Quantum Chemistry: Our Agenda

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Quantum Chemistry Our Agenda

• Postulates in quantum mechanics (Ch. 3)
• Schrödinger equation (Ch. 2)
• Simple examples of V(r)
• Particle in a box (Ch. 4-5)
• Harmonic oscillator (vibration) (Ch. 7-8)
• Particle on a ring or a sphere (rotation) (Ch.
  7-8)
• Hydrogen atom (one-electron atom) (Ch. 9)
• Extension to chemical systems
• Many-electron atoms (Ch. 10-11)
• Diatomic molecules (Ch. 12-13)
• Polyatomic molecules (Ch. 14)
• Computational chemistry (Ch. 16)

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Lecture 3. Simple System 1. Particle in a Box
References

• Engel, Ch. 4-5
• Molecular Quantum Mechanics, Atkins Friedman
  (4th ed. 2005), Ch. 2
• Introductory Quantum Mechanics, R. L. Liboff
  (4th ed, 2004), Ch. 4
• A Brief Review of Elementary Quantum Chemistry
• http//vergil.chemistry.gatech.edu/notes/quantrev
  /quantrev.html
• Wikipedia (http//en.wikipedia.org) Search for
• Particle in a box

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Free Translational Motion (V 0)
Solutions
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Free Translation (V 0) with BoundariesParticle
in a Box (Infinite Square Wall Potential)
?
?
The same solution as the free particle but with
different boundary condition.
A particle of mass m is confined between two
walls but free inside.
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• Applying boundary conditions

n cannot be zero.
(quantum number)
Normalization
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Final Solution (Energy Wave function)
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Rapidly changing ?
Higher E
node
quantized
zero-point energy
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Energy, Wave function Probability density
node
not constant over x
Quantum (confinement) effect
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Classical Limit Bohrs Correspondence Principle
n ??
by increasing E ( kT) or m or L
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What is the maximum value for n ?
Case I T 300 K, m me, L 1 nm Case II
T 300 K, m 1 kg, L 1 m
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Position, Momentum and Energy of PIB
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Two independent quantum numbers
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