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The magnitude of the curvature of a wave function relates to the kinetic energy of the
system, and the square of the wave function relates to probability density. The requirement
that the wavefunction not diverge in areas of negative kinetic energy can constrain total
energies to certain values, a property which is explored for the harmonic oscillator, the Morse
potential, and the Columbic potential. Consideration of the influence of mass reveals an
“isotope effect” on dynamics and on the energy, vibration frequency, and length of bonds.
Introducing the double minimum potential leads to the study of bonding.
Chemistry 125: Lecture 8Sept 18, 2009
One-Dimensional Wave Functions
For copyright notice see final page of this file
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 curvature recipe allows stepwise tracing of (x), from initial (0) [= 1], with initial slope [0], and a guessed E.
100 kcal/mole
2.5Å0
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)
Infinitesimal denominator allows large negative KE
without significant curvature.
Thus initial slope 0
100 kcal/mole
2.5Å0
20.74 kcal/mole
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
Much Harder for Many Particles
Is it worth our effort?
Reward for Finding
Knowledge of Everythinge.g.
Allowed EnergiesStructureDynamicsBonding
Reactivity
Harmonic Spacing
Even Energy Spacing for Hooke’s Law
E = k (n- )12
“We only wish that we could glean
an inkling of what could mean.”
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 !
“Normalization”
Scale so that total (integral of)
2 volume = 1
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)
Coulomb Three
E = kn2
50 Å
Higher levelsspread way out
Reward for Finding
Knowledge of Everythinge.g.
Allowed EnergiesStructureDynamicsBonding
Reactivity
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
What about U235
or a marble?
m = 1
m = 14
Mass Effect and Vibration
C
HHigher-energy H shifted to right in unsymmetrical
Morse Potential.
half-maximumprobability density
Mass Effect and Vibration
H
C
Mass Effect and Vibration
H
C
H
C ±0.05Å (3% of X-C)
±0.1Å (9% of X-H)
Dunitz et al. (1981)
Typically vibratingby ±0.050 Åin the crystal
Reward for Finding
Knowledge of Everythinge.g.
Allowed EnergiesStructureDynamicsBonding
Reactivity
Single- vs. Double MinimumFor Hooke's Lawthe Blue Energy
is too Lowand
the Red Energyis too High.
The Correct LowestEnergy must lie
between these values.
Single-Mimimum
Actuallythis is aDouble-
Minimum.
The Blue and Reds are correct!
What if the wells were further apart?
Closer wells givelowered minimum energy
and higher next energy
~ same as single-minimum
solution
“Splitting”
End of Lecture 8Sept 18, 2009
Copyright © J. M. McBride 2009. Some rights reserved. Except for cited third-party materials, and those used by visiting speakers, all content is licensed under a Creative Commons License (Attribution-NonCommercial-ShareAlike 3.0).
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The following attribution may be used when reusing material that is not identified as third-party content: J. M. McBride, Chem 125. License: Creative Commons BY-NC-SA 3.0