About time-independent non-degenerate perturbation expansion

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  • #1
Ishika_96_sparkles
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TL;DR Summary
Confusion regarding the state vector expansion over a set of basis functions and the perturbative expansion in the powers of $lambda$.
We know that any state vector in Hilbert space can be expanded as
$$ \Psi = \sum_{n=1}^{\infty} \alpha_i \psi_i $$

However, when we start the perturbation expansion of the eigenvalues and the eigenfunctions as
$$E_a = \lambda^0 E_a^0+\lambda^1 E_a^1+\lambda^2 E_a^2+\lambda^3 E_a^3+...+\lambda^n E_a^n+...$$
and
$$\psi_a = \lambda^0 \psi_a^0+\lambda^1 \psi_a^1+\lambda^2 \psi_a^2+\lambda^3 \psi_a^3+...+\lambda^n \Psi_a^n+...$$
respectively, is it the single state being expanded over its basis set or what? What exactly is this set ##{\psi_a^i}_{i=1}^n##?
How should I differentiate between the two?

PS: If my question is not clear, then please help me refine it as I really want to understand the basics of perturbation theory.
 
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  • #2
To reduce confusion, don't reuse the same symbols. For instance, consider
$$ \Psi = \sum_{n=1}^{\infty} \alpha_i \phi_i $$
where the ##\{ \phi_i \}## form some basis.

Then, in
Ishika_96_sparkles said:
$$\psi_a = \lambda^0 \psi_a^0+\lambda^1 \psi_a^1+\lambda^2 \psi_a^2+\lambda^3 \psi_a^3+...+\lambda^n \Psi_a^n+...$$
##\psi_a^0## would usually be an eigenstate of the unperturbed Hamiltonian ##\hat{H}_0##, and the other ##\psi_a^i## more generic functions (such that ##\psi_a## looks more and more like an eigenstate of the full Hamiltonian).

All the ##\psi_a^i## could be expressed in terms of basis states ##\{ \phi_i \}## (which would typically be the eigenstates of ##\hat{H}_0##, but could also be the eigenstates of the full Hamiltonian, or any other basis functions).
 
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  • #3
Thank you @DrClaude! for a clear writeup and reply.
DrClaude said:
ψai more generic functions (such that ψa looks more and more like an eigenstate of the full Hamiltonian).
which probably are the "first-order... second-order...correction" terms that modify the unperturbed eigenfunction to match the eigenfunction of the full Hamiltonian. did i get that right?

Furthermore, all the ##\psi_a^i## are expandable in the original basis set ##\{\phi_i\}##.
 
  • #4
Ishika_96_sparkles said:
Thank you @DrClaude! for a clear writeup and reply.

which probably are the "first-order... second-order...correction" terms that modify the unperturbed eigenfunction to match the eigenfunction of the full Hamiltonian. did i get that right?

Furthermore, all the ##\psi_a^i## are expandable in the original basis set ##\{\phi_i\}##.
Correct.
 
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