1
Show that the particleinabox wave functions, which are given by, n (x) =
! nx "
2
L
s
L
sin L , n = 1, 2, satisfy the relation, 0 n (x) m (x) = 0 for m = n where here the
complex conjugate is irrelevant since the functions are real.
!
"2
!
"
!
"
!
"
The theme I have for this problem solution set is getting used to picking out the salient features of
a solutions so that you can generalize and reuse the work you do.
To this end, Dirac notation is previewed as a shorthand. Prof. McCurdy will give a more
Chem 110A
Homework 5: Due Friday, May 8, 2015
Chemistry 110A 2015
Tomoyuki Hayashi, Ph.D.
Problems 2 and 3 on this homework set involve the same concepts as problems 610 and
613 in the text.
1. Rigid Rotor application of energy level formula: The intern
Chem 110A Homework 5: Due Friday, October 30, 2009 Chemistry 110A 2009 Professor McCurdy Problems 2 and 3 on this homework set involve the same concepts as problems 611 and 613 to 615 in the text. 1. The internuclear distance of the 12 C16 O molecule i
Chem 110A
Homework 7: Due Friday, May 13, 2016
Chemistry 110A
Professor McCurdy
1. Mathematica/MatLab Problem: Variational calculation on the hydrogen atom
using a linear combination of Gaussian basis functions.
5
Using the fiveterm linear trial function
Name SOLUTI O N
Answer any four of the following five questions. Please circle the numbers of the
problems that you want us to grade. Otherwise, we will just grade the first four
problems. You must show all your work to receive full credit. Each question
Chem 110A
Homework 6: Due Friday, November 14, 2014
Chemistry 110A
Professor McCurdy
1. Find the values of the following commutators for the hydrogen atom Hamiltonian.
Only very little algebra is required, because you may use the values of the angular
mom
CHE$110A$
HW#$5$
$
1.$
h ( m1 + m2 )
h
h
B=
=
=
2
2
2
8 I 8 r
8 2 r 2 m1m2
$
kg m 2
(1.996 1026 kg + 2.657 1026 kg )
s
=
= 5.78 1010 Hz
2
2
10
26
26
8 (1.1283 10 m ) 1.996 10 kg 2.657 10 kg
6.626 10 34
$
for$Joules$use$ E = h $ $ B = hBHz = 3.83 10 23 J $
Show that the particleinabox wave functions, which are given by, n (x) =
1
L
sin
, n = 1, 2, satisfy the relation, 0 n (x) m (x) = 0 for m = n where here the
complex conjugate is irrelevant since the functions are real.
nx
L
2
L
L
0 n (x) m (x) dx =
1
The following wave function (in which A and B are numbers not functions of x or
E1 t
E1 t
t) (x, t) = A1 (x)e
+ B2 (x)e
is a linear combination of two solutions of the time
En t
dependent Schrdinger equation each having the form n (x)e
o
with n=1 and 2,
Chem 110A
Homework 8: Due Wednesday, June 3 2015
Chemistry 110A 2015
Tomoyuki Hayashi, Ph.D.
These problems cover material in both chapter 8 and chapter 9 of McQuarrie and
Simon. Additional review problems are also suggested here
1. Properties of a Slater
CHEMISTRY 110A
Final Exam
Monday December 10th 2012
Name_
A, B = AB BA
x
sin 2x
d
du
(sin u) = (cos u )
dx
dx
2
sin xdx = 2 4
x 2 x sin 2x cos 2x
x sin xdx = 4 4 8 2
d
du
(cos u ) = (sin u)
dx
dx
2
x3 x 2
1
x cos 2 x
3 sin 2 x
x sin xdx =
6 4 8
4
Homework Week 2: Due Friday, October 17
Chemistry 110A 2014
Professor McCurdy
1. Properties of wave functions. Show by explicit integration that the particleinabox
wave functions, which are given by,
2 $ n x '
n (x) =
sin&
n = 1,2,3,
)
L % L (
satisfy
Chem 110A
Homework 5: Due Friday, May 8, 2015
Chemistry 110A 2015
Tomoyuki Hayashi, Ph.D.
Problems 2 and 3 on this homework set involve the same concepts as problems 610 and
613 in the text.
