### 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 may al­ways be writ­ten as a sum of the eigen­func­tions :

where, if the ap­prox­i­ma­tion is any good at all, the co­ef­fi­cient of the ground state is close to one, while , , ...are small.

The con­di­tion that is nor­mal­ized, 1, works out to be

since the eigen­func­tions are or­tho­nor­mal.

Sim­i­larly, the ex­pec­ta­tion en­ergy works out to be

Elim­i­nat­ing us­ing the nor­mal­iza­tion con­di­tion above gives

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

The ex­pres­sion above also shows that while the de­vi­a­tions of the wave func­tion from the ex­act ground state are pro­por­tional to the co­ef­fi­cients , 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 , then the amount of eigen­func­tion pol­lut­ing ap­prox­i­mate ground state must be no more than . 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:

That er­ror is no more than . To ver­ify it, note that

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

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.