University of Pennsylvania
Department of Chemistry
Chemistry 221
Third Mid-term Examination
Monday Nov 20, 2006
The available points on this test amount to 116, but you are not expected to answer everything. Full credit can be obtained
on answers totallin
Rotational Motion
Electrons in atoms
Molecular rotation
Rotation in the
x-y plane
generates angular
momentum about
the z-axis
L
m
Consider a mass, m, moving in a plane
around a point at a fixed radius r in the
absence of any external field. We can set
V
Quantum Mechanics I
Now lets move a little closer to the theory of Quantum Mechanics
Complex Numbers and Expressions for Trigonometric Functions
x 2 x3 x 4 x5 x 6 x 7
e = 1 + x + + + + + + + .
2! 3! 4! 5! 6! 7!
x2
x3 x 4
x5 x6
x7
ix
e = 1 + ix i + + i i +
Particle in a Box
Translational Motion
We start with translational motion and the example of the particle in a
box, which is often introduced at the freshman level.
Simple model for quantum mechanics
Predicts electronic transitions of some large molecules
Quantum Mechanics
Preamble
Energy formulas
We encountered the formula attributed to Planck and others.
E = h =
hc
= hc
Here, symbol indicates that the units are in cm-1, whereas the symbol
indicates sec-1.
Thus, a photon having a wavelength of 500 nm has
Approximation Methods 3:
Perturbation Theory
In the variation(al) approach, we modified the wavefunction, but used the
same Hamiltonian to handle cases that do not provide exact solutions of the
Schrodinger equation.
Another way to approximate the energy
Postulates and General
Principles of Quantum
Mechanics
9/16/2010
Principles of Quantum Mechanics
1
Postulates of Quantum Mechanics
1. The state of a quantum-mechanical system is described as fully as possible
by the wavefunction (r1, r2 ). (M&S section 4-
A few notes on hydrogen-like atoms.
10/14/2010
Hydrogen-like Atoms
1
Lets briefly review the Bohr arguments extended to a Hydrogen-like
element, where the nuclear charge is Ze. Lets treat the case of a single
electron (H, He+, Li2+, ). As shown earlier, t
Morse Oscillator
9/28/2010
Morse Oscillator
1
Morse Oscillator
One convenient form of the potential energy function that allows
significant correction to the harmonic model is the Morse equation:
V (r ) = De (1 exp( (r re ) 2
=
2
ke
=
2 De
2 De
In the ab
Electron Spin and Discussion of
Angular Momentum, including
Commutation Rules
Angular Momentum
1
Spin
A fourth quantum number is needed to describe atomic properties, including
the following:
1. The aufbau principle
2. The magnetic properties of atoms, mo
Name:_
Please indicate course number_
Physics 151/141
Instructor M. Cvetic
First Mid-term
October 20, 2005
Please show all work on these sheets. Your grade will depend both on your answer and on the
work you do to get that answer. If you need space use th
BluesyBluesing
BlackTropesinJazzStylesandPractices
What do you hear?-the elements of music
Melody: a succession of pitches-the emotional focal point
Rhythm: how music moves through time-no beat, regular beat, irregular pattern
(more later)
Harmony: two or
Jazz
WhatisItandHowDoesitMean?
won't you celebrate with me
won't you celebrate with me
what i have shaped into
a kind of life? i had no model.
born in babylon
both nonwhite and woman
what did i see to be except myself?
i made it up
here on this bridge bet
Statistical thermodynamics
THE FUNDAMENTAL statistical postulate of equal a priori probabilities can be
used as the basis of the entire theory of systems in equilibrium. In addition,
the hypothesis mentioned at the end of Sec. 23 (andibased on the assume
Print your name:
Problem 1
(12 points)
q
Three point charges q, q , and q (q = 3.2 C) are held
xed at the vertices of an equilateral triangle of side a =
1.8 m.
(1 C= 106 C)
(a) Find the electric eld E at the midpoint P of the line
connecting the two posi
Name:_
Please indicate course number 151 or 141_
Physics 151/141
Instructor M. Cvetic
Second Midterm Exam
November 15, 2007
Please show all work on these sheets. Your grade will depend both on your answer and on the work you do
to get that answer. If you
Hydrogen Atom 2
Some more detail on the calculations
Example
Given the normalized wavefunction for a 2pz orbital of hydrogen:
2p
z
1 Z
=
32 a0
3/ 2
Zr
r
exp
cos
a0
2a0
Let's first prove longhand that this wavefunction is indeed normalized.
To d
Hydrogen Atom
(see Chapter 6)
Schrdinger Equation for the Hydrogen Atom
The electron is retained in the hydrogen atom by an electrostatic
potential force. The full three-dimensional form of the Schrdinger equation,
expressed in spherical polar coordinates
Department of Chemistry
Chemistry 221
Second Mid-term Examination
Monday Nov 2, 2009
More than 100 points are available, so that you have some choice on this test. Try to pick out the questions
you can answer best in the time available. Use labeled diagra
University of Pennsylvania Department of Chemistry
Chemistry 221
Mid-term Examination 1
Monday, October 5, 2009
More than 100 points are available on this exam, so you have some choice of questions to attempt. Since time
is limited, use labeled diagrams a
University of Pennsylvania
Department of Chemistry
Chemistry 221
Mid-Term Examination 1
Wednesday, Oct. 4, 2006
Since time is limited, use labeled diagrams and/or equations to illustrate your answers. TA's will be
looking for appropriate explanation and j
Department of Chemistry
Chemistry 221
Third Mid-term Examination
Monday November 23, 2009
As before, extra points are available - you are not expected to answer everything. Because time is short, use
labeled diagrams and/or equations where possible. Try t
Tunneling
Reference to Problems 4.33 to 4.35 in M&S
Tunnelling
In the example of the particle in a linear box, we fixed the potential
energy to be zero everywhere in the box, and infinite beyond the edges of the
box. How does the wavefunction change if we
Applications of Variation
Theory
Application of Variation Theory to the Helium Atom Ground State
The Schrdinger equation can be solved exactly for the hydrogen
atom. This means that there exists a set of functions, n, that satisfy the
eigenvalue equation:
Variational Methods
Variation Theory
1
Application of Quantum Mechanics to Larger Atoms
The Schrdinger equation has been used so far for the hydrogen atom, which is a
central field problem - the only contribution to the potential energy is from an
attract
Supplementary notes on calculating the
rotational constant(s) for a linear polyatomic
molecule
Initial assumption here was that
the C was at the origin and we
input a couple of bond lengths.
Input:
Confirmation:
z
Input coordinates
But then, we need to as