chem 125 lecture 8 9/19/08 this material is for the exclusive use of chem 125 students at yale and...
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Chem 125 Lecture 89/19/08
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. 26 !Covers Lectures through Wednesday
Including:
Functional GroupsX-Ray Diffraction
1-Dimensional Quantum Mechanics & 1-Electron Atoms
(Sections I-V of quantum webpage& Erwin Meets Goldilocks)
Exam Review 8-10 pm Wednesday, Sept. 24, Room WLH ???Come with questions.
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.
100 kcal/mole
2.5Å0
Nodes and Quantizationin One Dimension
from Erwin Meets Goldilocks(for Wiki see Monday Problem Set)
Too Cold
Too Hot
Just Right!
20.74 kcal/moleGuess 21 kcal/mole
Guess 20 kcal/mole
DangerNegativeKineticEnergy
(Curve Away from Baseline)
DangerNegativeKineticEnergy
(Curve Away from Baseline)
Erwin Meets Goldilocks
100 kcal/mole
2.5Å0
20.74 kcal/mole
Erwin Meets Goldilocks
Could there be a lower-energy Psi?
4.15 kcal/mole
12.45 kcal/mole
Could there be an energy between?
NODES0 because of sign change
More Energy
More Curvature
More Nodes
Structure: 2 Probability Density
Max Born (June 25, 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!
Structure: 2 Probability Density
Aber eine innere Stimme sagt mir, dass das doch nicht der wahre Jakob ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt sie uns kaum näher. Jedenfalls bin ich überzeugt, dass der nicht würfelt.
But an inner voice tells me, that this is not the real thing. The theory yields a great deal, but it brings us no nearer to the secret of the Old One. Anyway I am convinced that He does not play dice.
Albert Einstein to Max BornDecember 4, 1926
Probability Density
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!)
ProbabilityPenetrates
the 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)
TotalPotential
Kinetic
Morse Quantization
Morse Potential : Quantized; Probability Spreads to Right
Energies not evenly spaced as for Hooke’s Law
7 Å
As the energy increases, alongwith the number of nodes, the
well widens more than it wouldfor a Hooke’s Law parabola.
Thus wavelengths become longer,and energies lower, than
expected for Hooke’s Law.
Morse Continuum
Morse Potential : Not Quantized above Dissociation Limit
~ sin(x)
(~ constant positivekinetic energy)
TotalPotential Kinetic
Coulombic Spacing
50 Å
One-dimensional
e- in Coulombic
Potential of a Proton
! High Curvature(Erwin Program
is approximate)
Change mass Single-bonded H
Increase mass from1 to 14 (H to C14)
C
Curvature of
m
Greater mass meansmore curvature forthe same energy.
Change mass Single-bonded H
Increase mass from1 to 14 (H to C14)
Need to lower energy(curvature) for m=14
C
Curvature of
m
Greater mass meansmore curvature forthe same energy.
Mass Effect and Vibration
C
HHigher-energy H shifted to right in unsymmetrical
Morse Potential.
half-maximumprobability density