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 [tex]\mathcal{H}[/tex], so that [tex]<\psi|\mathcal{H}|\psi> = \sum_n E_n |<\psi|\Psi_n>|^2[/tex]. Suppose that we want to calculate the excited state eigenvalue [tex]E_m[/tex]. If we constrain the trial function [tex]|\psi>[/tex] to satisfy [tex]<\psi|\Psi_{n'}> = 0[/tex] for all [tex]n'[/tex] such that [tex]E_{n'} \leq E_m[/tex], then it will follow that [tex]<\psi|\mathcal{H}|\psi>\leq E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>[/tex]. Hence we can calculateE_m by minimizig [tex]<\mathcal{H}> \equiv <\psi|\mathcal{H}|\psi>/<\psi|\psi>[/tex] subject to the constraint that [tex]|\psi>[/tex] be orthogonal to all state functions and energies lower than [tex]E_m[/tex].

Shouldn't

[tex]<\psi|\mathcal{H}|\psi>\leq E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>[/tex]

read

[tex]<\psi|\mathcal{H}|\psi> {\color{red}\geq} E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>[/tex]

?

--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

[tex]<\psi|\mathcal{H}|\psi> {\color{red}\geq} E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>[/tex]

?

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

I think it should indeed be [tex]\geq[/tex] , since otherwise it doesn't make sense
to "minimize" the ratio to get the eigenvalue. The [tex]\geq[/tex] 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.

 

[Entropia]

I cannot confirm or deny if there is a mistake in Ballentine's book without further investigation. However, it is possible that there is a typographical error in the equation mentioned. It is important to carefully review and double-check all equations and statements in scientific literature to ensure accuracy. If you have concerns about a potential mistake, it would be best to reach out to the author or publisher for clarification.