Derivation of the Schrödinger Wave Equation Using Variational Principle #3822216 (License: Personal Use)

This image presents a detailed handwritten derivation of the time-independent Schrödinger equation starting from the variational principle δ⟨H⟩ = 0. It includes key steps such as applying integration by parts, enforcing boundary conditions (ψ → 0 at ±∞), and simplifying the Euler-Lagrange expression to arrive at the canonical form -(ℏ²/2m)∇²ψ + Vψ = Eψ. The notation reflects standard quantum mechanics conventions, including expectation values and functional derivatives.

Commonly used in advanced undergraduate or graduate-level quantum mechanics courses, lecture notes, or research supplements to illustrate how the Schrödinger equation arises naturally from energy minimization. Matches user intent for deep conceptual understanding or reference during problem-solving.