Page 1 (multiple choice) should take a total of only ~5 minutes. Instructions: Select the
correct statement among the options provided below for each problem. (5 points each).
1. _
For the n=3 particle in the box wavefunction (plotted here):
A. The probab
Chem 442: Homework for lecture L28
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 20th at the beginning of lecture.
1. Problem 7.2 in the book.
2. Turn in Time, for once, to calculate one of those
Chem 442: Homework for lecture L22
(only turn in BOLD assignment; do all assignments by due date)
1. Show that if is an operator with real eigenvalues (hermitian), then
=
[Hint: it's just a slightly more general version of question (2) on H21]
2. Turn i
Homework H29 Solution
1. In class, we evaluated three terms in the Hartree-Fock energy expression. Evaluate the remaining
terms in the energy expression for 2 electrons in the ground state, to show that
!" = ! + ! + !" !"
[Hint: You may need to prove thin
Chem 442: Homework for lecture L21
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, March 30th at the beginning of lecture
1. In the previous lecture, Hilbert spaces were introduced with an example where y
Homework H21 Solution
1. In the previous lecture, Hilbert spaces were introduced with an example where you define the
axes as ! = cos and ! = sin . Then, according to the notation for basis
!
!
functions, what are |1 and |2 ?
!"
!"
Solution:
In Hilbert
Chem 442: Homework for lecture L19
(only turn in BOLD assignment; do all assignments by due date)
1. Problem 5.1 in the book. Remember from Lecture 13 that this is indeed one of the
two degenerate eigenfunctions for the particle on a ring/circle, which we
Homework H19 Solution
1. Problem 5.1 in the book. Remember from Lecture 13 that this is indeed one of the
two degenerate eigenfunctions for the particle on a ring/circle, which we combined
and generalized as !"# .
!
!
Solution: For normalization,
= 1
!
!
Homework H18 Solution
Turn in1. Show that if | and | and orthogonal and normalized (this is called
orthonormal for short), then
1
|! |! |! |!
2
is normalized.
Solution:
Note that the subscripts 1 & 2 denote the electron labels.
So, 1,2 =
!
!
|! |! |! |!
=
Homework H22 Solution
1. Show that if is an operator with real eigenvalues (hermitian), then
=
[Hint: it's a more general version of question (2) on H21]
Solution: Let be an operator with real eigenvalues E, or
H |n> = En |n>
Then we have
= ( ! >< | )
Homework H23 Solution
1. Use the link given below
http:/www.mathstools.com/section/main/system_equations_solver#.VRR1N1qprzK
to numerically diagonalize the matrices
10
4
5
4
and
4
10
4 15
Use the eigenvalues you get for each matrix to find the eigenvector
Chem 442: Homework for lecture L23
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 6th at the beginning of lecture.
1. Use the link given below
http:/www.mathstools.com/section/main/system_equations
Homework H27 Solution
1. Problem 8.5 in the book. [Hint: Remember that CH.3 is planar. Considering it to
be in the x-y plane, only 2s, 2px and 2py participate in hybridization (and not 2pz ,
which is orthogonal to the plane.)]
Solution: The carbon atom in
Homework H28 Solution
1. Problem 7.2 in the book.
Solution: For the first excited state, the symmetric and anti-symmetric spatial
wavefunctions are given respectively by,
! =
and
! =
1
2
1
2
! 1 ! 2 + ! 2 ! 1
! 1 ! 2 ! 2 ! 1
For 2 electrons being in two d
Chem 442: Homework for lecture L27
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 13th at the beginning of lecture.
1. Turn in Problem 8.5 in the book. [Hint: Remember that CH.3 is planar.
Consider
Homework H26 Solution
1. Show that you cannot get both electrons in the (bonding) orbital AND at the same
time have the same spin (e.g. 1 2), by constructing the determinant for the
wavefunction and multiplying it out.
Solution:
Writing the determinant fo
Chem 442: Homework for lecture L25
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 6th at the beginning of lecture.
1. Problem 8.8 in the book.
2. Turn in: Problem 8.9 in the book.
3. Problem 8.10 i
Homework H25 Solution
1. Problem 8.8.
Solution: Consider the 3 carbon atoms to have atomic orbitals ! . We want to find the molecular
orbitals , by setting up the secular determinant to zero. Under the Hckel approximation, the
diagonal terms in the Hamilt
Chem 442: Homework for lecture L26
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 13th at the beginning of lecture.
1. Show that you cannot get both electrons in the (bonding) orbital AND at the sa
Chem 442: Homework for lecture L24
(only turn in BOLD assignment; do all assignments by due date)
The Turn in homework is due Monday, April 6th at the beginning of lecture.
Turn in 1. What is the probability that a vibrating molecule in the n=0 state is i
Homework H24 Solution
Turn in 1. What is the probability that a vibrating molecule in the n=0 state is in the
tunneling region?
Solution: Recall that the energy of a vibrating molecule within the harmonic oscillator
approximation is
! =
!
where =
!
!
!
Th
Chem 442: Homework for lecture L18
(only turn in BOLD assignment; do all assignments by due date)
Due Monday March 9th 2015 at beginning of lecture.
1. Turn in Show that if | and | and orthogonal and normalized (this is called
orthonormal for short), then
Chem 442: Homework for lecture L17
(only turn in BOLD assignment; do all assignments by due date)
Due Monday March 9th 2015 at beginning of lecture.
1. Make a plot of the linear combination of the linear combination ! + ! in the x-y
plane ( = /2). What is
Homework H17 Solution
1. Make a plot of the linear combination of the linear combination ! + ! in the x-y
plane ( = /2).
Solution:
The form of the Spherical Harmonics we need here are:
!,! =
1 15 !" !
4 2
Setting = /2 and making a polar plot in the x-y
Chem 442: Homework for lecture L10
(only turn in BOLD assignment; do all assignments by due date)
rd
Due Monday February 23 2015 at beginning of lecture.
1. In lecture, we talked about how the vibrating molecule Hamiltonian (energy operator)
is symmetrica
Homework H10 Solution
1. In lecture, we talked about how the vibrating molecule Hamiltonian (energy operator)
is symmetrical in p and x because it depends on the square of both. We made use of this
symmetry to find out that a Gaussian is the lowest energy
Chem 442: Homework for lecture L9
(only turn in BOLD assignment; do all assignments by due date)
1. (Turn in) Use postulate (4) of quantum mechanics to prove very explicitly that the energy of
a quantum particle with wavefunction n(x) e
i
En t
!
is simpl
Homework H8 Solution
1. Turn in Problem 1.8 (Page 25)
Solution:
-|x|
Writing the wavefunction as (x) = e
space we have
!
!
=
=
!
and integrating the square modulus over all
!
! ! = 2
!
! = !
!
!
= 1 (1)
!
Thus the wavefunction is already normalized! NO
Homework 12 Solution
1. Problem 2.2 in the book.
! !
Solution: For the particle in a 1D box, ! = !
1 mole (6.02 x 10-23 molecules) weighs 28g =0.028 kg.
So, 1 N2 molecule weighs 0.028/(6.02 x 10-23) = 4.65 x 10-26 kg.
So, the energy separation between n=1