Iyengar Group
Theoretical and Computational Chemistry and Chemical Physics

C561 Class Notes

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• Basic Linear algebra
• Complex Variables

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Part I: Fundamental concepts

• The Stern-Gerlach Experiments To test the Stern-Gerlach experiments, download this nifty java applet. It will allow you to run the Stern-Gerlach experiments on your computer and understand it better. (You will need to have Java installed and enabled for this to run from the browser.) Click on the help menu to customize experiments.
• Analogy between plane-polarized light, circularly polarized and the Stern-Gerlach experiments drive home wave-particle duality and vectors as a mathematical theory to understand quantum mechanics
• A brief summary of Stern-Gerlach experiments
• Postulates of Quantum mechanics
• Rationalizing the results from the Stern-Gerlach experiments provides the necessary structure to proceed further with quantum mechanics: Pauli spin matrices as representations of the spin operators. Commutators between spin operators
• de Broglie’s Wave particle duality
• The time-dependent and time-independent Schrodinger Equations
• Probability current or flux: as introduced by considering the imaginary part of the Time independent Schrodinger Equation. The imaginary part of the TDSE is essentially a fluid continuity equation. Flux in minus flux out is rate of change of density.
• The Classical limit for the Time independent Schrodinger Equation.
• What makes it possible for two observations to be compatible? That is one not to destroy the other as we have seen during the Stern Gerlach?
• The Heissenberg Uncertainty Principle
• A visual treatment of uncertainty

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Part II: Analytically solvable problems

• Particle-in-a-box
• Resonance in poly-enes explained using particle-in-a-box
• Resonance in benzene explained using particle-in-a-ring. Understanding molecular (pi-) orbitals in benzene using particle-in-a-ring.
• Quantum Confinement: Applications to quantum dots
• One dimensional problems. Connections to chemical reactions and kinetics
• Harmonic Oscillator
• Harmonic Oscillator derivation
• Harmonic Oscillator using Second quantization (due to Dirac)

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Part III: Atomic and molecular systems

• As discussed in class, we are able to obtain spherical harmonics purely from the symmetry of the sphere. As noted, if one were to solve a 2D PIB problem, with periodic BCs across one axis and slope of the wavefunction equal to zero BCs across the other direction, we obtain these states. See if you can visualise the s, p, d and f orbitals from these pictures? Remember these are drawings on the surface of a sphere. I have assumed the two directions to be x and y, as opposed to theta and phi. f and h are functions along the two directions and the first index in denoting each funcion is the quantum number obtained from the respective BCs.
• Theory of Angular Momentum and Ladder operators
• Spherical Harmonics: Transforming the angular momentum problem in radial coordinates
• Hydrogen atom
• The Born Oppenheimer approximation
• Discussions on limitations of the Born Oppenheimer approximation will be carried out in class.
• Discussions on electronic structure theory: Permutation symmetry and the independent particle This leads to what is known as the Hartree-Fock approximation and also forms the basis for DFT.
• Variational principle.
• Group Theory

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Mathematical foundations

To be worked by students at home. Also do watch this section closely. As we go further into the semester, we will populate here with additional information.

• Theory of representations and Dirac notation
• The position and the momentum representation
• Dirac notation summary
• Operators
• Theory of Operators I
• Theory of Operators II