7.1.b: Heat capacities
Cv,m(T) = 3 R fE(T) with
f E (T ) E
T
v
v
2
E / 2T
e
/T
e E 1
2
At low temperatures, solid behaves as if
it has very few atoms since there is not
enough energy to excite large number of
oscillators.
At high temperatures, heat
Transmission Probability for particles with E > V0 & E = V0
v
For E > V0:
E > V0
v
For E = V0:
T vs. for various k2a
Particle in a finite depth box
V0 x < 0
V(x) = 0 0 x L
V0 x > L
V0
v
Region I
Region II
Region III
Two cases: E < V0 & E > V0
Case 1: E <
Vibrational Motion
One-dimensional one body harmonic oscillator
v
v
Classical treatment gave us:
x(t) = A cost with A as amplitude and
What are potential, kinetic and total energies of this system?
E = V + KE
One-dimensional two body harmonic oscillator
l
SquareofAngular
Momentum
1
1 2
L
sin sin sin 2 2
2
2
z-componentofAngular
Momentum
Interpretationofland
ml
[l(l+1)]1/2representsthemagnitudeof
angularmomentumLvector.
v
mlrepresentsthevectorsprojectionon
z-axis.
v
Wavefunctions
Interpretationofland
Wavefunctions of one-dimensional harmonic oscillator
N is the normalization constant
= 2/ = (k/2)1/2
v
(x) are orthogonal functions.
v
v
Wavefunctions of one-dimensional harmonic oscillator
doxdrum.wordpress.com
Large amplitude of (x) at classical turnin
v
v
Tunneling probabilities for harmonic oscillator
Probability that oscillator is stretched or compressed beyond classical limit is:
Tunneling probabilities:
are independent of k and .
Decrease quickly with increasing and vanishes completely as
What is
Wave functions of particle on a sphere
2 2
E
Schrdinger equation:
2m
2
2
2
2 2 2 2
x y z
2 2 1 1 2
1
2 2
sin
In spherical polar coordinates:2 2
r
r r r sin 2 sin
To find a solution use: (,) = ()()
Solving () and () independently we get:
ml = 0, 1
Interpretation of Veff
Term dependent on l is analogous to
centrifugal force in classical physics.
For l = 0, potential energy is purely Columbic
attractive force.
For l 0:
Centrifugal term has positive contribution
(repulsive force) to Veff.
Electro
Energies of a particle on a ring
m l2 2
Quantized energies:
E
2I
ml = 0, 1, 2, 3, .
Energy is independent of the sense of rotation (i.e., sign of m l).
Except for ml = 0, all other energy states are doubly degenerate.
Applicable to any body with momen
Tunnelling: Extension of particle in a finite depth box
E < V0
V0
V0
Classically
Forbidden
region
Tunnelling
v
v
Tunnelling: Penetration or leakage of a particle into a classically forbidden region.
Alternate view: Particle with energy less than the barri
Superposition of wavefunctions (sec. 7.5.e)
v
v
Postulate 2 says that the only observables in an experiment
are the eigenvalues.
What if the wavefunction is not an eigenfunction of the
operator corresponding to the property of interest?
Example: Consider
My Goal for this Course
Fundamentals of QM
(Chaps. 7 & 8)
Atomic and Molecular
Structure
(Chaps. 9 & 10)
Vibrational &
Rotational
Spectroscopy
(Chap. 12)
Principles and application
of spectroscopy
Electronic
Spectroscopy
(Chap. 13)
Magnetic
Resonance
Spec
2D Planar surface with
L1 and L2 dimensions
Particle in 2D and 3D Boxes
3D Box with L1, L2, L3
(aka a, b, c) dimensions
v
v
v
Key to solving these is separation of variables.
2D problem splits into two 1D problems.
3D problem splits into three 1D problems
Postulates of Quantum Mechanics (7.3 7.7)
(1) The state of the system is described by a function of the coordinates and the time.
, known as wave function or state function, contains all the information about the system.
should be a single-valued, contin
Postulate 2
To every physically observable property there corresponds a linear Hermitian operator.
This operator can be found by writing down its classical-mechanical expression in terms of
(x,y,z) and (px,py,pz) and then replace each coordinate by its op
Chapter 8
v
v
v
v
Translational Motion:
Particle in 1D. 2D and 3D
Tunnelling
Vibrational Motion
Rotational Motion:
Particle on a ring (2D)
Particle on a sphere (3D)
Spin
Translational Motion
Motion of a free particle along one dimension
General solution:
Property # 2 of Hermitian operator
v
(a) Two eigenfunctions of a Hermitian operator that correspond to
different eigenvalues are orthogonal.
(b) eigenfunctions of a Hermitian operator that belong to a
degenerate
eigenvalue can always be chosen to be ortho
Postulate 3
The only possible values that can result from measurements of the physically observable
property O are the eigenvalues oi in the equation fi = oifi. The eigenfunctions fi are
required to be well-behaved.
Problem was done in class. Refer to you
The Uncertainty Principle (sec.
7.5.e)
x is the uncertainty in x & px is the uncertainty in px
Examples:
1)
The Uncertainty Principle (sec.
7.5.e)
Calculate x for a 150 g baseball thrown at 40 m/s if we measure its
momentum to a millionth of 1.0%.
(2) Cal
Wave functions of particle on a sphere
2 2
E
Schrdinger equation:
2m
2
2
2
2 2 2 2
x y z
2 2 1 1 2
1
2 2
sin
In spherical polar coordinates:2 2
r
r r r sin 2 sin
To find a solution use: (,) = ()()
Solving () and () independently we get:
ml = 0, 1
Molecular Structure (Chapter 10)
v
Valence-bond theory
v
Molecular orbital theory
Born-Oppenheimer Approximation
Nuclei of atoms in a molecule are regarded as fixed at arbitrary locations and the
Schrdinger equation is solved for the wavefunctions of the
Chapter 9: Atomic Structure and Spectra
Hydrogenic atoms (9.1 9.3)
Atomic structure
Atomic orbitals and their energies
Spectroscopic transitions and selection rules
v
v
Many electron atoms (9.4 9.5)
Note: Basics of complex atoms spectra (9.6 9.10) will be
CHEM321L
FA2013
Dr. Nellutla
EXPERIMENT 3
ESTIMATION OF CARBON CARBON BOND LENGTH IN CONUGATED
CYANINE DYES: AN APPLICATION OF PARTICLE IN A 1D BOX
LAB REPORT DUE ONE WEEK FROM THE DAY YOU FINISH YOUR
EXPERIMENT
CAUTION: WEAR GLOVES ALL THE TIME EXCEPT WH
CHEM 321
FA2013
Dr. Nellutla
HOME WORK #3 (Due 10/29/13 by 4 pm)
(1) Show that Y11(,) is an eigen function of
carry out the mathematical manipulations.
. Use spherical coordinates to
(2) Show that normalization constant for Y12(,) is
. Do not use the
for
CHEM 321
FA2013
Dr. Nellutla
HOME WORK #2 (Due 10/16/13 by 11 am)
(1) The state of an electron in a one-dimensional cavity of length 1.0 nm in a
semiconductor is described by the normalized wave function:
1
1
1
(x ) 1 (x ) 2 (x )
4 (x)
2
2i
2
n(x) is th