Perturbation theory byeushhdhehehhshdhe.pptx

It Quantum mechanic

1. Perturbation theory Present by Group 1

2. Abstract Perturbation theory is a mathematical approach used to find an approximate solution to a problem by starting from the exact solution of a simpler, related problem. It’s widely used in quantum mechanics, physics, and engineering , allowing for easier calculation of complex phenomena by breaking them down into more manageable parts.

3. Introduction Perturbation theory is a general method to analyze complex quantum systems in terms of simpler variants. The method relies on the expectation values, matrix elements and overlap integrals just introduced, which we now use to break down complex quantum processes into simpler parts.

4. Understanding Perturbation Theory Perturbation theory studies small disturbances in a system’s Hamiltonian, which represents its total energy. It helps analyze a system’s energy when exact solutions are unknown by examining the effects of external factors. Perturbation theory can be applied to time-dependent and time-independent systems.

5. Perturbation Theory Overview 1.Expansion of Hamiltonian and State Functions

H=H0+λH1
Ψ=Ψ0+λΨ1
E=E0+λE1

6. 2.Perturbed Schrödinger Equation

HΨ=EΨ

Substituting the expansions, we get:
(H0+λH1)+(Ψ0+λΨ1)=(E0+λE1)+(Ψ0+λΨ1)

7. 3.Simplifying and Grouping Terms: H0Ψ0+λH0Ψ1+λH1Ψ0+λ2H1Ψ1=E0Ψ0+λE0Ψ1+λE1Ψ0+λ2E1Ψ1 Comparing terms of the same order in λ, we obtain:

H0Ψ0=E0Ψ0
H 0 Ψ 1 +H 1 Ψ 0 =E 0 Ψ1+E 1 Ψ 0

8. 4. Forming Inner Product to Solve for E1
Taking the inner product with Ψ0
⟨Ψ0 ∣H0 ∣Ψ1 ⟩+⟨Ψ0 ∣H1 ∣Ψ 0 ⟩=E 0 ⟨Ψ 0 ∣Ψ 1 ⟩+E 1 ⟨Ψ 0 ∣Ψ 0 ⟩
Noting that
⟨Ψ0∣𝐻0∣Ψ1⟩=𝐸0⟨Ψ0∣Ψ1⟩ and ⟨Ψ0∣Ψ0⟩=1,
We solve for 𝐸1:
E 1 =⟨Ψ 0 ∣H 1 ∣Ψ 0 ⟩
This gives the first-order correction to the energy due to the perturbation H1

9. Conclusion Perturbation theory is a method used in physics and engineering to find approximate solutions to complex problems by treating them as small deviations from simpler, solvable systems. It expands the solution in a series of corrections based on a small parameter, providing progressively more accurate results. This technique is essential for studying systems where exact solutions are difficult to obtain.

10. References www.google.com

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