Question on Time-independent perturbation theory: I am confused

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Discussion Overview

The discussion revolves around time-independent perturbation theory in quantum mechanics, specifically addressing the calculation of energy corrections and the relationship between the average energy of an atom and its wavefunction expansions. Participants explore the implications of different formulations and their resulting values.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the standard formulation of time-independent perturbation theory, questioning whether the average energy is given by a simple sum of energy corrections or by a more complex expression involving the perturbed wavefunction.
  • Another participant asserts that the two formulations should not yield different results up to the order of the wavefunction calculated, referencing a theorem that connects the order of the wavefunction to the energy calculation.
  • A subsequent reply emphasizes that the second formula includes terms that do not appear in the simpler energy correction sum, specifically mentioning the term <0|V|0(1)>, which complicates the comparison.
  • Another participant warns about the normalization of the original and perturbed wavefunctions, suggesting that the norms may differ and that this must be accounted for in the calculations.

Areas of Agreement / Disagreement

Participants express differing views on the equivalence of the two formulations for calculating average energy, indicating that the discussion remains unresolved with multiple competing perspectives.

Contextual Notes

There are limitations regarding the assumptions made about the wavefunctions and their norms, which may affect the validity of the comparisons being drawn. The discussion does not resolve these mathematical nuances.

ani4physics
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We all know from time-independent perturbation theory that if we have an atom in ground state [0>, and when a time-independent perturbation acts on it, the energy of the ground state gets shifted and the ground state wave function also gets modified. Using Time-independent Schroedinger eq.,

[H0 + lambda . V] [0> = [E0] [0>, where V is the perturbation hamiltonian.

Now we expand E0 as E0 = E0(0) + lambda . E0(1) + lambda^2. E0(2) +...
and [0> as [0> = [0>(0) + lambda. [0>(1) + lambda^2. [0>(2) + ...

Then we compare powers of lambda, left multiply with <0] to get the energy corrections and so on.

My question is, is the average energy of the atom E0 = E0(0) + E0(1) + E0(2) + ...

or is it [<0](0) + <0](1) + <0](2) + ...] [ H0 + lambda V] [[0>(0) + [0>(1) + [0>(2) + ...]

Which one is it? They give different result.
 
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They shouldn't be different to the order you calculated the wavefunction. However, it is a general theorem that the wavefunction of order n is sufficient to calculate the energy to order 2n+1 with the second formula you gave.
 
DrDu said:
They shouldn't be different to the order you calculated the wavefunction. However, it is a general theorem that the wavefunction of order n is sufficient to calculate the energy to order 2n+1 with the second formula you gave.

Yeah I was hoping they should be equal. But in the second formula we get terms like <0(1)] H0 [0(1)>, which do not appear in the equation E = E0(0) + E0(1) + E0(2) +..., because

E0(2) = <0] V [0(1)>
 
Be carefull, because your original wavefunction and the perturbed wavefunction do not have the same norm, i.e., although you may chose <0][sum_n O(n)>=1, <sum_n O(n)][sum_n O(n)> >1. That is you have to divide your last formula in #1 by <sum_n O(n)][sum_n O(n)>.
 

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