Week 1 The following problems from your text are assigned for practice. Do not turn them in. If you have questions, please bring them to office hrs. or discussion. Problems: 1-2, 1-2, 1-6, 1-8, 1-12, 1-14, 1-16, 1-18, 1-25, 1-27, 1-36 (You may need to rea

Physical Chemistry
Course Number: C362
9 The rotational problem
9.1 Reduced Mass 1. For a diatomic molecules, we can write the Hamiltonian as: h2 H(r1, r2) = - 2M1 where Cartesian coordinates:
2 2 r1
h2 - 2M2
2 r2
+ V (r1, r2)
(9.8)
2 2 2 = 2+ 2+ 2 x y z

Physical Chemistry
Course Number: C362
5 The time-dependent Schr dinger Equation o
(x, t) = H(x, t) (5.1) t where, H is called the Hamiltonian operator. It is the operator in quantum mechanics that corresponds to the energy of the system. In quantum mech

Physical Chemistry
Course Number: C362
4 Postulates of quantum mechanics
Mathematically, we define a quantity, , that completely describes the system. The quantity, , when represented in terms of coordinates of particles in space, leads to the wavefuncti

Physical Chemistry
Course Number: C362
16 Additional reading: Probability Current
1. Consider the time-dependent Schr dinger Equation and its complex conjuo gate: h2 2 i (x, t) = H(x, t) = - h + V (x, t) t 2m x2
(16.10)
2. Multiply Eq. (16.10) by (x, t)

Physical Chemistry
Course Number: C362
7 Particle inside a three-dimensional box: The time-independent Schr dinger Equation in three dimensions o
Consider the figure of the metal porphyrinin in problem 3-27. The electrons are delocalized and one might con

Physical Chemistry
Course Number: C362
6 Particle-in-a-box (PIB)
1. Consider a linear poly-ene. 2. The electrons are completely delocalized inside the poly-ene, but cannot leave the molecular framework. 3. Let us approximate this system by a one-dimension

Physical Chemistry
Course Number: C362
1 The mathematical ideas you will encounter while learning quantum mechanics
We first outline a few mathematical ideas that one needs to tackle quantum mechanics. This section includes a brief review of: Vectors and

Physical Chemistry
Course Number: C362
12.5 The hydrogen atom energy only depends on the "principal qauntum number" 1. But now depends upon E the energy as given in the first two of Eq. (12.34). Using Eqs. (12.34) n2 = 2 = and hence 2 Z 2 e4 Z 2 e4 =- 2 h

Physical Chemistry
Course Number: C362
12.4 Using the boundary conditions to enforce physical bounds on the quantum numbers the structure of the periodic table 1. Now, as i , Ci+1 ci /i. For large i, F () diverges. Note: for large Ci+1 ci /i which yields

Physical Chemistry
Course Number: C362
12.3 Solution to Eq. (12.32) 1. Lets first analyze the equation. The term involving the derivatives can be interpreted as a second derivative operator and hence a radial kinetic energy (notice the angular kinetic ene

Physical Chemistry
Course Number: C362
12.2 The hydrogen atom Schr dinger Equation in spherical coordinates o 1. As noted in the spherical coordinates handout it is always convenient to choose a coordinate system that represents the symmetry of the proble

Physical Chemistry
Course Number: C362
12 The Hydrogen Atom
12.1 Factorization of the hydrogen atom Schr dinger Equation into center o of mass and relative coordinates 1. As we noted in the Born-Oppenheimer approximation, solving the timeindependent molec

Physical Chemistry
Course Number: C362
12 The Hydrogen Atom
1. As we noted in the Born-Oppenheimer approximation, solving the timeindependent molecular Schr dinger Equation: o H(ri, RI ) = E(ri, RI ) where
N n Ze2 h2 N 1 2 h2 n 2 H = - - + - 2 I=1 MI I 2m

Physical Chemistry
Course Number: C362
8 Harmonic Oscillator
1. For the case of the harmonic oscillator, the potential energy is quadratic and hence the total quantum Hamiltonian looks like: 1 h2 d2 + kx2 (8.1) H=- 2m dx2 2 where k is the force constant f

Physical Chemistry
Course Number: C362
15 Group Theory Basics
1. A good reference: "Group Theory and Quantum Mechanics" by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian. We

Physical Chemistry
Course Number: C362
15 Group Theory Basics
1. A good reference: "Group Theory and Quantum Mechanics" by Michael Tinkham. 2. We said earlier that we will go looking for the set of operators that commute with the molecular Hamiltonian. We

Physical Chemistry
Course Number: C362
Why do we want to learn Quantum Mechanics? Quantum mechanics is a mathematical theory that can be used to predict properties of atoms, molecules, nano-materials, and condensed phase systems. It can also be used to un

Physical Chemistry
Course Number: C362
3 de Broglie's wave particle duality
1. Louis de Broglie (for his Ph.D. thesis) proposed that particles have a wave character and waves have a particle character. 2. The latter was already suspected, in regards to li

Physical Chemistry
Course Number: C362
14 Extra-credit Homework: Classical mechanics as a special case of quantum mechanics
This homework will help you see how the classical Newtonian equations arise as a special case of the time-dependent Schr dinger Equ

Physical Chemistry
Course Number: C362
11 The Born-Oppenheimer approximation, the Many Electron Hamiltonian and the time-independent molecular Schr dinger o Equation
1. We have now seen a few problems where we have solved the time-independent Schr dinger

Physical Chemistry
Course Number: C362
13 Theory of Angular Momentum
1. Why do we want to study angular momentum? 2. We need to study the properties of chemical systems, how is angular momentum relevant? 3. Lets consider the Hamiltonian for any molecule.

Physical Chemistry
Course Number: C362
13 Theory of Angular Momentum
1. Why do we want to study angular momentum? 2. We need to study the properties of chemical systems, how is angular momentum relevant? 3. Lets consider the Hamiltonian for any molecule.

Physical Chemistry
Course Number: C362
10 A one-dimensional collision problem
We will now treat a one-dimensional collision problem. 1. Consider a step potential: V (x) = 0 V (x) = V0 x<0 x0
(10.1)
So this is an unbound system. But what does this remind u