PCLS Lecture 04: SchrodingerEqtn скачать в хорошем качестве

PCLS Lecture 04: SchrodingerEqtn

Lecture Description: The Schrödinger Equation as an Operator and Eigenvalue Problem In this lecture for Physical Chemistry for the Life Sciences (PCLS), we explore the claim that the Schrödinger equation is both an operator equation and an eigenvalue problem. Understanding this dual perspective is essential for making sense of how quantum mechanics is formulated and applied. We develop this idea through six major topics: Connection to Classical Waves We begin with a worksheet that shows how the Schrödinger equation is related to the classical wave equation, which we solved previously. This helps establish continuity between classical and quantum descriptions. What Is an Operator? We introduce the concept of an operator and show that we have already been using operators all along—just without calling them that—through familiar examples. The Schrödinger Equation as an Operator Equation We rewrite the Schrödinger equation in operator form and introduce the Hamiltonian operator, which plays a central role in quantum mechanics. The Linearity Test Because operators in quantum mechanics must be linear, we learn how to test an operator for linearity and why this property is essential. Eigenvalue Problems We introduce eigenvalue problems and examine familiar examples, including how functions of the form eax e ax can be eigenfunctions of the derivative operator. Casting the Schrödinger Equation as an Eigenvalue Problem Finally, we express the Schrödinger equation using the Hamiltonian operator H^ H ^ and arrive at the familiar form H^Ψ=EΨ H ^ Ψ=EΨ This shows explicitly how the Schrödinger equation is both an operator equation and an eigenvalue problem.