Introduction to Quantum Mechanics: Understanding the Radical Shift from Classical Physics, Study notes of Physics

Download Introduction to Quantum Mechanics: Understanding the Radical Shift from Classical Physics and more Study notes Physics in PDF only on Docsity! Lec11 • 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’s mechanics predicts atoms should be unstable – Electrons classically have any energy but see only discrete energies – Energy of electromagnetic waves in vac- uum infinite! Diffraction of electrons.. – Photoelectric effect – light waves like particles ! 2 Schrödinger’s equation − 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 Allowed energies Rewrite equation: d2φ dt2 = [ 2m (V (x) − E) h̄2 ] φ • For a bound state need V > E at large x. • In which case this equation develops expo- nential solutions φ ∼ eκx, e−κx with κ2 = 2m(V (x)−E) h̄2 . • For small x V < E and generate oscillatory solutions φ ∼ sinκx, cosκx 7 What happens • Find only discrete set of E work. So both wavefunction and E are output from cal- culation! • E0 > Vmin. Otherwise cannot match with large x asymptotics. Particle cannot be stationary at minimum of potential. Quan- tum fluctuations even for zero tempera- ture. • E increases with number of oscillations. • Find E using bisection algorithm ... 10 Summary • In QM speak only of probabilities. Throw out notion that particles have simultane- ously well-defined positions and momenta (Heisenberg uncertainty principle: δxδp ≥ h̄.) • Probabilities gotten by solving Schrödinger equation. Can be solved using same algo- rithm as used for Newton’s equations! • For bound state problems find discrete spec- trum of allowed energies/states. • Many other things we haven’t talked about: scattering, making quantum observations, relativity, many particles, approximation meth- ods, operators, connection to classical physics, ... 11