Lecture_1 - gwu.edu

Physical Chemistry (II)
CHEM 3172-80
Lecture 1
Introduction to Quantum Mechanics
Lecturer: Hanning Chen, Ph.D.
01/12/2015
Who am I ?
Name:
Title:
Hanning Chen
Assistant Professor of Chemistry
Ph.D., University of Utah
Postdoctoral Fellow, Northwestern University
Research Interests: Theoretical and Computational Chemistry
Campus Address: Room 4510, Science and Engineering Hall
Campus Phone#: 202-994-4492
EMail: [email protected]
Office Hours:
Research Group Website:
3:00 PM-4:00 PM, Tuesdays
http://www.chenlabgwu.net
Syllabus
Course Name:
Physical Chemistry II
Course Number: CHEM 3172
Course Credits: 3, undergraduate level
Course Textbook:
Physical Chemistry, 10th Edition, Atkins and de Paula,
ISBN 978-1429290197
Lecture Room:
Room 111, Corcoran Hall
Lecture Schedule: 11:15 AM-12:25 PM, Mondays and Wednesdays
From 01/12/2015 to 04/27/2015
except for university holidays
03/04/2015: midterm examination
05/04/2015: final examination
Course Objectives
The probabilistic nature of quantum mechanics.
Electrons are indistinguishable and delocalized in chemical systems.
Electronic structures determined by Schrödinger equation.
Investigate molecular translations, vibrations and rotations.
Valence-bond theory and molecular orbital theory for many-electron systems.
Group theory and molecular symmetry.
Molecular spectrocopy and corresponding selection rules.
Electron spin resonance and nuclear magnetic resonance.
Course Organization
a total of 26 lectures, each of which is 75-minute long
10 min
55 min
10 min
Quiz: assess your understanding of the
immediate past lecture
Lecture
Homework Review: answer your questions
on the most recently graded assignments
Grading
Maximum Point: 100
In-class quiz: 25
(a total of 25)
(a total of 25)
Homework: 25
(Lecture 1 to 13) Midterm exam: 25
(Lecture 14 to 27)
Final exam: 25
1 point/quiz
1 point/assignment
Final letter grades determined by a curve fitting
with the mean grade around B+
No time extension or make-up will be given to tardy or absent students
for quiz and homework
No student will be excused from taking an exam without my written permission.
Class Policies
Students are expected to do their own quizzes, homework and
examinations without group discussion or plagiarizing.
Any such dishonorable acts will be reported immediately
to the Office of Academic Integrity of GWU.
You are always encouraged to report such activities to me, or to
Professor Michael M. King, the chair of the Chemistry Department.
Your privacy will be strictly protected.
In case you are not satisfied with my teaching, please talk to me and provide your thoughts.
Your comments are always appreciated !
What is Quantum Mechanics ?
Quantum mechanics is the fundamental law of interactions
between elementary particles.
incomplete list of elementary particles (61 in total)
6 quarks
Fermions
(half spin)
Bosons
(integer spin)
6 leptons
up, down, charm, strange, top, bottom (fractional charge)
electron, electron neutrino, muon, muon neutrino, tau, tau neutrino
(integer charge)
photon, gluon, W boson, Z boson, Higgs boson
Higgs boson: the last experimentally observed elementary particle
58-year wait for the creator of mass (God Particle)
Standard Model of Elementary Particles
“God particle”
elementary particles are interconvertible,
but they are unbreakable !
photon
most chemically important
electron
pair-production
positron
Physics, 1947
History of Classical Mechanics
from wiki
Newton’s Law of Motion
2
dv
d x
F = ma = m = m 2
dt
dt
a particle’s trajectory is deterministic
F(t) → x(t)
a particle’s presence is localized.
δ x(t) = 0
Newton’s apple
Isaac Newton (1642-1727)
founder of Classical Mechanics
a particle’s response is instantaneous.
2
d x(t)
m
=
F(t
')
δ
t
−
t
'
(
)
2
dt
Supposedly Newton’s Apple Tree
Cambridge University Botanic Garden
Successful Stories of Classical Mechanics
Astrophysics
Roller Coaster
At any given time, a particle’s position is unique !
Energy spectrum is continuous !
ΔE → 0
Is it always true???
t → t + Δt
x → x + Δx
E → E + ΔE
Black Body
three important properties
1. thermodynamic equilibrium, T > 0
2. perfect emitter,
λemission ∈( 0,+∞ )
3. isotropic emission, irrelevant to direction
an idealized physical object
that can absorb all incident radiation
1900
the birth of quantum theory Max Planck
Breakdown of Classical Mechanics
from wiki
blackbody radiation
infra-red
T≠0
T=300K
thermal energy electromagnetic radiation
Classical Rayleigh-Jeans Law:
8π kBT
I λ (T ) =
4
λ
λ → 0, I → ∞
???
what went wrong?
