I About time-independent non-degenerate perturbation expansion

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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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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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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\}##.
 
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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Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. Towards the end of the first lecture for the Qiskit Global Summer School 2025, Foundations of Quantum Mechanics, Olivia Lanes (Global Lead, Content and Education IBM) stated... Source: https://www.physicsforums.com/insights/quantum-entanglement-is-a-kinematic-fact-not-a-dynamical-effect/ by @RUTA
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