Wavefunctions and Bound Systems Chapters Q9 (3 rd ed)

What’s a Wavefunction? Looking a head - Schroedinger’s Equation: H H = E  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: –Quantum: 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 Momentum operator Wavefunction (eigenfunction) Eigenvalue for momentum

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?

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 “Coulomb”“Spring”“Box” The quanton’s wavefunction must “fit” within these potentials – that sets and hence the energy and momentum of the quanton

Suggested Study Problems from Chp 7 Q7B.1, Q7B.4, Q7S.4