L6.2 Degenerate Perturbation Theory: Problem 6.6 Detailed Solution Part (a) 2/2
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0:00 - Introduction to Matrix Elements in Quantum Mechanics
0:12 - Overview of Energy Terms: E⁺ and E⁻ Multiplication
0:37 - Expanding Key Quantum Expressions
1:03 - Defining Variable Relations in Quantum Systems
1:25 - Breaking Down Multiplication of Energy Terms
2:06 - Applying Algebraic Formulas: A² - B²
2:39 - Simplifying Terms in Quantum Expressions
3:40 - Detailed Expansion of Modulus Squared Terms
4:48 - Handling Negative and Positive Term Cancellations
6:06 - Final Proof for Energy Term Multiplication
7:04 - Proving the Addition of Energy Terms
9:33 - Correction and Simplification of Final Results
Lecture Notes:
https://drive.google.com/file/d/1bACN...
Board Image:
https://drive.google.com/file/d/1WCth...
Welcome to this detailed solution of Problem 6.6 from David J. Griffiths' Introduction to Quantum Mechanics (2nd Edition). In this video, we'll walk through the problem step by step, exploring the orthogonality of unperturbed states, matrix elements of the perturbation Hamiltonian, and energy shifts.
Problem Statement:
Let the two "good" unperturbed states be:
ψ₀₊ = α₊ψₐ₀ + β₊ψ_b₀,
where α₊ and β₊ are determined (up to normalization) by Equation 6.22 (or Equation 6.24). Show explicitly that:
(a) ψ₀₊ and ψ₀₋ are orthogonal (⟨ψ₀₊|ψ₀₋⟩ = 0);
(b) ⟨ψ₀₊|H'|ψ₀₋⟩ = 0;
(c) ⟨ψ₀₊|H'|ψ₀₊⟩ = E₊¹ with E₊¹ given by Equation 6.27.
In this lecture on quantum mechanics, we delve into the intricacies of energy relations, focusing on E⁺ and E⁻ in matrix mechanics. We systematically derive key equations, simplify complex expressions, and explore the interplay of variables like Wab, Waa, and Wbb. Whether you're a student or an enthusiast, this lecture is designed to help you grasp the mathematical elegance of quantum mechanics. Stay tuned to master the algebraic techniques used in solving quantum equations and deepen your understanding of fundamental physics.
Quantum mechanics lecture
Energy terms in quantum mechanics
Matrix mechanics derivation
Quantum physics algebra
Simplifying quantum expressions
E⁺ and E⁻ energy relations
Quantum variables Wab, Waa, and Wbb
Advanced quantum mechanics tutorials
Mathematical quantum mechanics
Physics problem-solving techniques
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