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Where do the energy equations come from? The motion of atoms, molecules, electrons … is described by Quantum Mechanics. The central equation of Quantum Mechanics is the Schrödinger Equation. Solving the Schrödinger equation for a ‘problem’, results in an expression for the energy of the particle(s) in the problem and a wavefunction. The wavefunction is a mathematical function of the coordinates of the particle(s) and provides information on the location of the particle(s). (Note: In Classical Mechanics, one solves Newton’s 2nd Law equation to get the energy and position of the particle(s).) A prescription for obtaining the Schrödinger equation of a problem for one particle of mass m, moving in 1D x. 1) H ( x) ( x) E ( x) is the completely general form of the time-independent Schrödinger equation where is the wavefunction, a function of the coordinate x (x) E H (x) is the Energy of the particle is the Hamiltonian operator (Note: want to determine (x) and E by solving the equation.) 2) Write the Hamiltonian in Classical terms as the sum of Kinetic Energy (T) and Potential Energy (V) H ( x ) T ( x ) V ( x ) p / 2m V ( x ) 2 (Note: T = mv2 / 2 = p2 / 2m and V(x) depends on the problem being considered.) 3) Make the change from Classical Mechanics to Quantum Mechanics by introducing: p i p 2 2 2 2 d dx 2 2 Note: V(x) seldom (never in this course) depends on p and therefore this change only affects the p2 / 2m term. H ( x) ( x) E ( x) d 2m dx V ( x) ( x) E ( x) 2 2 2 This is the time-independent one dimensional Schrödinger Equation Note: Hamiltonian operator because it involves a differential operator (d2 / dx2). The ‘relationship’, p i , is the essential difference between Classical and Quantum Mechanics. The operator means that there has to be something, a function, for it to operate on – the wavefunction.