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

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