Homework 1 Solution
1. Read the postulates, and if there is any weird vocabulary, look
it up in the Quantu Vocabulary handout. If its not shown there,
deand in the next lecture that Prof. Gruebeleadd it!
Don't worry if the postulates dont ake crystal-clea
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 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 t
Homework H13 Solution
1. In lecture L11, Gruebele showed that any time-dependent wavefunction (x,t) can be
!
written as a sum over stationary states (eigenstate times ! ! ! factor):
!
! ! !
! ! ! (!) !
! !, ! =
(1)
still satisfies the time-dependent Schr
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,
!
Homework H14 Solution
1. Write the normalized wavefunction of a particle in the ground state of 1D box. Call it
!. Now consider a 3D box, with the particle in the ground state. [There is just one
ground state]. You can write this wavefunction as a product
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
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 ! =
!
!
!
Homework H20 Solution
Turn in 1: The molecule H2- contains 3 electrons. The wave function can be written
approximately as a product over individual electron wavefunctions, or (r1, r2, r3) =
a(r1). b(r2). c(r3).
a. Show that this function does not satisfy
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 Hami
Homework H15 Solution
!
1. Using the formulas for x, y, and z show that ! = tan-1
and ! =!"! !
!
Solution:
Transformation between rectangular and spherical polar coordinates give,
!=!
(1)
!= !
(2)
and !=!.
(3)
!
!
!
!
Therefore ! = tan !, ! = !"!
and, ! =
Homework H16 Solution
1. Turn in
a) Prove that !1-1 - !11 is also an eigenfunction of !"# .!Hint: if each function in a sum
has the same eigenvalue of energy, what is the energy of the sum?
b) Plot !1-1 - !11 in the x-y plane ( = /2, plot as function of )
Homework H7 Solution
1. Turn in Go to the QM demo from lecture L4 online, and start the 1D quantum applet.
(As always, wait a minute after clicking the link, and hit Run when it asks you to run it.)
Now solve the time-dependent Schrdinger equation using t
Homework H6 Solution
1. Turn in: An electron has mass me9.1.10-31 kg.
a. Its velocity has a range of 1 m/s. What is the range of positions that will be measured?
b. In a hydrogen atom, the range of velocities is closer to one million meters/second. (That
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 alread
Hour Exam 1 Solution
1a. (5 pts) On an x-p plot, draw the trajectory of a ball vibrating on a spring, shown on the right.
Draw the trajectory for the ball starting at x0>0 and p=0. (x=0 is the equilibrium position of the
spring.) Draw one cycle of the vib
Homework 2 Solution
1. A spring has restoring force F=-kx, where k is the spring constant. x=0 when the spring is not
stretched or compressed. Assume that x(t=0) = 0 and v(t=0)=0 initially. Plug into the Verlet
algorithm to show that x(t=t) = 0 also. If t
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
Homework H9 Solution
1. (Turn in) Use postulate (4) of quantum mechanics to prove very explicitly that the
! !
energy of a quantum particle with wavefunction ! ! ! ! is simply En with 100%
probability.
Remember what postulate (4) says in GENERAL: if an ob
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 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
! ! ! =
Hour Exam 2 Solution
1. (10 pts) Calculate the energy in Joules required to excite an electron in the hydrogen
atom from the ! = 2 to ! = 4 state, given that the Rydberg constant converted to Joule
units is 2.18 10!" !.
Solution:
To find the energy requir
Lecture 23
Born-Oppenheimer approximation
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Scie
Lecture 32
General issues of spectroscopies. II
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
Nationa
Lecture 28
Point-group symmetry I
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Science Foun
Lecture 27
Molecular orbital theory III
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National Scienc
Lecture 33
Rotational spectroscopy: energies
(c) So Hirata, Department of Chemistry, University of Illinois at Urbana-Champaign. This material has
been developed and made available online by work supported jointly by University of Illinois, the
National S
Lecture 2
Wave-particle duality
Wave-particle duality
Light has particle-like characteristics and
electrons have wave-like characteristics. The de
Broglie relation between wavelength and
momentum = h / p quantifies these competing
characteristics.
It is
Lecture 3
The Schrdinger equation
The Schrdinger equation
We
introduce the Schrdinger equation as
the equation of motion of quantum chemistry.
We cannot derive it; we postulate it. Its
correctness is confirmed by its successful
quantitative explanations
Lecture 4
Complex numbers, matrix algebra,
and partial derivatives
Basic mathematics for QM
Complex
numbers
Matrix algebra
Partial derivatives
Derivative operators in the Schrdinger equations
Kinetic-energy operators in the Cartesian and
spherical coor
Lecture 8
Particle in a box
The particle in a box
This
is the simplest analytically solvable
example of the Schrdinger equation and
holds great importance in chemistry and
physics.
Each of us must be able to set up the
equation and boundary conditions,
Lecture 1
Quantization of energy
Quantization of energy
Energies
are discrete (quantized) and not
continuous.
This quantization principle cannot be derived;
it should be accepted as physical reality.
Historical developments in physics are
surveyed that