New Project for Undergraduate or MS New Project for Undergraduate or MS Student

Pair Hamiltonian analysis of a many-electron problem

Idea: Real materials are composed of many identical electrons. In general, describing these electrons accurately, using the laws of quantum mechanics, is mathematically very difficult. In this project, we will study a model many-electron system for which the mathematics is tractable. This will enable us to draw some general conclusions about the behavior of many-electron systems and to access some mathematical and computational methods for studying them.

Prerequisites: Elementary quantum mechanics (PHY 141 or CHM 342/344), Differential equations (MTH 251), and some computer programing experience and interest.

Dates: The starting date is flexible. Summer of 2001 would be ideal.

Some details of idea

We can write Schrödinger Equation for a many electron system as follows:

H Ya = Ea Ya,
where,
H º
å
 i 

æ
ç
è 		- (h/2p)2
2m
 Ñi2 + V(ri) ö
÷
ø
h(i) 
 +
å
 i < j 
 e2
|ri - rj|

and

Ya(x1,x2,x3,...xN) = - Ya(x2,x1,x3,...xN).

Energy eigenvalue of this Hamiltonian can be expressed as an expection value:

Ea = áYa|H|Yañ º ó
õ 		d1 ó
õ 		d2 ó
õ 		d3 ¼ ó
õ 		dN   Y*aH Ya,

which can be written and a very suggestive form:

Þ Ea = Trace { r2a   K} .

Here, the ``reduced" or ``pair" Hamiltonian is defined according to:

K(1,2) º æ
ç
è 		1
N-1
 ö
÷
ø 		( h(1) + h(2)) + e2
|r1 - r2|
 ,
and the two-particle density matrix is given by:
r2a(1,2;1¢,2¢) º ó
õ 		d3 ó
õ 		d4¼ ó
õ 		dN Y*a(1,2,3,4...N)Ya(1¢,2¢,3,4...N)

Consider, more carefully, the pair Hamiltonian:

K(1,2) º æ
ç
è 		1
N-1
 ö
÷
ø 		( h(1) + h(2)) + e2
|r1 - r2|
Suppose that it is possible to find the eigenvalues en and corresponding eigenstates |nñ:
K |nñ = en |nñ.
The eigenstates Ea of the many-electron system can be expressed in terms a pair states state expansion:
Þ Ea =
å
 n 
 enWan
where       Wan º án|r2a|nñ    and      
å
 n 
 Wan = 1.

Proposed project: Systematic study of the pair state expansion for many-electron ``harmonic" atoms.

real atom:  V(r1) = -Z e2
r1
     Þ       ``harmonic"atom:  V(r1) = 1
2
 KZ r12.
The ``harmonic" atom model is useful because:

  • K |nñ = en |nñ can be solved exactly !
  • r2a can be approximated using techniques developed by quantum chemists.

Some questions to be answered as we study Ea = ån en Wan for increasing numbers of identical electrons, N, in our model system:

  • Is there an approximate shell structure for our ``harmonic" atoms as there is for the periodic table of real atoms?
  • Are there only a small number of pair states participating in the states of the ``harmonic" atoms? That is, is it true that Wan » 0 for n > N(N-1)/2 ?

File translated from TEX by TTH, version 2.20.
On 24 Nov 1999, 17:40.