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1 year ago in Quantum Mechanics By Krirthi

What is the mathematical relationship between the Sturm-Liouville problem and the time-independent Schrödinger equation?

We solved Sturm-Liouville problems in my PDEs class, emphasizing eigenfunctions and orthogonality. Now in Quantum Mechanics, we're solving the Schrödinger equation and getting the same concepts. I sense a deep equivalence but need help formally connecting the dots. What specific terms in the Sturm-Liouville operator correspond to the Hamiltonian's kinetic and potential energy parts?

MathematicalPhysics Sturm-LiouvilleTheory SchrOdingerEquation

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By Rohini Singh Answered 1 year ago

You've identified one of the most elegant unifications in theoretical physics. The time-independent Schrödinger equation, Hψ = Eψ, is precisely a Sturm-Liouville problem. In the standard S-L form, the Hamiltonian's kinetic energy term (-?²/2m d²/dx²) corresponds to the derivative term [d/dx (p(x) d/dx)], with p(x) being ?²/2m. The potential V(x) becomes the q(x) term. The eigenvalue E is your λ. This formal mapping is why we inherit the powerful S-L theorems: your wavefunctions ψ_n are guaranteed to be orthogonal and form a complete basis for your Hilbert space, which is the foundation of quantum expansion methods.

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What is the mathematical relationship between the Sturm-Liouville problem and the time-independent Schrödinger equation?

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