L22.1 Hydrogen Molecule Ion (H₂⁺) part 6/7

#variationalMethod #groundstateenergy #hydrogenmoleculeion
00:00 - Introduction to Hamiltonian and Kinetic Energy Terms
00:10 - Breaking Down the Hamiltonian Components
00:20 - Applying Terms to Wave Functions (ψ₀)
00:33 - Simplifying the Hamiltonian Expression
01:29 - Defining and Solving Exchange Terms
02:59 - Explanation of Cross Terms and Orthonormality
05:50 - Introduction to Exchange Terms in Quantum Mechanics
07:55 - Simplifying the Hamiltonian Equation
09:15 - Defining Parameters (D and X) for Calculations
11:00 - Calculating the Value of D
13:48 - Calculating the Value of X
14:34 - Rewriting the Hamiltonian with D and X
15:08 - Ground State Energy (E₁) and Bohr Radius
17:03 - Proton-Proton Repulsion and Coulombic Potential
18:24 - Finalizing the Hamiltonian Equation

Lecture Notes:
https://drive.google.com/file/d/1kNzB...

Board Image:
https://drive.google.com/file/d/1UyPC...

In this quantum mechanics lecture, we dive deep into the Hamiltonian operator, focusing on its components, exchange terms, and proton-proton repulsion. Starting with the kinetic energy term, we explore how the Hamiltonian acts on wave functions (ψ₀) and simplify the resulting equations. Key topics include:

Exchange Terms: Understanding cross terms and their role in quantum calculations.
Orthonormality: How wave functions interact and contribute to the Hamiltonian.
Proton-Proton Repulsion: Calculating Coulombic potential energy and its relation to ground state energy (E₁).
Bohr Radius: Converting terms to simplify calculations.

This lecture provides a step-by-step breakdown of complex quantum mechanics concepts, making it ideal for students and enthusiasts. Don’t forget to check the lecture notes for detailed calculations!

quantum mechanics, Hamiltonian operator, exchange terms, proton-proton repulsion, wave functions, orthonormality, Coulombic potential, ground state energy, Bohr radius, kinetic energy, quantum calculations, physics lecture, quantum physics, advanced quantum mechanics.