Is There a Mistake in Ballentine's Description of the Variational Principle?

[Derivator]

Hi folks,

I'm just reading Ballentine's book on quantum mechanics and was wondering whether he really made a mistake. It's about the variational principle.

In chapter 10.6 (p. 296 in the current edition) he says:

Although the variational theorem applies to the lowest eigenvalue, it is possible to generalize it to calculate low-lying excited states. In proving that theorem, we formally express the trial function as a linear combination of eigenvectors of $$\mathcal{H}$$, so that $$<\psi|\mathcal{H}|\psi> = \sum_n E_n |<\psi|\Psi_n>|^2$$. Suppose that we want to calculate the excited state eigenvalue $$E_m$$. If we constrain the trial function $$|\psi>$$ to satisfy $$<\psi|\Psi_{n'}> = 0$$ for all $$n'$$ such that $$E_{n'} \leq E_m$$, then it will follow that $$<\psi|\mathcal{H}|\psi>\leq E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>$$. Hence we can calculateE_m by minimizig $$<\mathcal{H}> \equiv <\psi|\mathcal{H}|\psi>/<\psi|\psi>$$ subject to the constraint that $$|\psi>$$ be orthogonal to all state functions and energies lower than $$E_m$$.

Shouldn't

$$<\psi|\mathcal{H}|\psi>\leq E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>$$

read

$$<\psi|\mathcal{H}|\psi> {\color{red}\geq} E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>$$

?

--derivator

 

[strangerep]

I'm just reading Ballentine's book on quantum mechanics and was wondering whether he really made a mistake. It's about the variational principle. [...]

In chapter 10.6 (p. 296 in the current edition) he says:
[...]

Shouldn't
[...]

read

$$<\psi|\mathcal{H}|\psi> {\color{red}\geq} E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>$$

?

I sure hope it's a typo. (Otherwise I don't understand it either. :-)

I think it should indeed be $$\geq$$ , since otherwise it doesn't make sense
to "minimize" the ratio to get the eigenvalue. The $$\geq$$ is also what he
wrote in the previous Variational theorem on pp291-292.

Googling for "ballentine quantum errata" produced a few hits, but nothing
comprehensive, afaict. I sure wish Prof Ballentine and/or the publishers
would compile an errata list. One of the later chapters seemed to have
an elevated number or errors, as I recall. Maybe it didn't get good proofreading.