PHGN 450 / EGGN 502
Fall 2005
Project 6: Matrix Representations of the Time-translation Operator
Due Oct. 19
Background
The time-dependent Schrodinger equation (SE) determines the quantum evolution of a particle subject to some potential
energy function, V (x). In one dimension the time-dependent SE takes the form:
h 2 @ 2
@ (x; t)
= 2m @x2 + V (x) (x; t);
(1)
{h
@t
where m is the particle's mass. Recall that the wave function is the \probability density amplitude". That is, the modulus
square of the wave function (j(x; t)j2 = (x; t)(x; t)) gives the probability per unit distance (volume, if in 3-dimensions)
of nding the particle at some location, x, at time, t.
The evolution of the wavefunction can be thought of as a transformation operation on the wavefunction at the original
time to the wavefunction at some future time: (x; t + t) = U^ (t)(x; t). Assuming all the potentials are real then this
transformation must conserve the total probability. Such transformations are called "unitary" and satisfy the property:
U^ y U^ = 1.
Using Taylor's theorem, we can understand that the time-translation operator, U^ (t) is generated by the time derivative
operator:
2
2
x; t)
+ :::
(2)
(x; t + t) = (x; t) + t @ (@tx; t) + t2 @ (
@t2
Or, equivalently:
2
2
(x; t + t) = (1 + t @t@ + t2 @t@ 2 + : : :)(x; t) = exp(t @t@ )(x; t)
(3)
where we dene the exponent of the derivative operator by its Taylor expansion. Thus, U^ (t) = exp(t @t@ ).
The SE can be thought in terms of operators acting in the linear vector space of wavefunctions (called Hilbert space).
Specically, the SE equation states that the time derivative operator is equal to ( {=h ) times the Hamiltonian operator, H^ .
h 2 @ 2
@
{^
{
= h H = h 2m @x2 + V (x) :
(4)
@t
Thus, if satises the SE, then U^ (t) = exp( {H^ t=h ), and we say that the Hamiltonian is the \generator" of timetranslations. The purpose of this project is to use an approximate matrix representation of U^ (t) to evolve some initial wave
packet incident on a potential to determine the transmission and reection probabilities.
Matrix Representation
In the by now standard fashion we discretize the (nite) domain of interest, say D=2 ! +D=2, by N equally spaced
points: xi = D=2 + (i 1)x; 0 i N . We will represent the discretized by the column vector:
0
1 (t) 1
e (t) = B
(5)
@ ... C
A
N (t)
where i (t) = (xi ; t). Using column vectors to represent the wavefunction, allows us to use N N matrices to represent
operators. Specically, the potential energy operator becomes the diagonal matrix:
0
1
V1 0 0 B 0 V2 0 C
C
Ve = B
(6)
B
@ ... ... . . . ... C
A
0 0 VN
V~ (x) is diagonal because V (x) is assumed to be local, that is, it acts on (x) at the same location, x. In general one
could have non-local potentials such as the Hartree-Fock potential in atomic physics. In this case the potential matrix will
1
have o-diagonal
elements. Using the three-point nite dierence approximation for the second derivative, the kinetic energy
operator ( 2hm2 @x@22 ) has the matrix representation:
0
2 1 0 1
B 1 2 1 C
2
C
gE = h 2 B
K
(7)
2mx B
@ ... ... . . . ... C
A
0 1 2
We will assume an initial gaussian wavefunction with some initial momentum, k:
2 1=4
(x; t = 0) = exp({kx) exp( (x x0 )2 =)
(8)
This represents a gaussian wave packet, centered at x0 , and moving in the positive x-direction with speed, hk=m.
a. (30 points) Write an algorithm to construct the matrix representation of the Hamiltonian. Use your Hamiltonian matrix
to construct a matrix representation of the time translation operator expanded to second order in the time step:
t
U~ (t) = exp( {H~ t=h) ' 1 { H~
h
t2 ~ 2
H
2h2
(9)
b. (30 points) Apply your time step operator to the problem of one dimensional scattering of a wave packet from a square
well potential of depth, V02 and width, L (15 points). For numerical purposes use mp= 1=2, h = 1, and dx = :1. These will set
h
an energy scale: E0 = 2mdx
:25E0 = 5, and = 200dx2 = 2. (To see
2 = 100. Next try V0 = :3E0 = 30, L = 10dx = 1, k =
the wavefunction clearly I used a domain from -10 to 10.) Note that since the kinetic energy is a little less than the barrier
height, classically the particle cannot make it over the barrier. However, quantum mechanically the particle can \tunnel"
through the barrier by virtue of its quantum mechanical wave nature. Evolve the initial wavefunction enough to be able to
extract the transmission and reection probabilities to 3 digits (10 points). Test to see how well your algorithm maintains
conservation of probability (5 points).
c. (10) Construct an algorithm to nd the inverse of a diagonally banded matrix using Gauss-Jordan. Demonstrate that it
works by nding the inverse of the 3 by 3 matrix:
0
1
a11 a12 a13
Ae = @ a21 a22 a23 A :
(10)
a31 a32 a33
d. (30 points) Repeat part (b) using your matrix inversion routine applied
translation operator.
1 (1
U~ (t) = exp( {H~ t=h) '
(1 + { 2th H~ )
2
to the Cayley approximation for the time{
t ~
2h H ):
(11)