1. Rigid Rotor application of energy level formula: The intern
Homework Week 3: Due Friday, October 16 Chemistry 110A 2009 Professor McCurdy 1. Problem 425 in McQuarrie and Simon The algebra should be very short and its a good idea to do this one before problem 2. 2. For the particleinbox, the system is prepared i
Homework Week 2: Due Friday, October 9 Chemistry 110A 2009 Professor McCurdy 1. Problem 316 in McQuarrie. 2. (a) Find the average value of position x , momentum p , and square of the mometum p 2 for the ground and first excited state of the particleina
Chem 110A Homework Week 4: Due Friday, October 23 Chemistry 110A 2009 Professor McCurdy 1. Make the harmonic approximation to each of the following diatomic potential energy functions to obtain the corresponding harmonic potential energy function of the k
1a
Using atomic units, for the quantities in the general formula for the energy
Z2
e2
levels of a one electron atom, En = 2n2 a0 4 0 , show that the energy of the helium ion
(He+ , with Z = 2) in its ground state (1s state) in atomic units is 2. These en
Variational calculation of approximate
ground state energy for H atom with
trial function = Exp[ r^2]
Evaluate Integrals in variational expression
Use Gaussian units so qe^2 =
In[1080]:=
H r
r
e^2
4*Pi*0
Clear[, r, hbar, mu, qe, PsiTrial, num, den];
Ps
Chem 110A
Homework Week 4: Due Friday, October 31
Chemistry 110A 2014
Professor McCurdy
Three very short problems only, since we have an exam on Wednesday, these will be
good practice for the exam
1. Warm up: The 1D harmonic oscillator eigenfunctions hav
17 Sirius, one of the hottest known stars, has approximately a blackbody spectrum
with Jim = lnm. Estirnate the surface temperature of Sirius.
. mammary: 10 J 5 2.9979241: 103 me .  5
50mm: A = 2173th =5 T = region 3 : l'm'mssxm m"
T 3 W e 11145K
Chem 110A
Homework 8: Due Monday, December 1, 2014
Chemistry 110A
Professor McCurdy
1. Atomic units:
(a) Using atomic units, for the quantities in the general formula for the energy levels of a
Z 2 " e2 %
+
one electron atom, En = 2 $
' , show that the en
Building Blocks of the wave functions
of hydrogen and all other one electron atoms
Spherical Harmonics  built in function
SphericalHarmonicY[l,m,]
Clear[l, m, , ];
l = 1;
m = 1;
example = SphericalHarmonicY[l, m, , ]

1
2
3
2
Sin[]
2
H_atom_eigenfunct
Chem MOA
1. (25 points, Harmonic oscillator and first order perturbation theory) The motion of
a particle of mass / is described by the Hamiltonian
H = H, + Hxcfw_x)
with theunperturbed Hamiltonian being thatof the Harmonic Oscillator:
h1 dl
k ,
 *H^"e'
Chem 11OA
Name
Spring 2015
KS^
T. Hayashi
Student ID
TA Name
Midterm Exam I  April 24,2015
Chemistry 110A
Tomoyuki Hayashi, Ph.D.
Instructions: You may use only a pencil or pen. No other materials are allowed.
Write all answers on the exam in the spaces
CHE110A  Simulation of final
06/08/2016
Review questions
Answer the following questions and justify your answers (1pt for each right
answer + 3pt for the correct explanation):
1. Is the probability density (x, t)2 for a stationary state timeindependen
CHE110A: Homework 2
Due on 4/15/2016
1
Particle in a onedimensional box
Find the expectation value (i.e. the average) for the momentum hpi of a
quantum particle of mass m in a onedimensional box of size L, for a pure
state of quantum number n. The eige
CHE110A  Homework 1
Due: 04/08/2016
1
Black body radiation
Integrate the spectral radiance of the black body, which is:
R()d =
c 8 h 3
d
4 c3 e kh
BT 1
to get Stefans law, P/A = T 4 , and derive the units of .
Hint:
Z
4
x3
= .
dx x
e 1
15
0
Express Plan
CHE110A: Homework 3
Due on 4/22/2016
1
Timedependent Schr
odinger equation
Show that for a particle of mass m in onedimension, with Hamiltonian
2
2
= h d + V (x)
H
2m dx2
the time derivative of the expectation value of the position operator is
dhxi
hpx
1
True or False
These come from Levines Quantum Chemistry textbook used in CHE 210A.
Thinking through all these problems until you really understand them is very
good exam preparation. If a question is only sometimes true, answer false.
is a linear opera