Emission Intensity Profile
(“ultraviolet catastrophe”)
Derivation of Rayleigh-Jeans Law
black body
L→∞:
to ensure a complete spectrum of molecular vibrations
λ ∈( 0,∞ )
Under thermal equilibrium:
L
E = kBT
all vibrational models have the same energy
k BT
regardless of their wavelengths
number of states:
8π
g (λ ) = 4 V
λ
volume of a 3-D
black body
Total energy per unit volume, energy density, for a given vibrational mode,
g ( λ ) k BT 8π k BT
U classical ( λ ) =
=
4
V
λ
Planck’s Law
Max Planck:
(1858-1947)
2
2hc
1
(“empirically” derived)
I λ (T ) = 5 hc /( λ kB T )
λ e
−1
Long Wavelength:
λ k BT
2ckBT
λ → ∞, hc / ( λ kB T )
→
, I λ (T ) →
4
hc
λ
e
−1
1
Rayleigh-Jeans Law
Short Wavelength:
λ → 0,
1
e
hc / ( λ kB T )
−1
→e
hc
−
λ kB T
2
2hc
, I λ (T ) → 5 e
λ
hc
−
λ kB T
Wien Approximation
Perfect agreement with experiments is achieved !
Derivation of Planck’s Law
L → ∞ : to ensure a complete spectrum of molecular vibrations
planck constant
h
2π c
!
=
6.63 × 10 m kg / s
ω=
ω ∈ 0, ∞
2π
λ
(
black body
)
−34
2
For a given vibration mode, ω, the allowed energies are discrete
En = n!ω
L
n : any postive integer
Under thermal equilibrium:
Pn (ω ) =
e
∞
En
−
kB T
∑e
En
−
kB T
n =1
canonical partition function
zero point
E (ω )
0 when T → 0
1
!ω
U (T )
= ∑ En Pn (ω ) = !ω + !ω / kB T
2
e
−1
n =1
∞
thermally averaged energy for a quantized oscillator
Further Derivation of Planck’s Law
ω
density of states: g (ω ) =
V
2 3
4c π
2
volume of a 3-D
black body
1
1
How many oscillation modes are there within a given energy range, from E − δ E to E + δ E ?
2
2
Total energy per unit volume, energy density, for a given vibrational mode,
g (ω ) E (ω )
!ω
U quantum (ω ) =
= 2 3 !ω / k T
B
V
4c π e
−1
3
(
)
convergent when
ω→∞
If the available energy levels are continuous as assumed by classical mechanics
g (ω ) k BT ω k BT
U classical (ω ) =
=
2 3
V
4c π
2
divergent when
ω→∞
Heat Capacities
Petit’s Law:
U = 3NkBT
each vibrational degree of freedom:
molar constant volume heat capacity:
CV ,m
CV.m → 24.9 JK mol
irrelevant to vibrational frequency when
hvvib << kbT
equal access to all vibrational modes
⎛ ∂U m ⎞
=⎜
=
3R
⎟
⎝ ∂T ⎠ V
High temperature : very good
−1
1
kBT
2
for an N-atom system
−1
Low temperature : large deviation
T → 0, CV.m → 0
Einstein Formula
⎞
⎛ θE ⎞ ⎛ e
= 3RfE (T ) = 3R ⎜ ⎟ ⎜ θ E /T
⎟
⎝ T ⎠ ⎝ e − 1⎠
2
CV ,m
θ E /2T
2
a correction must be added to account for the temperature dependence
on the accessible vibrational modes
Einstein temperature:
at high temperature:
T ≫ θE
Taylor’s expansion:
θ E = !ω / k b
θ E /2T
e
θE
= 1+
+ ...
2T
θE ⎞
⎛
2
2
1+
θE ⎞
⎛ θE ⎞ ⎜
⎛
⎟
2T
fE (T ) ≈ ⎜ ⎟ ⎜
=
1+
≈
1
⎜
⎟
⎟
θE
⎝T ⎠
⎝ 2T ⎠
⎜⎝ 1+ − 1 ⎟⎠
T
2
no correction is needed !
