Quantum Mechanics in Chemistry
CHEM 3890 – Honors Physical Chemistry I
Quantum physics is mind-bending, but it is also
one of the most successful scientific theories ever put forward.
–New Scientist
Fall 2013
Bard 132
Monday, Wednesday, Friday 11:15am-12:05pm
Instructor:
Office Hours:
M. Luke McDermott Baker B38
Friday 2:30 pm and by request
Teaching Assistants: TA#1
e-mail
Office Hour
Recitation Section
TA#2
e-mail
Office Hour
Recitation Section
Grading:
Letter Grades Only, 4 credits
I. Rationale:
From the College:
CHEM 3890 is an introduction to the quantum mechanics of atoms and molecules. The
fundamental principles of quantum mechanics are introduced, and applications of the theory to
atomic and molecular structure are covered in detail. CHEM 3900 is a continuation of CHEM
3890 and discusses the thermodynamic behavior of macroscopic systems in the context of
quantum and statistical mechanics. After an introduction to the behavior of ensembles of
quantum mechanical particles (statistical mechanics), kinetic theory and the laws of
thermodynamics are covered in detail.
Translation:
CHEM 3890 provides the science student with a rigorous foundation for Physical Chemistry.
More generally, CHEM 3890 allows students to understand why quantum mechanics matters,
how the subject developed, and where they see it daily.
Prerequisites:
MATH 2130 or MATH 2310 or MATH 2220; PHYS 2208; CHEM 2080 or CHEM 2160 or
permission of instructor.
1
II. Course Aims and Objectives:
Aims
By the end of this course, students will have:
 Presented why Quantum Mechanics was needed to
supplement Classical Mechanics
 Described the development of Quantum mechanical
theory through significant historical breakthroughs
 Solved Schrödinger’s wave equation for constrained
particles
 Explained the origin of general chemistry principles
such as molecular orbitals
 Understood state-of-the-art Physical Chemistry
research
Specific Learning Objectives:
Students will demonstrate a mastery of fundamental principles
of Quantum Mechanics by learning vocabulary and
mathematical methods, solving classic problems,
diagramming historically significant experiments, and
analyzing current scientific articles in Physical Chemistry.
Specifically, students should be able to:
 Remember fundamental experiments and theories in
quantum mechanics.
 Understand how quantum mechanics supplements
classical mechanics.
 Apply Schrödinger’s wave equation to classic
problems: (1) the Particle in a Box; (2) The Harmonic
Oscillator and the Rigid Rotor; (3) The Hydrogen
Atom; and (4) Multielectron Atoms.
 Analyze complex problems related to the above classic
problems and create paths to solving these complex
problems.
 Evaluate modern research in quantum mechanics,
especially molecular modeling.
 Create mathematical computer code to solve problems
quantum mechanically.
III. Format and Procedures:
Class time is centered on finding answers. Students are
expected to come to class ready to solve problems. To interact
with the rest of the class, you will need a web-connected
device (e.g. smartphone, tablet, laptop). If you do not have
access to such a portable device, we can loan you one for each
class.
Using these devices to interact (see sidebar), class time will
follow this cycle:
2
Learning Catalytics
Brain wave studies
show that sitting
passively in a lecture
can induce as little
brain activity as
watching television. To
counteract the passive
nature of lectures, bring
your web-connected
device to class every
day and log on to our
LCatalytics.com class
session.
During class, we will
have all kinds of
interactive questions,
where you will answer
through the website.
Then you will have a
chance to work with
your neighbors to
convince each other
who is right.
Then put your new (or
old) answer into the
question again.
With Learning
Catalytics, we can give
you instant feedback
and find out where the
class should go.
Make a username now
and log on with our
course session ID:
CHEM3890. Our session
password is
QMfall2013.
1. I will explain a Concept
2. I will pose a Question on the Concept
3. You will each make an educated guess to Answer the Question on our Learning
Catalytics website (learningcatalytics.com)
4. You will attempt to convince others around you of the correct answer
5. You will adjust your educated guess to Answer the Question
6. I will go over the correct Answer
7. I will review and summarize the Concept
Other class activities, which you will produce in groups for class time (see Section VII.):


