Chem 125 Lecture 99/25/06

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Exam 1 - Friday, Sept. 29 !Covers Lectures through next Wednesday

Including:

Functional GroupsX-Ray Diffraction

1-Dimensional Quantum Mechanics(Sections I-IV of webpage

& 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 120Thursday 7-10:30 PM, WLH 114

Function of What?

Named by "quantum numbers"(e.g. n,l,m ; 1s ; 3dxy ;

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 classicalkinetic energy

over all particles of interest.

(adjujsts for desired units)

mi vi2

i

Const 12

Kinetic Energy!2

xi2

2

yi2

2

zi2

+ +1mi

i

h2

82

d2

dx21

mC

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 particlestheir 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)

= ex

KineticEnergy

= e-x

C/mparticle infree space

a2 C/m shorter wave higher energy

’’

-C/m

-C/m

Const PE > TE

”Not just a mathematical curiosity.

Actually happens for electrons bound to nucleiat large distance, where 1/r ceases changing much!

(negative kinetic energy!)

Rearranging Schrödinger to give a formula for curve tracing.

C

Curvature of

m

+ V = E

CCurvature 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 Everythinge.g.

Allowed EnergiesStructureDynamicsBonding

Reactivity

Harmonic Spacing

Even Energy Spacing for Hooke’s Law

E = k (n- )12

“…an inkling of what could mean.”

Structure: 2 Probability Density

Max Born (1926)

If one wishes to translate this result intophysical 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)

Oops!

Probability Density

dens

ity

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 energymeans low curvature

7 Å

~ Exponential Decay (e-x)(~ constant negative kinetic energy)

end