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]
=
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.
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 !
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)