〈ψE|φ〉−E〈φ|ψE〉−E〈ψE|φ〉= 0

We find that the derivative is zero around any eigenfunction, proving that variations of the energy
are second order in variations in the wavefunction.

That is,E′is stationary (2nd order changes only) with respect to variation inψ. Conversely, it can
be shown thatE′is only stationary for eigenfunctionsψE. We can use thevariational principle
to approximately findψEand to find an upper bound onE 0.

ψ=

∑

E

cEψE

E′=

∑

E

|cE|^2 E≥E 0

For higher states this also works if trialψis automatically orthogonal to all lower states due to some
symmetry (Parity,ℓ...)

• See Example 25.6.1:Energy of 1D Harmonic Oscillator using a polynomial trail wave function.*


• See Example 25.6.2:1D H.O. using Gaussian.*

25.5 Variational Helium Ground State Energy

We will now add one parameter to the hydrogenic ground state wavefunction and optimize that
parameter to minimize the energy. We could add more parameters but let’s keep it simple. We will
start with the hydrogen wavefunctions but allow for the fact thatone electron “screens” the nuclear
charge from the other. We will assume that the wave function changes simply by the replacement

Z→Z∗< Z.

Of course theZin the Hamiltonian doesn’t change.

So our ground state trial function is

ψ→φZ

∗

100 (~r^1 )φ

Z∗

100 (~r^2 ).