operator of the total energy () {}
TRANSCRIPT
PC 3902 – Elements of Quantum Many-Body Physics, Lectures 3 & 4
1
Operator H of the total energy Many-body Hamiltonian H :
H = T ±V R( ) = !!2
2m"
i
2
i=1
N
# + V rij( )
i=1
N
# (3)
H! R( ) = T +V R( ){ }! R( ) = T! R( ) +V R( )! R( )
= "!2
2m#
i
2! R( )i=1
N
$ + V rij( )! R( )
i=1
N
$
Stationary (time-independent) many-body Schrödinger equation. Eigenvalue equation for the energy eigenvalues E
n and eigenfunctions
!nR( ) = !
nr1,r2,...,r
N( ) of the many-body Hamiltonian H (eq. 3)
H!nR( ) = E
n!nR( ) (4)
Index n simply numerates the distinct energy eigenfunctions !nR( ) . Absolute ground
state of the many-body system at temperature T = 0 is denoted by n = 0 . E0 is the
ground-state energy, and !0R( ) is the ground state wavefunction.
E0< E
n for al other excited states. Ground state is energy eigenstate with the lowest
energy eigenvalues.
k n = !k* R( )!
nR( )dR
VN"
= ... !k* r
1,r2,...,r
N( )!n r1,r2,...,r
N( )dr1dr2 ...drNV"
V"
V"
There are N 3-dimensional integrals, integrating over r1, r
2, …, r
N.
k n = ... !k * R( )!n R( )dx1dy
1dz
1dx
2dy
2dz
2...dxNdyNdzN
V"V"V"
The above is the scalar product of !kR( ) and !
nR( ) . 3N -dimensional volume
integral or N three-dimensional volume integrations, where ri is integrated over the
three-dimensional volume V occupied by the many-particle system. k H n = !
k* R( ) H!
nR( )"# $%dRV
N&= H!
kR( )"# $% *!n R( )dR
VN&
(6)
As Hamiltonian is self-adjoint (Hermitian , symmetric). The asterisk denotes complex conjugation.
! R( ) = Re ! R( ){ } + i Im ! R( ){ }
! R( )* = Re ! R( ){ } " i Im ! R( ){ }
Multiply (4) with !k* R( ) and integrate over R :
!k* R( ) H!
nR( ) = E
n!k* R( )!
nR( )
!k* R( ) H!
nR( )dR
VN" = E
n!k* R( )!
nR( )dR
VN"
Using equations (5) and (6), k H n = E
nk n
PC 3902 – Elements of Quantum Many-Body Physics, Lectures 3 & 4
2
H!kR( ) = E
n!kR( )
H!k* R( ) = E
n!k* R( )
So H , equation (3), is real. All eigenvalues E
n of H are real, since H is self-adjoint.
En* = E
n
H!k* R( )"# $%!n R( )dR
VN& = H!
kR( )"# $% *!n R( )dR
VN& = k H n
= Ek
!k* R( )!
nR( )dR
VN& = E
kk n
!Ekk n = k H n
! En" E
k( ) k n = 0 k n = 0 for E
n! E
k
Eigenfunctions of different eigenvalues are orthogonal. In general, degeneracy (different eigenfunctions have same energy eigenvalues). Eigenfunctions can always be made orthogonal to each other by means of the Gram-Schmidt orthogonalization method. From now on always assume that eigenfunctions !nR( ) of H form a complete normalized orthogonal system of basis functions of the
Hilbert space. Orthogonality: k n = 0 for k ! n
Normalization: n n = !nR( )
2
dRVN" = 1
If !!nR( ) is not normalized, then !
nR( ) :
!nR( ) =
!!nR( )
!!n!!n
!!n!!n
= !!nR( ) dR
VN"
Orthonormalised: k n = !kn
(7)
Kronecker delta !kn=0 k " n
1 k = n
#$%
Completeness: any wave function ! R( ) may be expanded in terms of !
n R( ) .
! R( ) = cn"nR( )
n
#
Multiply with !k* R( ) and integrate,
!k* R( )" R( )dR
VN# = k " = c
n!k* R( )!
nR( )dR
VN#
n
$
= cnk n
n
$ = cn%kn
n
$
ck= k ! " c
n= n !
PC 3902 – Elements of Quantum Many-Body Physics, Lectures 3 & 4
3
! R( ) = n ! "nR( )
n
#
n ! = "n* R( )! R( )dR
VN# (8)
1.2 The Many-Body Density Operator
Temperature T > 0 , ! =1
kBT
W =1
Ze!" H (9)
This is the density operator. Z is the partition function, and is also known as the sum of states.
e!" H#
nR( ) = e!"En#
nR( ) (10)
where H!nR( ) = E
n!nR( ) . The eigenfunctions of e!" H are the eigenfunctions !
nR( )
of H and its eigenvalues are e!"En . e!" H# R( ) = e!" H n # $
nR( )
n
%
= n # e!" H$
nR( )
n
%
= n # e!"E
n$nR( )
n
%
e!" H# R( ) = e
!"En n # $
nR( )
n
% (11)
Z = Tr e!" H{ }
where Tr{ } denotes the trace, i.e. the sum of eigenvalues (energies).
Z = e!"E
n
n
# (12)
W!nR( ) =
e"#E
n
Z!nR( ) (13)
Eigenvalues of the canonical density operator W are given by e!"E
n
Z.
Tr W( ) =e!"E
n
Zn
# =1
Ze!"E
n
n
# = 1
This gives unit-normalization of the density operator. Macroscopic thermodynamics (statistical physics) may be obtained from Z , e.g.
F = E ! TS = !kBT ln Z( )
where F is the Helmholtz free energy, E = Ekin + Epot is the total energy and S is the entropy.