A.7 Ac­cu­racy of the vari­a­tional method

This note has a closer look at the ac­cu­racy of the vari­a­tional method.

Any ap­prox­i­mate ground state so­lu­tion $\psi$ may al­ways be writ­ten as a sum of the eigen­func­tions $\psi_1,\psi_2,\ldots$:

\psi = c_1 \psi_1 + \varepsilon_2 \psi_2 + \varepsilon_3 \psi_3 + \ldots

where, if the ap­prox­i­ma­tion is any good at all, the co­ef­fi­cient $c_1$ of the ground state $\psi_1$ is close to one, while $\varepsilon_2$, $\varepsilon_3$, ...are small.

The con­di­tion that $\psi$ is nor­mal­ized, $\langle\psi\vert\psi\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1, works out to be

1 = \langle c_1\psi_1 +\varepsilon_2\psi_2+\ldots \vert
...s\rangle = c_1^2 +
\varepsilon_2^2 + \varepsilon_3^2 + \ldots

since the eigen­func­tions $\psi_1,\psi_2,\ldots$ are or­tho­nor­mal.

Sim­i­larly, the ex­pec­ta­tion en­ergy $\langle{E}\rangle$ $\vphantom0\raisebox{1.5pt}{$=$}$ $\langle\psi\vert H\psi\rangle$ works out to be

\left\langle{E}\right\rangle = \langle c_1\psi_1 +\varepsil...
..._1^2 E_1 +
\varepsilon_2^2 E_2 + \varepsilon_3^2 E_3 + \ldots

Elim­i­nat­ing $c_1^2$ us­ing the nor­mal­iza­tion con­di­tion above gives

= E_1 + \varepsilon_2^2 (E_2-E_1) + \varepsilon_3^2 (E_3-E_1) + \ldots

One of the things this ex­pres­sion shows is that any ap­prox­i­mate wave func­tion (not just eigen­func­tions) has more ex­pec­ta­tion en­ergy than the ground state $E_1$. All other terms in the sum above are pos­i­tive since $E_1$ is the low­est en­ergy value.

The ex­pres­sion above also shows that while the de­vi­a­tions of the wave func­tion from the ex­act ground state $\psi_1$ are pro­por­tional to the co­ef­fi­cients $\varepsilon_2,\varepsilon_3,\ldots$, the er­rors in en­ergy are pro­por­tional to the squares of those co­ef­fi­cients. And the square of any rea­son­ably small quan­tity is much smaller than the quan­tity it­self. So the ap­prox­i­mate ground state en­ergy is much more ac­cu­rate than would be ex­pected from the wave func­tion er­rors.

Still, if an ap­prox­i­mate sys­tem is close to the ground state en­ergy, then the wave func­tion must be close to the ground state wave func­tion. More pre­cisely, if the er­ror in en­ergy is a small num­ber, call it $\varepsilon^2$, then the amount $\varepsilon_2$ of eigen­func­tion $\psi_2$ pol­lut­ing ap­prox­i­mate ground state $\psi$ must be no more than $\varepsilon$$\raisebox{.5pt}{$/$}$$\sqrt{E_2-E_1}$. And that is in the worst case sce­nario that all the er­ror in the ex­pec­ta­tion value of en­ergy is due to the sec­ond eigen­func­tion.

As a mea­sure of the av­er­age com­bined er­ror in wave func­tion, you can use the mag­ni­tude or norm of the com­bined pol­lu­tion:

\vert\vert\varepsilon_2 \psi_2 + \varepsilon_3 \psi_3 + \ldots\vert\vert
= \sqrt{\varepsilon_2^2+\varepsilon_3^2 +\ldots}

That er­ror is no more than $\varepsilon$$\raisebox{.5pt}{$/$}$$\sqrt{E_2-E_1}$. To ver­ify it, note that

\varepsilon_2^2 (E_2-E_1) + \varepsilon_3^2 (E_2-E_1) + \ld...
..._2-E_1) + \varepsilon_3^2 (E_3-E_1) + \ldots
= \varepsilon^2.

(Of course, if the ground state wave func­tion would be de­gen­er­ate, $E_2$ would be $E_1$. But in that case you do not care about the er­ror in $\psi_2$, since then $\psi_1$ and $\psi_2$ are equally good ground states, and $E_2-E_1$ be­comes $E_3-E_1$.)

The bot­tom line is that the lower you can get your ex­pec­ta­tion en­ergy, the closer you will get to the true ground state en­ergy, and the small er­ror in en­ergy will re­flect in a small er­ror in wave func­tion.