Debates on quantum mechanics interpretations.
Perspectives on historical scientists.
IV. Time Commitment:
Per week:
3 hours: Class time is your opportunity to interact with other students and me.
1-3 hours: Recitation and TA Office Hours are your greatest resource for having questions
answered. TAs will lead recitations so that you will do problems in groups during the session and
the TA will work with you as your group encounters problems.
1-3 hours: Group homework sessions are the best way to complete problem sets. Teaching and
learning from peers is proven to help students learn more effectively than interaction with
professors.
V. My Assumptions
Quantum Mechanics is taught in many fields, but in this course, you will learn the chemist’s
perspective. At the end of this syllabus, you will find an outline of what part of quantum
mechanics history we will cover with this course, as well as how quantum mechanics relates to
the rest of chemistry. Also attached is a formula sheet that will let you see all of the equations
that you will know how to use at the end of the course.
VI. Required Texts
- Introduction to Quantum Mechanics, 2nd edition, by
D.J. Griffiths This book is short, readable, and full of
helpful equations and problems (Solution manuals can be
purchased).
- Physical Chemistry: A Molecular Approach, by D.A.
McQuarrie & J.D. Simon (University Science Books).
http://www.uscibooks.com/mcq.htm .You will need this
book for Chem 3900 as well.
VII. Assignments
Problem Sets
A large portion of the learning for this class will be done in the context of problem sets. The
reality is that quantum mechanics is best understood and used as a rigorous, mathematical tool.
3
In modern research, scientists use the same equations as you to solve their research problems.
When you see how the equations we discuss in class accurately describe molecular-level
problems, the power of quantum mechanics will be clear to you. The goal of these problem sets
is to empower you and help you are now able to calculate how chemistry works at the most
fundamental level.
The Final Exam
The final exam is your opportunity to test the variety of your abilities at the end of this class.
When you walk out of the exam, my hope is that you will feel excited by how much you are
capable of.
Group Presentations and Journal Article Review
These assignments bring you into the historical and modern community of scientists. When
scientists talk about breakthroughs, they often discuss it in the context of who made the
breakthrough. Or scientists talk about how their own work in the context of how it relates to
other scientists’ work. Your presentations and reviews will show you the history and community
of quantum mechanics.
VIII. Grading Procedures:
1. 7 Problem Sets (40%)
These problems can be time-consuming, but collaborative work and computer
coding can lead to memorable learning experiences. Above all: work together
2. One Final Exam (30%)
The concepts tested by
Research shows that practice tests
problem set will be tested
again during the final exam.
are one of the most effective learning
The problems will not be in
strategies.
exactly the same format, as
no computers will be
available during the exam, so take the practice exams that will be available.
3. Group Presentation (20%)
You will be assigned to one of 12 groups. 10 groups will be assigned an
important scientist in quantum mechanics. They will make a 20-minute
presentation including his most significant work and an assessment of class
understanding. Points will be awarded for completeness, creativity, and class
understanding. Two groups will be assigned to take part in a 45-minute debate on
a significant perspective on quantum mechanics. Points will be awarded for
clarity and completeness. Bonus points are available for exceptional
presentations.
4. Journal Article Review (10%)
You will summarize two scientific journal articles, one of which must be a
review article. Points will be awarded for connections drawn to course material.
IX. Academic Integrity
Each student in this course is expected to abide by the Cornell University Code of Academic
Integrity. Any breach of academic integrity will receive a zero.
4
For this course, collaboration is allowed for all assignments. However, each character of
mathematical code, each word of explanation, and all submitted work must be written by your
own hands.
However, this permissible cooperation should never involve one student having possession of
