Peter Atkins • Julio de Paula Atkins’ Physical Chemistry Eighth Edition Chapter 9 Quantum Theory: Techniques and Applications Copyright © 2006 by Peter Atkins and Julio de Paula

Chap 9 Quantum Theory: Techniques and Applications Objectives: Solve the Schrodinger equation for: Translational motion (Particle in a box) Vibrational motion (Harmonic and anharmonic oscillator) Rotational motion (Particle on a ring & on a sphere)

∴ For acceptable ψ, boundary conditions Fig 9.1 Particle in a one-dimensional box Particle is not free ∴ For acceptable ψ, boundary conditions must be set: ψ must vanish at x = 0 and x = L Implies quantization!

Fig 9.2 Allowed energy levels for a particle in a one-dimensional box Normalized wavefunction: n = 1, 2, 3, … n ≠ 0 so: is called the zero-point energy

Fig 9.3 First five normalized wavefunctions of PIB

Fig 9.4 First two normalized wavefunctions of PIB with probability distributions

Real world PIB: a delocalized π electron in a conjugated system 1 β-Carotene

Correspondence Principle: Classical mechanics emerges from quantum mechanics as high quantum numbers are reached i.e., particle may be found anywhere as n → ∞

Fig 9.5 Probability of two wavefunctions ψ1 and ψ3 are orthogonal or orthonormal In Bra-ket notation: 〈1|3〉 = 0 when n ≠ n'

Fig 9.6 Two dimensional square well

Fig 9.7 Contours for particle in 2-D rectangular well n1 = n2 =1 n1 = 1, n2 =2 n1 = 2, n2 =1 n1 = 2, n2 =2

Fig 9.8 Contours for particle in 2-D square surface Here, L1 = L2 = L are said to be degenerate

Fig 9.9 Tunnelling of a particle through wall when V < ∞ Leakage by penetration through a classically forbidden region

Fig 9.13 Wavefunction of a heavy particle decays more rapidly than that of a light particle Light particles have higher probability of tunnelling

Tunneling Chemical effects of tunneling: Isotope-dependence of reactions rates Transfer of a proton in an acid-base reaction Mechanism of enzyme-catalyzed reactions Electron transfer in redox reactions Scanning tunneling microscopy (STM)

Fig 9.16 Tip of a Scanning Tunnelling Microscope (STM) Pt-Rh or W Ultrahigh vacuum

Title : The Making of the Circular Corral Media : Iron on Copper (111) We can predict what goes on in the corral by solving the classic eigenvalue problem in quantum mechanics -- a particle in a hard-wall circular box.

Title : Stadium Corral Media : Iron on Copper (111)