Download 1 Lecture 3: The many body problem 2 The Schrödinger equation

1
Lecture 3: The many body problem
In this lecture the many body problem is introduced in the context of first and second
quantisation.
2
The Schrödinger equation
We consider N particles; in many cases of interest in this course the hamiltonian for
the particles takes the form
N
X
N
1 X
H=
T (xk ) +
V (xk , xl ),
2
k=1
k6=l=1
(1)
where xk is the position of the kth particle, T is the kinetic energy, and V is the
potential energy of interaction, counted once, between the particles. (It is often
convenient to assume that the position variable xk includes not only the particle’s
coordinates in R3 , but also its internal configurations such as spin, etc.) The timeindependent Schrödinger equation then reads
i~
∂
Ψ(x1 , x2 , . . . , xN , t) = HΨ(x1 , x2 , . . . , xN , t),
∂t
(2)
together with some appropriate choice of boundary conditions for the wavefunction.
Suppose that ψj (x), j = 1, 2, . . ., is a complete set of orthonormalised single-particle
wavefunctions. For example, ψjk (xk ) could be the eigenfunctions of the harmonic
oscillator. Then any product ψj1 (x1 )ψj2 (x2 ) · · · ψjN (xN ) is a valid many-body wavefunction. Further, these products are complete, in that any many-body wavefunction
can be expressed as a linear combination of them:
X
Ψ(x1 , x2 , . . . , xN , t) =
C(j1 , j2 , . . . , jN , t)ψj1 (x1 )ψj2 (x2 ) · · · ψjN (xN ). (3)
j1 ,j2 ,...,jN
(This is a consequence of the tensor product rule. That this should be true follows from
general information-theoretic considerations emerging from the assumption that it is
possible to perform tomography of the entire wavefunction by separately measuring
the particles.)
Substituting (3) into the Schrödinger equation (2) yields
N
XX
∂
i~ C(j1 , j2 , . . . , jN , t) =
∂t
k=1 l
Z
dxk ψ jk (xk )T (xk )ψl (xk )C(j1 , . . . , l, . . . jN , t) (4)
Z
N
1 X XX
+
dxk dxk0 ψ jk (xk )ψ jk0 (xk0 )V (xk , xk0 )ψl (xk )ψm (xk0 )×
2 k6=k0 l m
(5)
C(j1 , . . . , l, . . . , m, . . . jN , t).
(6)
1
We now accommodate the indistinguishability of the particles into the wavefunction
by demanding that an exchange of particles cannot lead to an observable consequence.
This means that
Ψ(. . . , xj , . . . , xk , . . .) = eiφjk Ψ(. . . , xk , . . . , xj , . . .),
(7)
i.e., the wavefunction must be the same up to a phase. Since a further exchange
of particles j and k yields the original wavefunction the phase eiφjk = ±1 in order
to ensure that Ψ is not multiple valued. (In geometries with nontrivial topologies,
or in two dimensions the wavefunction may possibly be multiple valued, yielding the
possibility of anyons, and other exotic particles.) A necessary and sufficient condition
to ensure the (anti-)symmetry of the wavefunction is that the coefficients in (3) satisfy
C(. . . , jk , . . . , jl , . . .) = ±C(. . . , jl , . . . , jk , . . .)
(8)
Exercise: prove this.
2.1
Bosons
The particles described by a wavefunction Ψ that is completely symmetric under
interchange are known as bosons. The symmetry of the coefficients allows us to
reorder the indices of the coefficient C in the summation (3). Suppose the state 1
occurs n1 times, and the state 2 occurs n2 times, etc. Then all such terms with
the same numbers nk have the same coefficient. It is thus convenient to rename the
coefficient function
· · · 2} · · · , t)
C(n1 , n2 , . . . , nk , . . . , t) ≡ C(11
· · · 1} 22
| {z
| {z
n1
(9)
n2
We now define another coefficient function
r
f (n1 , n2 , . . . , nk , . . . , t) ≡
N!
