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Wavefunctions and Bound Systems Chapters Q9 (3 rd ed) What’s a Wavefunction? • Looking a head Schroedinger’s Equation: Hy= E y Wavefunction “psi” solves Schroedinger’s equation and contains, in its components, all of the information we need to determine values of observables… Wavefunctions are only part of the story… • The information in the wavefunction is “coded” in its components. Actual values for observables depend on “how you ask” the wavefunction • Operators tell us what we want to know: • Example: momentum – Classical: p mv – Quantum: d p i dx This is not as strange as it seems – you got used to thinking of momentum as “em” times “vee” – this is just another way to think about it! example (cont…) Consider a free particle traveling in space… An eigenvalue equation Eigenvalue for py py momentum d i ( kx t ) i ( kx t ) i ( Ae ) pAe dx Momentum operator Wavefunction (eigenfunction) What wavefunctions are (and are not!) • Wavefunctions are mathematical ideas that depict probability distributions (Born interpretation) • Wavefunctions can be described using the mathematics of waves but are not “real” • Wavefunctions obey strict mathematical rules: – continuous, differentiable, finite • Consider the wave function below: This is the wavefunction This is the probability distribution Applications… • Particle in a Box • Quantized Spring! • Bohr Atom Molecular Examples of Particle in a Box… From Carrots to Quantum Physics! • Why are carrots orange? C40 H56 beta Carotene and Cyanine • CY-5 – Bond length 0.14 nm – # bonds 6 – Level n = 3 4 • b-Carotene ? – Bond length 0.14 nm – # bonds 18 – Level n = 9 10 How a quanton is “bound” depends on the Potential Energy Function V(x) • V(x) is where the chemistry happens! • The Shape of the potential sets the shape of possible wavefunctions (energy eigenfunctions) • Energy Eigenfunctions are stationary states (ie – standing waves!) Three Potentials in Chp 7 “Box” “Spring” “Coulomb” The quanton’s wavefunction must “fit” within these potentials – that sets l and hence the energy and momentum of the quanton Suggested Study Problems from Chp 7 Q7B.1, Q7B.4, Q7S.4