- #1

Ishika_96_sparkles

- 57

- 22

- 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.

$$ \Psi = \sum_{n=1}^{\infty} \alpha_i \psi_i $$

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.