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Derivator
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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:
Shouldn't
read
[tex]<\psi|\mathcal{H}|\psi> {\color{red}\geq} E_m \sum_n |<\psi|\Psi_n>|^2 = E_m <\psi|\psi>[/tex]
?
--derivator
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
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