Hierarchy of Systems in Quantum Mechanics
1.
Simple systems with analytic solutions to SE
Particle in box
One-dimensional harmonic oscillator
Particle on sphere
Hydrogen atom
2. Interacting systems with numerical solutions to SE
Orbitals in polyatomic mo
Towards Atomic Electronic Structure
Particle on a Ring
Particle on a Sphere
Uses particle on a ring
as a starting point: the
sphere is a stack of rings
Particle on a sphere Hamiltonian also applies
to rotation of diatomic molecule (Chapter 8).
The Hydroge
Towards Atomic Electronic Structure
Particle on a Ring
Particle on a Sphere
Uses particle on a ring
as a starting point: the
sphere is a stack of rings
The Hydrogen Atom
Uses particle on a sphere
as a starting point: the
atom consists of nested
spheres
Ch
Ch11. Quantum States of Many-Electron Atoms
Many-electron spin and orbital angular momentum operators are given by
sums of single electron operators
L lj
j
S sj
j
For the z components we have
Lz ljz
j
S z s jz
j
which leads to convenient expressions for t
Example: Evaluate the following eigenvalue equation for a wave function of the
electron configuration 2p13p1
1 2 p 1 3 p 2 2 p 2 3 p 1 1 2
L2
1
1
1
1
2
1
?
2 p1 1 3 p1 2 2 p1 2 3 p1 1 1 2
2
Example: Evaluate the following eigenvalue equation for a w
Realistic Numbers for HeH+ Molecule
a=He
72.1 36.7
H eV
36.7 47.1
1.00 0.45
S
0.45 1.00
b=H
d c c 2ca cb Sab
2
2
a
2
b
86%
He
2%
H
12%
Roots of the secular determinant are found using
H aa
H ba Sba
H ab Sab
2
H aa H bb H ab Sab 0
H bb
2
2
2 1
Ch12. Chemical Bonds in Diatomic Molecules
The Hamiltonian of H2+ sums on the kinetic energies of each particle. The
potential energy terms account for all pair-wise Coulombic interactions.
e2 1 1
e2
H
2a b2 2m e2 4 r r 4 R
2m p
e
0 ae
be
0
2
Kinetic e
Ch12. Chemical Bonds in Diatomic Molecules
The Hamiltonian of H2+ sums on the kinetic energies of each particle. The
potential energy terms account for all pair-wise Coulombic interactions.
e2 1 1
e2
H
2a b2 2m e2 4 r r 4 R
2m p
e
0 ae
be
0
2
Kinetic e
Chemistry 482: Physical Chemistry 2
September 11, 2013, Quiz #1
Instructions: Write your name on this piece of paper and staple it together with your solutions when
you are finished. Show your work on problems 4 and 5 on a separate sheet of paper.
(1) (a)
Chemistry 482: Physical Chemistry 2
November 11, 2013, Quiz #5
Instructions: Write your name on this sheet of paper and staple it together with your solutions when
you are finished.
(1) (1point) (a) True or False. The lowest energy term symbol associated
Chemistry 482: Physical Chemistry 2
October 28, 2013, Quiz #4
Instructions: Write your name on this sheet of paper and staple it together with your solutions when
you are finished.
(1) (1point) (a) True or False. The following linear combination of spheri
Chemistry 482: Physical Chemistry 2
October 9, 2013, Quiz #2
Instructions: Write your name on this sheet of paper and staple it together with your solutions when
you are finished.
(1) (1point) (a) True or False. Integrand has odd symmetry.
2 x x3 0 x dx
Vibrational Quantum States in Polyatomic Molecules
Key Idea: Vibrations delocalize onto bond stretching and bending
motions with similar intrinsic frequencies
How to Treat Interesting Systems With Hamiltonian Matrices
Most systems cannot be treated analyt
Ch10. Helium: The Smallest Many-Electron Atom
The Schrdinger equation cannot be solved analytically for atoms possessing
more than one electron. Approximations must be introduced.
The Hamiltonian sums the kinetic energies of the electrons, their attractio
Ch3. Quantum Mechanical Postulates
Postulate 1
The state of a quantum mechanical system is completely specified
by its wave function. The probability that a particle will be found at
time t0 in a spatial interval of width dx centered at x0 is given by
x0
Ch4. Quantum Mechanics on Simple Systems
In chapter 2, we obtained the Schrdinger equation for stationary states
d 2 x
V x x E x
2
2m dx
2
Eq. 4.3
Lets first treat a system in which the potential energy is equal to zero everywhere
d 2 x
2mE
2 x k 2 x
Mathematical Arguments are the Foundation of
Quantum Mechanics
Chapter 2 argues for the plausibility that the Schrodinger equation
describes wave-particles
Chapter 3 presents the postulates (i.e. assumptions) on which quantum
mechanics is based. The postu
Quantum Mechanics Arose Out of the Interplay Between
Experiments and Theory
Early in the 20th century, a number of experiments found that the
predictions of classical physics were inconsistent with observations
Chapter 1 emphasizes two properties that d
Potential Energy
Tunneling Through a Potential Energy Barrier
V0
Zone I
V(x)=0
Zone II
V(x)=V0
0
Zone III
V(x)=0
L
The goal of this problem will be
to obtain the probability that a
particle approaching from the
left is transmitted through the
barrier into
Ch6. Commutation Relations
Two observables can be known simultaneously only if the outcomes of the
measurements are independent of the order in which they are conducted.
Mathematically, this means that the observables corresponding to operators, A
and B,
Ch7. Harmonic Oscillator
Write the Hamiltonian and expand the potential energy in a Taylor series
p2
H
V x
2m
2
2
p V dV x 1 d V x 2
H
0
2m
2 dx 2 0
dx 0
(1)The constant leading term, V0, is dropped because it does not impact
250000
vibrational motion.
Chemistry 482: Physical Chemistry 2
September 25, 2013, Quiz #2
Instructions: Write your name on this sheet of paper and staple it together with your solutions when
you are finished.
(1) (1point) (a) True or False. The Stern-Gerlach experiment shows that