Problem Set 3. Due Feb. 6 in class.
ea
a 1 can be used to solve the
a2
2
i 2
definite integral (where m = 0, 1, 2, 3): e im
(8 pts)
m
0
1. Math practice! Show that the indefinite integral: ea
Hint: when m is = 0, 1, 2, 3, then eim2 1 .
im
Answer: Firs
Lecture 23: The Kinetic Theory of Gases
The Maxwell-Boltzmann Distribution of Molecular Speeds (ideal gas so far) Last time, starting from the microcanonical ensemble, we have derived the Boltzmann distribution of (distinguishable) particles across variou
Lecture 34: Theoretical Kinetics & Reaction Dynamics
Simple transition-state theory We can minimize the flux FStotal and get closer to R by maximizing the total energy E needed for the passage of the AB-pair (reactants) over the dividing surface. This can
Lecture 33: Theoretical Kinetics & Reaction Dynamics
Monte Carlo integration methods they can be used to approximately evaluate integrals and other expressions (convenient for finding R, if PAB is known).
We can evaluate <f(x)> by summing over a large and
Lecture 32: Theoretical Kinetics & Reaction Dynamics
Collision Theory we need to find the collision rate and the probability of reaction (many parameters), and integrate their product. Consider: D 2 + H DH + D
clas. vib.
General reaction rate: - total rat
Lecture 31: Phenomenological Kinetics
Stationary-state approximation used to simplify the solutions of complex reactions
B, B C, . Consider again the consecutive reactions: A When the intermediate is slowly generated, but it decays fast: [B] < [A] and [C
Problem Set 4. Due Feb. 13 in class.
x
1. Math practice! Operate on e using the operator
1
1
1
x
2 2
x x x
x x x
(6 pts)
x
e
Hint: 3
Answer. Starting with the first operation on the left (the derivative) and note the operation on the right
x
x
x
x
Problem Set 1. Due Jan. 23 in class.
1. Math practice: evaluate the following:
2 cos(k x )
2 sin( x )
cos(k x )
sin( x )
a.
b.
c.
d.
x 2
x 2
x
x
x eikx
2eikx
1
f.
g.
h. eikx x
i. Prove i
2
x
x
i
e.
eikx
x
(18 pts)
Note: ii = 1
g. eikx x i keikx
f. k
Problem Set 2. Due Jan. 30 in class
2
2
2
1 2
which is: E E0 f (x) f ( y) f (z) f (t ) ,
2
2
2
2
2
x
y
z
c t
t
the time dependent function was found to be: f ( t ) sin 2 c
a. What does ct represent?
t
b. What are the units of c ?
c. Why is there a 2?
t
d
Quiz 3.
and p y
, can you show if p x , p y
i y
i x
k
k x
( x ) and p y ( y) y ( y)
commute or not? Hint: p x ( x )
i
i
1. (4 pts) If (x, y) (x) ( y) and p x
Answer: Im mostly asking if you know what a commutator is, p x , p y p x p y p y p x
When
Quiz 8.
1. Why are electronic transitions classified as being vertical?
(3 pts)
Answer: Because the electrons are so light, they can re-arrange
themselves faster than the nuclei can move upon absorption of
light.
2. How does a laser work?
(3pts)
Answer: H
Quiz 5.
1. In the derivation of the H-atom Hamiltonian, used the fact that
1 2
2 2
; can you
r
r r 2
r 2 r r
prove this is true (starting from the left hand term)?
(3 pts)
Answer: fyi we did this question twice in class. Starting with the left and side
Quiz 6.
1. Say two things about: Spin-orbit coupling.
(3 pts)
Answer: Spin-orbit coupling results from an electrons angular momentum generating a magnetic field
that can favorably interact with the spin magnetism of the same electron (antiparallel) or unf
Quiz 2.
x 2
1. Normaize the following wavefunction from 0<x<: e
Answer: The proper setup is:
x e
*
0
x2
2
e
x2
2
2
x e
0
2x 2
x And using the identity:
2
0
1 2
1
2
you just have to recognize that a= 2 making the integral =
. If you got this
2 a
2
2
0
fa
Quiz 4.
1. For the ground state vibrational wavefunction here,
describe what motions in the a and b directions are.
Describe what is happening at x=0.
(3 pts)
Answer: a. the oscillator is expanding. b. the oscillator
is contracting. x=0, the oscillator is
Quiz 7.
1. How is rotation of benzene different from HCl?
(2 pts)
Answer: You pick up a new quantum number because the asymmetry is like adding a new dimension.
2. How is a parabolic vibrational potential different from a more realistic potential?
(4 pts)
Each question on a quiz will be worth three points.
Quiz 1
1. For a series of waves show here:
described by:
x
sin n
L
a. What does n do?
Weak answer, full credit: N causes the
wavelength to get shorter.
Best answer: N is the mode index; it causes the wa
Lecture 30: Phenomenological Kinetics
Temperature dependence of the rate coefficient Reactions need less than the (H2) dissociation energy: H2 + D HD + H (9.3% of it) k(T) ~ P(EEa) molecules need to gain the activation energy Ea to react; Arrhenius equati
Lecture 29: Phenomenological Kinetics
We will study chemical reaction rates. The systems are out of equilibrium and often evolve fast (they require special experimental techniques). In principle, we can calculate the reaction rates by combining quantum an
Lecture 13: The electronic structure of atoms The Zeeman effect
Electric current in a loop has a magnetic dipole moment ( gyromagnetic ratio)
In the H atom, magnetic moment of electron couples to external magnetic fields B:
The spherical harmonics remain