Chem 125 Lecture 9 9/25/06 Projected material This material is for the exclusive use of Chem 125 students at Yale and may not be copied or distributed further. It is not readily understood without reference to notes from the lecture.

Exam 1 - Friday, Sept. 29 ! Covers Lectures through next Wednesday Including: Functional Groups X-Ray Diffraction 1-Dimensional Quantum Mechanics (Sections I-IV of webpage & Erwin Meets Goldilocks)Erwin Meets Goldilocks IMPORTANT PROBLEMS therein due Wednesday Exam Review 7-9 pm Tuesday, Room WLH 208 Other Help Available Wednesday 8-10 PM, WLH 120 Thursday 7-10:30 PM, WLH 114

 Function of What? Named by "quantum numbers" (e.g. n,l,m ; 1s ; 3d xy ;  Function of Particle Position(s) [and time and "spin"] We focus first on one dimension, then three dimensions (one electron), then many-electron atoms, then many atoms, & finally functional groups. N particles  3N arguments! [sometimes 4N+1]

Schrödinger Equation H  = E  (for “stationary” states) time-independent ( E times  )(NOT H times  )

= H  = E  Kinetic Energy + Potential Energy = Total Energy Given - Nothing to do with  (Couloumb is just fine) Hold your breath! H  = E 

Kinetic Energy? Sum of classical kinetic energy over all particles of interest. (adjujsts for desired units) m i v i 2  i Const  1 2

Kinetic Energy! 22 xi2xi2  22 yi2yi2  22 zi2zi2  ++ 1 mimi  i h2h2 8282  d2d2 dx2dx2  1 m C  C  Curvature of  m One particle;One dimension: Note: H works with the shape of , not just its value.

Solving a Quantum Problem Given : a set of particles their masses & their potential energy law [ e.g. 1 Particle/1 Dimension : 1 amu & Hooke's Law ] To Find :  a Function of the position(s) of the particle(s) Such that H  /  is the same (E) everywhere AND  remains finite!!! (single-valued, continuous,  2 integrable)

The Jeopardy Approach Answer Problem  = sin (x)  = sin (ax)  = e x Kinetic Energy  = e -x C/m particle in free space a 2 C/m shorter wave  higher energy ’’ - C/m Const PE > TE ” Not just a mathematical curiosity. Actually happens for electrons bound to nuclei at large distance, where 1/r ceases changing much! (negative kinetic energy ! )

Rearranging Schr ö dinger to give a formula for curve tracing. C  Curvature of  m + V = E C  Curvature of  m (V- E) = Curves away from 0 for V>E; toward 0 for V<E. Since m, C, V(x) are given, this recipe allows tracing  (x) in steps, from initial  (0) [= 1], with initial slope [0], and a guessed E.

Much Harder for Many Particles Is it worth our effort?

What we can learn from Erwin Meets Goldilocks

Reward for Finding  Knowledge of Everything e.g. Allowed Energies Structure Dynamics Bonding Reactivity

Harmonic Spacing Even Energy Spacing for Hooke’s Law E = k (n- ) 1 2

“…an inkling of what  could mean.”

Structure:  2  Probability Density Max Born (1926) If one wishes to translate this result into physical terms, only one interpretation is possible,  signifies the probability [of the structure] 1 ) Correction in proof: more careful consideration shows that the probability is proportional to the square of the size of . 1)1) Oops!

Probability Density density height Suppose the total mass in the flask is 1 kg. How much (or what fraction) is exactly 1 cm from the bottom? Multiply density by volume for mass (or fraction, or probability). 0 !

“Normalization” Scale  so that total (integral of)   2  volume = 1

Harmonic Probability Ultimately Probability Builds Up at the Extremes 1.5 Å (not normalized!) Classically ‘Forbidden’ Region

Morse Quantization Morse Potential : Quantized; Probability Spreads to Right Because low kinetic energy means low curvature 7 Å ~ Exponential Decay (e -x ) (~ constant negative kinetic energy)

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