a copy of all or part of work done by someone else, in the form of an e-mail attachment,
file, or hard copy.
X. Accommodations for students with disabilities
In compliance with the Cornell University policy and equal access laws, I am available to discuss
appropriate academic accommodations that may be required for student with disabilities.
Requests for academic accommodations are to be made during the first three weeks of the
semester, except for unusual circumstances, so arrangements can be made. Students are
encouraged to register with Student Disability Services to verify their eligibility for appropriate
accommodations.
5
XI. Tentative Course Schedule
Lecture 1
 Historical Development of Quantum Mechanics
Lectures 2-4
 Experimental Evidence for Quantum Mechanics
Lectures 5-7
 Machinery of Quantum Mechanics (Mathematics)
 Operators, Eigenvalues, Eigenfunctions
Journal Article Review Due
Lectures 6-12
 The Wavefunction and the Schrödinger
Equation
 Solving the Schrödinger Equation
 Particle in a Box
 Free Particle
Planck’s explanation of
blackbody radiation
Lectures 14-18
 Angular Momentum and Magnetic Fields
 Barrier Problems
 The Harmonic Oscillator
 The Hydrogen Atom
Lectures 19-21
Presentations
Lectures 22-25
 Approximation Methods
 Variational Principle
 Perturbation Theory
Lectures 26-30
 Chemical Bonds
 Spectroscopy
Lectures 31-33
Presentations and Debate
Lectures 34-36
 Final Exam Review
Journal Article Review Due
Final Exam
Quantum Tunneling
6
XII. Quantum Mechanics Map
1850-1900
Boltzmann, Maxwell, Hertz, Rydberg, Röntgen, Curie (1903 Nobel Prize), Rutherford
Statistical Mechanics, Molecular Orbitals
1900-1910
Planck, Lewis, Einstein, Rutherford, Taylor
Energy Quantization, Multiple Bonds (Octet Rule), Photoelectric Effect, Wave-Particle Duality
1910-1920
Millikan, Bohr
Oil Drop Experiment, Avogadro constant, quantum atomic levels,
1920-1930
Stern, Gerlach, de Broglie, Bose, Pauli, Hund, Heisenberg, Fermi, Dirac, Schrödinger, Neumann,
Born, Raman,
Spin, De Broglie Wavelength, Bosons, Fermions, Bose-Einstein statistics, Pauli exclusion
principle, Hund’s Rule, Heisenberg Uncertainty Principle, Fermi-Dirac statistics, wave
equations, quantum mechanics in terms of Hermitian operators on Hilbert spaces, Copenhagen
interpretation (probability and wavefunction collapse)
1930-now
Not covered in detail!
7
Quantum Mechanics Formulas
Constants
Expectation Values and Operators
h̄ ≡
h
2π
p̂ = −ih̄∇
Ê = ih̄
De Broglie–Einstein Relations
E = h̄ω
Ĥ = −
p = h̄k
ωlight (k) = ck
∀V
h̄k 2
2M
Harmonic Oscillator
V (x) =
Heisenberg Uncertainty Principle
∆px ∆x ≥ h̄/2
ψ0 (x) =
∆E∆t ≥ h̄/2
­ ®
2
∆x = x2 − hxi
L=
E0 =
h̄2 2
∂Ψ(r, t)
∇ Ψ(r, t) + V (r, t)Ψ(r, t) = ih̄
2m
∂t
En =
Time-Independant SWE
Ψ(r, t) = ψ(r)ϕ(t)
h̄
∇2 ψ(r) + V (r)ψ(r) = Eψ(r)
2m
ϕ(t) = e−iEt/h̄
Probability Current Density
j=−
ψ0 (Q) =
1 1/4 −Q2
e
π
h̄
mω
h̄ω
2
µ
¶
1
n+
h̄
2
Q ≡ x/L
µ
¶
d2
E
2
− 2 + Q ψ = ψ = ĥψ
dQ
E0
µ
¶
1
d
+
â ≡ √
−
+Q
dQ
2
µ
¶
1
d
−
â ≡ √
+Q
2 dQ
2
−
1
mω 2 x2
2
mω 1/4 −x2 /2L2
e
πh̄
r
Schroedinger Equation
−
h̄2 2
∇ + V (r)
2m
ρ(r, t) = |Ψ|2 = Ψ∗ (r, t)Ψ(r, t)
Z
Ψ∗ (r, t) fˆ Ψ(r, t) dV = hΨ|fˆ|Ψi
hf i =
Dispersion Relations
ωelectron (k) =
∂
∂t
ih̄
(Ψ∗ ∇Ψ − Ψ∇Ψ∗ )
2m
Hn (Q) = 2QHn−1 (Q) − 2(n − 1)Hn−2 (Q)
2/2
ψn (Q) = An Hn (Q)e−Q
Free Electron Beam
ψ(x) = A1 e
ikx
−ikx
+ A2 e
s
; k=
A2n =
2me (E − V0 )
h̄2
r
A∗ A2
R = 2∗
A1 A1
ψn (Q) =
1
A2n−1
2n
r
"
2
Qψn−1 (Q) −
n
#
n−1
ψn−2 (Q)
2
Stationary Perturbation Theory
Hydrogen Atom
Degenerate
e2
4π²0 r
V (r) = −
Ĥ = Ĥ (0) + Ŵ
Ĥ (0) |ni = En(0) |ni
R∞
En = − 2
n
Quasi Classical Approximation
ψ = e−i/h̄
TQC = e−2/h̄
Rb
a
R
= e−2/h̄
1
Ψ(x, t) = √
2π
Z
+∞
a(k) exp[i(kx − ωt)] dk
−∞
Z
1
Ψ(x, t) = √
2π
1
A(k, t) = √
2π
+∞
A(k, t)eikx dk
+∞
−∞
|Ψi =
HN 2
. . . HN N
aN
Hmn = hm|Ĥ|ni
H12
H22 − E
..
.
HN 2
...
...
..
.
aN
¯
¯
¯
¯
¯
¯=0
¯
¯
− E¯
H1N
H2N
..
.
. . . HN N
En0 = En + hn|Ŵ |ni
p = h̄k
E = Eu0 + hu|Ŵ |φi
|ψi = |ui +
∞
X
an = hn|Ψi,
an |ni
∆E (2) =
|an |2 = 1
|an |2 An
n
 
a1
a2 
|Ψi =  
..
.
|nihn| =
hm|Ŵ |ni
hn|ψi
0
E − Em
X
X |hm|Ŵ |ui|2
Eu − Em
m6=u
n=1
X
|mi
E ≈ Eu0 + ∆E (2)
Ĥ|ni = En |ni
∞
X
X X
m6=u n6=m
n=1
n
HN 1
|ψi = |ui + |φi
hφ|ui = 0, hψ|ui = 1
Expansion Principle and Hilbert Space
X
...
...
..
.
Nondegenerate
Ψ(x, t)e−ikx dx
1
Φ(p, t) = √ A(k, t);
h̄
hAi =
H12
H22
..
.
−∞
Z
hΨ|Ψi =
 
 
H1N
a1
a1
 a2 
 a2 
H2N 
 
 
..   ..  = E  .. 
 . 
.  . 
H11
 H21

 ..
 .
¯
¯H11 − E
¯
¯ H21
¯
¯
..
¯
.
¯
¯ HN 1
Wave Packets
an |ni
n=1

p(x) dx
|p(x)| dx
N
X
|ψi ≈
P̂nn = 1̂
n
2