1. Quantum Chemistry: Our Agenda Postulates in quantum mechanics (Ch. 3) Schrödinger equation (Ch. 2) Simple examples of V(r) Particle in a box (Ch. 4-5) Harmonic oscillator (vibration) (Ch. 7-8) Particle on a ring or a sphere (rotation) (Ch. 7-8) Hydrogen atom (one-electron atom) (Ch. 9) Extension to chemical systems Many-electron atoms (Ch. 10-11) Diatomic molecules (Ch. 12-13) Polyatomic molecules (Ch. 14) Computational chemistry (Ch. 16)

2. Lecture 3. Simple System 1. Particle in a Box References Engel, Ch. 4-5 Molecular Quantum Mechanics, Atkins & Friedman (4 th ed. 2005), Ch. 2 Introductory Quantum Mechanics, R. L. Liboff (4 th ed, 2004), Ch. 4 A Brief Review of Elementary Quantum Chemistry http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html Wikipedia (http://en.wikipedia.org): Search for Particle in a box

3. Solutions Solving Schrödinger Equation – 1 st Example. A Free (V = 0) Particle Moving in x (A Particle Not in a Box)

4. The Uncertainty Principle When momentum is known precisely, the position cannot be predicted precisely, and vice versa. When the position is known precisely, Location becomes precise at the expense of uncertainty in the momentum

5. Free Translation (V = 0) Confined within Boundaries: A Particle in a Box (Infinite Square Wall Potential) m A particle of mass m is confined between two walls but free inside. The same solution as the free particle but with different boundary condition.  

6. n cannot be zero. Applying boundary conditions Normalization (quantum number) 2

7. zero-point energy node Final Solution (Energy & Wave function) quantized 2 Rapidly changing  Higher E

8. node Energy, Wave function & Probability density Quantum (confinement) effect not constant over x

9. Particle in a Box: Classical vs. Quantum From Wikipedia (particle in a box)

10. Classical Limit: Bohr’s Correspondence Principle n  by increasing E (~ kT) or m or L

11. Case I: T = 300 K, m = m e, L = 10 nm Case II: T = 300 K, m = 1 kg, L = 1 m What is the maximum value for n ?

12. Once  (r, t) is known, all observable properties of the system can be obtained by applying the corresponding operators (they exist!) to the wave function  (r, t). Observed in measurements are only the eigenvalues {a n } which satisfy the eigenvalue equation. (Operator)(function) = (constant number)  (the same function) (Operator corresponding to observable)  = (value of observable)  eigenvalueeigenfunction Postulate 2 of Quantum Mechanics (measurement)

13. A Free Particle Moving along x, Two Cases constant number the same function It’s an eigenfunction of the momentum operator p x Only a constant momentum p x (eigenvalue) is measured. We know the momentum exactly. (  p x = 0) The position x is completely unknown. (  x =  ) It’s not a momentum operator’s eigenfuncition. The momentum is either or. (  p x  ) The position x is partially known. (  x  L) A free particle not confined A free article confined in a box of size L  x   p x  h  0 Heisenberg’s uncertainty principle ++ --

14. Alice, Bob, and Uncertainty Principle…

15. Postulate 4 of Quantum Mechanics (average) For a system in a state described by a normalized wave function , the average value of the observable corresponding to is given by = For a special case when the wavefunction corresponds to an eigenstate,

16. Position, Momentum and Energy of PIB momentum p

18. Two independent quantum numbers