Lecture IX
Classical particles The classical state of a system is specified as giving, at each instant in time, the complete set of coordinates and momenta of the particles, i.e., r1 (t ),.,rn (t ), p1 (t ),., p n (t ). If N=2, this would be
r1 (t ), r2 (

G25.2666: Quantum Mechanics II
Notes for Lecture 6
I. SOLUTION OF THE DIRAC EQUATION FOR A FREE PARTICLE
The Dirac Hamiltonian takes the form H = c P + mc2 where
Using P = ( /i) , in the coordinate basis, the Dirac equation for a free particle reads h i

G25.2666: Quantum Mechanics II
Notes for Lecture 7
I. A SIMPLE EXAMPLE OF ANGULAR MOMENTUM ADDITION
Given two spin-1/2 angular momenta, S1 and S2 , we dene S = S1 + S2 The problem is to nd the eigenstates of the total total spin operators S 2 and Sz and i

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G25.2666: Quantum Mechanics II
Notes for Lecture 5
I. REPRESENTING STATES IN THE FULL HILBERT SPACE
Given a representation of the states that span the spin Hilbert space, we now need to consider the problem of representing the the states the span the full

Lecture II The H2+ molecule ion is an example of a simple one-electron problem that can be solved exactly and leads to the energy eigenfunctions associated with an actual chemical bond. In fact, this molecule is the only one for which analytical solutions

G25.2666: Quantum Mechanics II
Notes for Lecture 11
I. AN EXAMPLE OF THE VARIATIONAL THEORY: THE H+ MOLECULE ION 2
The H+ molecule ion is the simplest example of a chemical bond. It is a particularly important example to study 2 because it is an analytica

Lecture 8: Introduction to Density Functional Theory
Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washington Square East New York University, New

Lecture 9:
Advanced DFT concepts: The Exchange-correlation functional and time-dependent DFT
Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science 100 Washing

Lecture XI Solving the electronic eigenvalue problem: In the Born-Oppenheimer approximation, the electronic eigenvalue problem takes the form.
[Te + Vee + Ven ] (x , R ) = (r ) (x , R )
Vee = 1 1 r r 2 i, j i j
i j
VeN =
ZI i =1 J =1 ri R I
N
M
N
Let Vex

G25.2666: Quantum Chemistry and Dynamics
Notes for Lecture 10
I. THE BORN-OPPENHEIMER APPROXIMATION
The next few lectures will treat the problem of quantum chemistry, a subeld of quantum mechanics also known as molecular quantum mechanics. The idea of qua

G25.2666: Quantum Chemistry and Dynamics
Notes for Lecture 5
I. EXPERIMENTAL EVIDENCE FOR ELECTRON SPIN
Up to now, we have considered quantum particles to have three degrees of freedom, x, y, and z. This, then, leads to three quantum numbers that characte