C hemistry 4 60
Dr. Jean M. Standard
Problem Set 9 Solutions
1 . Determine t he e xplicit f orm o f t he t otal H amiltonian o perator f or t he B eH m olecule.
The total Hamiltonian operator for BeH (in atomic units) is
1 2
H = 1 + 2 + 2 + 2
2
3
4
2
4
4
C hemistry 4 60
S pring 2 013
Dr. Jean M. Standard
February 19, 2013
Name _
Quantum Chemistry Exam 1 100 points
1.) (16 points) Explain or provide an example illustrating each of the following concepts.
a.) Born probability interpretation
b.) Normalizatio
C hemistry 4 60
S pring 2 013
Dr. Jean M. Standard
April 9, 2013
Name _ _KEY_
Quantum Chemistry Exam 2 Solutions (Take-home portion)
5.) ( 25 p oints) I n t his p roblem, t he l inear v ariation m ethod w ill b e e mployed t o d etermine
approximations to
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 1 Solutions
1 . The translation operator, T h , is defined by the rule:
represents t he d istance o f t ranslation.
T h f ( x ) = f ( x + h) , where h
a . ) Show whether r not T h is a linear operator.
o
Fo
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 2 Solutions
1 . In class, we considered a particle in a potential of width L and depth V0 :
V ( x ) = 0
V
0
x<0
0 x L .
x > L
V=V0
II
I
x=0
III
x=L
We showed in class that requiring the wavefunction t
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 3 Solutions
1 . Consider a particle impinging on a finite barrier shown in the figure below.
V=V0
I
II
x=0
III
x=L
In class, we discussed the case E < V0 . I n t his p roblem, y ou s hould o btain t he s ol
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 4 Solutions
1 . One of the important properties of Hermitian operators is that their eigenfunctions form a
complete set. This means that any arbitrary function f ( x ) may be exactly expressed as a
linear c
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 5 Solutions
1 . Use the raising and lowering operators for the harmonic oscillator to evaluate the
expectation v alue o f px in the n th s tate.
We need to evaluate the expectation value
px
n px n ;
=
ther
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 6 Solutions
1.
The Hamiltonian operator in atomic units for the hydrogen atom in its ground state is
given b y
1 2
2
1
H = 2 +
.
2 r
r r
r
The radial coordinate r is the distance between the electron an
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 7 Solutions
1 . The equations relating C artesian coordinates
( r, , ) are
( x, y, z )
and spherical polar c oordinates
x = r sin cos
y = r sin sin
z = r cos .
S olving f or
(r, , )
in terms of (x , y , z
C hemistry 4 60
Dr. Jean M. Standard
Problem Set 8 Solutions
1 . Determine t he e xpectation v alues Sx , Sy , and Sz for an electron in the spin up
state. Use the spin raising and lowering operators to evaluate the first two of these
expectation v alues.
C hemistry 4 60
S pring 2 013
Dr. Jean M. Standard
February 19, 2013
Quantum Chemistry Exam 1 Solutions
1.) ( 16 p oints) E xplain o r p rovide a n e xample i llustrating e ach o f t he f ollowing c oncepts.
a . ) Born p robability i nterpretation
The Bor