Quantum Mechanics: Numerical Solution of Schrödinger's Equation, Study notes of Physics

Download Quantum Mechanics: Numerical Solution of Schrödinger's Equation and more Study notes Physics in PDF only on Docsity! Lec9 • Intro to Quantum Mechanics • Numerical solution of Schödinger’s equa- tion 1 Why/when quantum ? • Newton’s laws give a very accurate descrip- tion of the behavior of everyday objects/motions. • But they fail miserably to describe atoms ! • This was a crisis for physics at turn of cen- tury ... • Eg Laws of EM + Newton mechanics pre- dicts atoms should be unstable • Electrons classically have any energy but see only discrete energies • Energy of electromagnetic waves in vac- uum infinite! 2 Basics • Discard notion that microscopic objects like electrons can a well defined position, veloc- ity etc. • Not a practical issue but one of principle! • Instead think of them as being described by a wavefunction Ψ(x,t). Like a usual wave in sense that electron is not localized like a classical particle. • But this is a probability wave. |Ψ(x, t)|2 yields probability of finding particle at (x, t). • Dynamics: replace Newton’s laws (simple ordinary differential equations) by Schrödnger’s equation (partial differential equation) 4 Points to note − h̄2 2m ∂2Ψ ∂x2 + V Ψ = ih̄ ∂Ψ ∂t • Equation looks like a funny wave equation (but only first order in t) • Involves square root of minus 1 (i). In gen- eral Ψ is complex! Hence need |Ψ|2 for positive real probability • New fundamental constant introduced h̄ = 1.05 × 10−34 Js. Planck’s constant. 5 Stationary states • Put Ψ = φ(x)e−iEt/h̄. Plug into equation. Find time independent Schrödinger equa- tion − h̄2 2m ∂2φ ∂x2 + V φ = Eφ • Such a wavefunction describes the allowed state of say electron in an atom with E being its energy. • Try to solve schematically. In general only certain energies allowed! 6 How to solve numerically ? Rewrite equations: dφ dx = p dp dx = [ 2m (V (x) − E) h̄2 ] φ Consider case V (−x) = V (x). Can show φ(x) is either even or odd function. In former case choose φ(0) = 1 and p(0) = 0 Choose some E and integrate equations using Euler/leapfrog. Look at |φ(x)| for large |x|, If it is growing will not be able to impose ∫ φ2 = 1. Choose another E and try again. Shooting method. 9 Summary • Find only discrete set of E work. So both wavefunction and E are output from cal- culation! • E0 > Vmin. Particle cannot be stationary at minimum of potential. Quantum fluc- tuations. • E increases with number of oscillations. 10