Low Temperature Heat Capacity
T ≪ θE
at low temperature:
2
2
⎞ ⎛ θE ⎞ ⎛ e
⎞
⎛ θE ⎞ ⎛ e
⎛ θ E ⎞ −θ E /T
fE (T ) = ⎜ ⎟ ⎜ θ E /T
≈
=
e
⎜
⎟
⎜
⎟
θ E /T ⎟
⎟
⎜
⎝ T ⎠ ⎝ e − 1⎠ ⎝ T ⎠ ⎝ e
⎝T ⎠
⎠
2
θ E /2T
2
θ E /2T
2
T → 0, f (E) → 0
Debye formula:
Debye temperature:
⎛T ⎞
f D (T ) = 3 ⎜ ⎟
⎝ θD ⎠
3 θD
∫
0
4 x
x e
(e
x
− 1)
dx
2
molecules vibrate over a wide range of frequency up to
hvD
θD =
kb
hvD
Photoelectric Effect
!ω
−
−
−
−
−
Electron removal only occurs when
Emax, kinetic
−
−
−
!ω ≥ W
minimum energy
to dissociate an
electron from the
surface
work function
c
= !ω − W = h − W
λ
The maximum kinetic energy is irrelevant to the incident radiation intensity.
Wave-Particle Duality
“For matter, just much as for radiation, in particular light,
we must introduce at one and the same time
the corpuscle concept and the wave concept?”
⎯ Louis de Broglie, 1929, Nobel Prize Speech
wave-particle duality
de Broglie relation
p
the wavelength of a particle
!
p
heavy, fast particle
short wavelength
λ
h
λ=
p
−34
Planck constant 6.63 × 10 m kg / s
p = mv
the momentum of a wave
!
p
light, slow particle
long wavelength
λ
2
Physical Meaning of de Broglie Wavelength
h uncertainty of a particle’s position
de Broglie wavelength:
λ=
p
!
p
wavelength
divergent position !!!
d
λ
d<λ
×
light diffraction
single-slit experiment
!
p
wavelength
λ
well-defined position
d>λ
d
light propagation
Uncertainty of a Particle’s Position
You can reduce your own spatial uncertainty by running fast !
The world’s fastest man:
Mr. Bolt weights 210 pounds (95 kilograms)
What is the de Broglie’s wavelength of Mr. Bolt?
λ Bolt
−34
h
h
6.63 × 10
=
=
=
100
pBolt mBolt × VBolt
95 ×
9.58
λ Bolt = 6.69 × 10
Usain Bolt (Jamica)
100-meter world’s record:
9.58 second
August 16th, 2009
Berlin, Germany
−37
meter
Mr. Bolt’s spatial uncertainty is negligible !
6.69 × 10
−37
meter << 100 meter
the Importance of de Broglie Wavelength
The de Broglie wavelength of an electron:
at ground state, the kinetic energy of the electron:
nucleus
H
+
e
Ek = 13.6 eV
−
the mass of electron:
hydrogen atom
h
h
h
λe =
=
=
pe meve
2Ek me
1
2
Ek = meve
2
me = 9.109 × 10
−31
kg
what is the electron’s de Broglie wavelength?
λ =?
λ = 3.32 × 10
−10
meter = 3.32Å
Does it matter?
Delocalization Characteristics of Elementary Particles
e
H-H bond length:
d = 0.74Å
H
+
−
d
H
e
+
spatial uncertainty of an electron
λe = 3.32Å
−
hydrogen molecule
H2
d < λe
It is even impossible to tell which atom owns which electron !!!
It would be more appropriate to describe an elementary particle
by a delocalized wave rather than by a localized particle.
Limited Applicability of Classical Mechanics
Classical Mechanics is a special limit of Quantum mechanics, and it is only valid for
high temperature, fast motion and heavy particles
Any energy levels with a gap over kbT should be quantized !
De Broglie particle-wave duality
wavelength
wave
h
h
λ=
=
mv p
momentum
particle
p → ∞, λ → 0 :
particle
Classical
p → 0, λ → ∞ :
wave
Quantum
Electron Microscopy
λ
e
e
ele
ctr
o
−
n
D
the condition of diffraction:
λe < D
h
h
h
λe = =
=
p
2me Ek
2meeΔφ
n
o
i
t
c
a
r
f
f
i
d
(~ 1Å)
Δφ : acceleration
potential
A strong electric field helps improve the resolution of electron miroscopy
Homework 1
Reading assignment:
Homework assignment:
Chapter 7A
Exercises 7A.2
Problems 7A.3
Homework assignments must be turned in by 5:00 PM, January 13th, Tuesday
to my mailbox in the Department Main Office
located at Room 4000, Science and Engineering Hall

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