Chem. 540
Instructor: Nancy Makri
Postulates of Quantum Mechanics
1.
Wavefunction
The state of a system is described by a vector in a Hilbert space. The state of a system
can be fully specified by its wavefunction in position space, (r; t ) , or by its wa
Chem. 540
Instructor: Nancy Makri
PROBLEM 42
In class we calculated the time-dependent Green function or propagator for a free particle of
mass m moving in one dimension.
a)
Using the one-dimensional result, find the propagator for a free particle moving
Chem. 540
Instructor: Nancy Makri
PROBLEM 41
The survival amplitude for a system initially described by the state (0) is defined as
C (t ) = (0) (t )
i.e., C (t ) is a measure of the overlap of the propagated state with the initial one. The survival
proba
Chem. 540
Instructor: Nancy Makri
PROBLEM 40
The variational principle derived in class applies to the ground state energy only. In general,
it is not easy in practice to use variational theories for excited states; but at least in principle it is
possibl
Chem. 540
Instructor: Nancy Makri
PROBLEM 39
Show that (ignoring terms of third or higher order in the perturbation) the energy through
second order in perturbation theory is equal to the expectation value of the Hamiltonian with re spect to the first ord
Chem. 540
Instructor: Nancy Makri
PROBLEM 38
Consider again the Hamiltonian defined in problem 37, but now with V12 = V21 = 0 . Once
again, we are interested in the energy corrections through second order in the perturbation. Notice,
however, that this pe
Chem. 540
Instructor: Nancy Makri
PROBLEM 37
A Hamiltonian H 0 for a three-state problem has eigenvalues equal to 0 , 0 , and 1 . The
two degenerate eigenstates are chosen to be orthogonal. A perturbation V is applied, which has
the following matrix eleme
Chem. 540
Instructor: Nancy Makri
PROBLEM 36
Consider a particle (e.g. an electron) in a one-dimensional box. Use first order perturbation
theory to predict how the energy eigenvalues will change if we apply an external electric field lin ear in x, i.e.,
Chem. 540
Instructor: Nancy Makri
PROBLEM 35
Consider a particle moving on a sphere, with wavefunction in spherical polar coordinates
given by
( , ) = Acfw_ Y1,+1 ( , ) + Y1,1 ( , ) ,
where Ylm is the usual spherical harmonic.
a) Determine the value of A
Chem. 540
Instructor: Nancy Makri
PROBLEM 34
Consider the eigenstates of S z for a spin 1 2 particle. In this basis, any spin state is spe cified by two coefficients, i.e., by a two-component vector. As usual, the vector representation of
the two eigensta
Chem. 540
Instructor: Nancy Makri
PROBLEM 33
By carrying out the appropriate integrals over the spherical polar angles, show that the
00 and 10 eigenstates of the angular momentum operators are orthogonal.
Chem. 540
Instructor: Nancy Makri
PROBLEM 32
Consider the angular momentum operators defined in class.
a) Calculate the following commutators: [ L+ , Lz ], [ L+ , L ], and [ L+ , L2 ] .
b) Which of these angular momentum operators are hermitian?
Chem. 540
Instructor: Nancy Makri
PROBLEM 31
Consider a quantum mechanical particle moving in two dimensions with the following
Hamiltonian:
2
2 p
= p x + y + V ( x, y ) ,
H
2m 2m
where
V ( x, y ) =
1
1
2
m x x 2 + m 2 y 2 .
y
2
2
a) Sketch this potentia
Chem. 540
Instructor: Nancy Makri
PROBLEM 43
Consider a quadratic perturbation,
1
V ( x) = bx 2 ,
2
on a harmonic oscillator
p2 1
H0 =
+ m 2 x 2 .
2m 2
Calculate the corrections to the energy eigenvalues through second order in the perturbation. Show
that
Chem. 540 Instructor: Nancy Makri
PROBLEM 44 The Stark effect is the shift of an atom's energy levels caused by a uniform electric field. In this problem you will use 1st order perturbation theory to explore the Stark effect on the hydrogen atom. Suppose
Chem. 540
Notes on Integrals
Functions of the form
f ( x ) = e ax
2
+bx
, Re a > 0
are called Gaussian. Integrals of Gaussian functions occur frequently in quantum mechanics. It can be
shown that
e ax
2
+ bx
dx =
a
eb
2
/ 4a
.
One not so well known techni
Chem. 540
Nancy Makri
Symmetries, Commutators and Conservation Laws
In class we proved a theorem that relates the time derivative of an observable to the commutator between
the corresponding operator and the systems Hamiltonian. Thus, if an operator (whic
Chem. 540, Fall 2010
Instructor: Nancy Makri
Computer Assignment 4
The Variational Principle
In this assignment you will apply the variational principle to calculate approximations to the ground
state energy of one-dimensional systems described by the Ham
Chem. 540, Fall 2010
Instructor: Nancy Makri
Computer Assignment 3
Eigenstates of General Hamiltonians in 1d
In this assignment you will calculate energy eigenvalues using a symbolic algebra program, but
this time the Hamiltonian matrix is not readily ava
Chem. 540, Fall 2011
Instructor: Nancy Makri
Computer Assignment 2
Eigenstates of Discrete Hamiltonians
The Hamiltonian operator for a two-state system has the form
H = 1 1 1 + 2 2 2 h ( 1 2 + 2 1
)
where 1 , 2 are orthonormal states that form a complete
Chem. 540, Fall 2010
Instructor: Nancy Makri
Computer Assignment 1
Wavefunctions and Matrix Elements
In this assignment you will plot wavefunctions, normalize them, and calculate bra-kets, i.e.,
overlaps of states and matrix elements of operators.
%
%
Con
Chem. 540
Nancy Makri
Notes on Complex Numbers
A very useful representation of a complex number z is in terms of its modulus (absolute value) z
and phase . If
z = x + iy ,
then
z = z ei
where z = x 2 + y 2 . This is the trigonometric form of the complex n
Chem. 540
Nancy Makri
Dirac bra-ket notation
The symbol n
(or n ) is called a ket and denotes the state described by the wavefunction
n . The complex conjugate of the wavefunction, , is denoted by the bra n (or n ). The ket
n
denotes a state in the most
CHEMISTRY 540: QUANTUM MECHANICS
Instructor: Nancy Makri
Suggested references
The course does not follow any particular book. Any of the following books (on reserve in the
Chemistry Library) covers the bulk of the course material and should provide a valu
Chem. 540
Instructor: Nancy Makri
PROBLEM 30
Find the explicit form of the first three ( n = 0,1, 2 ) harmonic oscillator eigenfunctions in position
space.
Chem. 540
Instructor: Nancy Makri
PROBLEM 29
Use your results from the previous problem and the definition of the uncertainty in position
and in momentum space from problem 27 to calculate the uncertainty product xp . Does your
result obey the uncertainty
Chem. 540
Instructor: Nancy Makri
PROBLEM 14
Consider a state of a system, which may or may not be an eigenstate of the time-inde
pendent Hamiltonian H . Show that the energy of the state, i.e., the expectation value of the
Hamiltonian, does not change w