C(n1 , n2 , . . . , nk , . . . , t).
n1 !n2 ! · · ·
The condition that the wavefunction Ψ is normalised becomes
X
|f (n1 , n2 , . . . , nk , . . . , t)|2 = 1,
(10)
(11)
n1 ,n2 ,...
where the coefficients nj must satisfy
∞
X
nj = N.
(12)
j=1
Exercise: prove this.
We can now use the f coefficients to rewrite the original wavefunction in terms of a
new convenient complete orthonormal basis
X
Ψ(x1 , x2 , . . . , xN , t) =
f (n1 , n2 , . . . , t)Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ), (13)
n1P
,n2 ,...,nk ,...
k nk =N
2
where
r
Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ) =
n1 !n2 ! · · · X
ψj1 (x1 )ψj2 (x2 ) · · · ψjN (xN ), (14)
N!
j ,j ,...,j
1
2
N
(n1 ,n2 ,...,)
and the summation is over all indices jk with the given pattern (n1 , n2 , . . . , ) of 1s, 2s,
etc. Note that the functions Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ) are completely symmetrised.
Exercise: prove that the functions Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ) form a symmetrised
complete orthonormal set.
Here is an example of a Φn1 n2 ···nk ··· function:
1
Φ210···0··· (x1 x2 x3 ) = √ (ψ1 (x1 )ψ1 (x2 )ψ2 (x3 )+ψ1 (x1 )ψ2 (x2 )ψ1 (x3 )+ψ2 (x1 )ψ1 (x2 )ψ1 (x3 )).
3
(15)
Substituting the expansion (11) into the Schrödinger equation (2) leads to
i~
X
∂
f (n1 , n2 , . . . , nk , . . . , t) =
hj|T |jinj f (n1 , . . . , nk , . . . nk , . . . , t)+
∂t
j
(16)
q
hj|T |j i nj (nj 0 + 1)f (n1 , . . . , nj − 1, . . . , nj 0 + 1, . . . nk , . . . , t)+
(17)
X
0
j6=j 0
q
1 X
0
0
hjj |V |kk i nj nj 0 (nk + 1)(nk0 + 1)×
2 j6=j 0 6=k6=k0
(18)
f (n1 , . . . , nj − 1, . . . , nj 0 − 1, . . . , nk + 1, . . . , nk0 + 1, . . . , t) + etc.,
(19)
where there is an additional term for all possible sets of occupation numbers
so that
√
0
0
two particles are removed, multiplied by an overall factor hjj |V |kk i× productof ns,
and then subsequently added back in at different places. This expression appears very
complicated, however, there is a way to write it in an equivalent form which is much
more compact.
2.2
Fermions
If a minus sign is chosen in (5) then the Cs are antisymmetric under the exchange of
any two particles:
C(. . . , jk , . . . , jl , . . .) = −C(. . . , jl , . . . , jk , . . .),
(20)
which shows that jl must be different from jk or else the coefficient vanishes. This,
in turn, shows that the occupation numbers nl can only be 0 or 1. Any coefficients
with the same states occupied are equal up to a minus sign, allowing us to define a
new coefficient C
C(n1 , n2 , . . . , t) = C(· · · jk < jl < jm · · · , t),
where all of the numbers j1 , j2 , . . . are first arranged in increasing order.
3
(21)
Exactly as in the bosonic case the many particle wavefunction Ψ can be expanded as
X
Ψ(x1 , x2 , . . . , xN , t) =
f (n1 , n2 , . . . , t)Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ),
(22)
n1P
,n2 ,...,nk ,...
k nk =N
where now
r
Φn1 n2 ···nk ··· (x1 , x2 , . . . , xN ) =
ψj (x1 ) ψj (x2 ) · · · ψj (xN ) 1
1
1
n1 !n2 ! · · · ..
..
.
.
, (23)
.
.
.
N!
ψjN (x1 ) ψjN (x2 ) · · · ψjN (xN )
and j1 < j2 < · · · < jN . These Slater determinants form a complete set of orthonormal antisymmetric wavefunctions.
We postpone the task of writing out the Schrödinger equation until next section where
we’ll develop a much more compact representation.
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Many particle hilbert space; creation and annihilation operators
In this section we introduce a new orthonormal basis for hilbert space describing
the number of particles in each state. This must be initially understood as an abstract construction until such time we can show it is equivalent to the first-quantised
treatments of the previous section. The basis we introduce is denoted
|n1 n2 · · · nk · · · i,
nj ∈ Z+ ,
(24)
which is meant to mean that nj particles are in the single-particle eigenstate ψj . This
basis is demanded to be complete and orthonormal, meaning that
hn01 n02 · · · n0k · · · |n1 n2 · · · nk · · · i = δn01 n1 δn02 n2 · · ·
(25)
and
X
|n1 n2 · · · nk · · · ihn1 n2 · · · nk · · · | = I.
(26)
n1 n2 ···
3.1
Bosons
In the bosonic case, associated with this occupation number basis, we introduce
the annihilation and creation operators bk , b†k , satisfying the canonical commutation
relations (CCR)
[bj , b†k ] = δjk , [bj , bk ] = [b†j , b†k ] = 0.
(27)
These operators are taken to act in the standard way on the number basis, e.g.,
p
(28)
b†k |n1 n2 · · · nk · · · i = nj + 1|n1 n2 · · · nk + 1 · · · i,
4
etc. The mode number operators nk are defined to be
nk = b†k bk .
(29)
We now use the occupation number basis to rewrite the Schrödinger equation. Form
the following state
X
|Ψ(t)i =
f (n1 , n2 , . . . , nk , . . . , t)|n1 n2 · · · nk · · · i,
(30)
n1 n2 ···
where the f s are taken to be the expansion coefficients of (11) which satisfy the
coupled differential equations (14). This state vector in our abstract hilbert space
obeys the differential equation
i~
X
∂
∂
|Ψ(t)i =
i~ f (n1 , n2 , . . . , nk , . . . , t)|n1 n2 · · · nk · · · i
∂t
∂t
n n ···
(31)
1 2
Relabelling dummy indices, using the properties of CCR, and rewriting the appropriate operations in terms of the annihiliation and creation operators leads us to the
consequence that the abstract state vector |Ψ(t)i satisfies the Schrödinger equation
i~
∂
b
|Ψ(t)i = H|Ψ(t)i,
∂t
(32)
b is given by
where the operator H
b=
H
X
b†j hj|T |kibk +
j,k
1 X † †
b b hjk|V |lmibl bm .
2 j,k,l,m j k
(33)
These equations restate the Schrödinger equation in second quantisation. All the
statistics and operators properties are expressed via the CCR. The physical problem
is unchanged by the new formulation, and the coefficients express the connection
between first and second quantisations. Thus any solution to the Schrödinger equation
in first quantisation yields a solution in second quantisation and vice versa.
3.2
Fermions
In the fermionic case we introduce the annihilation and creation operators ak , a†k ,
satisfying the canonical anticommutation relations (CAR)
{aj , a†k } = δjk ,
{aj , ak } = {a†j , a†k } = 0.
(34)
The action of these operators on the vacuum state |0i is expressed by
|n1 n2 · · · nk · · · i = (a†1 )n1 (a†2 )n2 · · · (a†k )nk · · · |0i.
If we now introduce, as before, the abstract state vector |Ψ(t)i
X
|Ψ(t)i =
f (n1 , n2 , . . . , nk , . . . , t)|n1 n2 · · · nk · · · i,
n1 n2 ···
5
(35)
(36)
where the f s are taken to be the expansion coefficients of (17) obeying the coupled
differential equations coming from substitution into the Schrödinger equation, then
we find that
∂
b
i~ |Ψ(t)i = H|Ψ(t)i,
(37)
∂t
b is given by
where the operator H
b=
H
X
j,k
a†j hj|T |kiak +
1 X † †
a a hjk|V |lmial am .
2 j,k,l,m j k
6
(38)