Download Many-body Quantum Mechanics

Many-body Quantum Mechanics
These notes will provide a short introduction to the construction of manybody Hilbert spaces and the operators using the method which for historical
reasons is referred to as second quantization. They were originally written
as a comment on section 1.1 - 1.5 in the textbook Quantum Many-Particle
Systems (Addison-Wesley, 1988) by J. W. Negele and H. Orland, and I use
(almost) the same notation. Another concise and well written summary of this
subject can be found in Appendix 2A of A. J. Legget’s book Quantum Liquids Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Oxford
University Press, 2006.
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Basis for N Fermions or Bosons
First recall that a many body wave function Ψ(~r1 , ~r2 , . . . ~rN ) is related to the
corresponding state vector by,
Ψ(~r1 , ~r2 , . . . ~rN ) = h~r1 , ~r2 , . . . ~rN |Ψi
(1)
where h~r1 , ~r2 , . . . ~rN | is a normalized N-particle position eigenstate. For simplicity we shall consider the case without spin. The generalization to particles
with spin is straightforward, but note that in the text one uses the notation
x̂ ≡ |~rστ i where σ is a spin and τ some other internal index, e.g. isospin or
a layer or band index. The state vector |Ψi is an element in the N-particle
Hilbert space HN = H ⊗ H ⊗ H ⊗ . . . ⊗ H, where H is the one-particle Hilbert
space. If |αi is an ON basis of H , the states of HN are spanned by the tensor products |α1 , . . . αN ) = |α1 i ⊗ |α2 i ⊗ . . . ⊗ |αN i. Note that α denotes all
the quantum numbers characterizing the one-particle state, which can be e.g.
momentum, or energy and angular momentum.
Next recall that quantum statistics requires,
Ψ(~r1 , ~r2 , . . . ~r, ~r 0 . . . ~rN ) = ±Ψ(~r1 , ~r2 , . . . ~r 0 , ~r . . . ~rN )
(2)
where the plus and minus signs are for bosons and fermions respectively.1
The N-particle Hilbert spaces for bosons, BN , and fermions, FN are subspaces of the full Hilbert space HN , obtained by projecting on the subspaces
where the corresponding wave functions satisfy (2). The explicit construction BN and FN
1
In the discussion of the quanum Hall effect at the end of the course, we shall encounter
a more general kind of quantum statistics which can occur only in two dimensions. The
particles which obey such so called fractional statistics, are called anyons.
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of these spaces are given in the referred text, and here I just give the final
expressions for the ON basis spanning these spaces, (Equ. 1.46):
X
1
|α1 . . . αN i = √ Q
ζ P |αP1 i ⊗ |αP2 i ⊗ . . . ⊗ |αP2 i
N ! α nα ! P
(3)
where the summation is over permutations and ζ P is the sign of the permutation P . It is important that you understand the logic in arriving at (3).
In a Hamiltonian formalism, the projected Hilbert spaces, and the corresponding projected operators are the basic building blocks of the theory. In
a Lagrangian approach, the pertinent path integral is constructed by using
the basis (3) as intermediate states. In the latter approach it is important to
realize that the closure relation in the projected Hilbert spaces are written in
terms of the states (3).
2
The Fock space for Fermions and Bosons
In the previous section we considered systems with a fixed number of particles,
but in many processes the particle number does change. Examples are electron
hole annihilations in metals or semiconductors, electron-phonon processes, and
photon absorbtion or emission. Also, in order to formulate statistical mechanics in terms of a grand canonical ensemble, we must be able to treat states
with different number of particles.
The bosonic and fermionic Fock spaces are the direct sums of the cor∞
responding N particle Hilbert spaces, B = ⊕∞
n=0 BN and F = ⊕n=0 FN re- B and F
spectively. For the Fock space to be an interesting concept, there must be
operators connecting the different N-particle sectors - these are the creation
and annihilation operators. The former is defined by,
a†λ
√
(4)
a†λ |λ1 . . . λN i ≡ nλ + 1|λλ1 . . . λN i
where nλ is the number of particles in |λ1 . . . λN i which are in the singleparticle state λ. The interpretation of this formula is the following: We start
with a (properly symmetrized or antisymmetrized) N-particle state where the
first particle is in the state λ1 the second in state λ2 etc.. For bosons there can
be any number of particles in the same state, so the same label λi can occur
many times. For fermions each λi can occur only once. By acting with the
creation operator a†λ we create a new state with one extra particle in the state aλ
2
labeled by λ. Finally, the annihilation operator aλ is defined as the hermitian
conjugate of a†λ .
In the text you will learn how to construct the bosonic and fermionic Hilbert
spaces from the zero particle, or vacuum, state |0i by consecutive action with
the creation operators, and also how to derive the very important commutator
algebra,
[aλ , a†µ ]± = δλµ
;
[aλ , aµ ]± = 0 ,
(5)
where the minus (plus) sign refers to bosons (fermions), and [A, B]± = AB ±
BA.
There are two different ways to specify vectors in the Fock space. The
first, given above, amounts to making a list of all the particles, and specify the
corresponding one-particle states λi . In the other approach one concentrates
directly on the one-particle states, and specifying their occupation numbers,
i.e. how many particles there are in each one of them. The corresponding n states are given by,
(a† )nα1 (a†α2 )nα2
q
. . . |0i
|nα1 , nα2 , . . .i = qα1
nα1 !
nα2 !
(6)
An important property of these states is,
aλ |nβ1 , nβ1 , . . . nλ . . .i =
√
nλ |nβ1 , nβ1 , . . . (nλ − 1) . . .i .
(7)
This and many other properties of the n-states are derived in the text. Finally
we notice that the the creation and annihilation operators in the position basis
are referred to as quantum field operators, or simply quantum fields. They quantum fields
are related to the aα and a†α operators via a basis change, and conventionally
denoted by ψ̂ and ψ † ,
ψ̂(x) =
X
hx|αi aα
(8)
α
ψ̂ † (x) =
X
hα|xi a†α .
α
The quantum field operators satisfy the commutation relation,
[ψ̂(x), ψ̂ † (y)] = δ(x − y) .
3
(9)
3
Representation of operators
As explained in the text, all operators in the Fock spaces of bosons and
fermions, can be expressed in the corresponding annihilation and creation
operators. The simplest and most obvious example, is the density operator,
given by
ρ̂(~r) = ψ̂ † (~r)ψ̂(~r)
(10)
Make sure that you understand why this is the correct expression (use (8)!).
From the density operator we can form the potential part of the Hamiltonian,
Z
1Z 3 3 0
3
d rd r V (~r, ~r 0 )ρ̂(~r)[ρ̂(~r0 ) − δ 3 (~r − ~r 0 )] (11)
Û + V̂ = d r U (~r)ρ̂(~r) +
2
where U (~r) is an external potential, and V (~r, ~r 0 ) a two-particle interaction
term. The δ-function in the last term eliminates the self interaction as explained in the text.
It is a little bit more tricky to derive the expression for the kinetic energy.
This is done in the text, and the result is
h̄2 Z 3 †
(12)
d r ψ̂ (~r)∇2 ψ̂(~r) .
2m
These are all examples of so called one- and two-body operators, which are
discussed in general in the text.
T =−
Historical note: Taking as the Hamiltonian H = T + V , with the potential
containing only the U term, and using the commutation relations (9), the
Heisenberg equation of motion for the operator ψ̂ takes the form
d
h̄2 2
ih̄ ψ̂(~r) = −
∇ + U (~r) ψ̂(~r)
dt
2m
!
(13)
which is identical to the usual Schrödinger equation, but with the wave function replaced by a quantum operator. For this reason one sometimes refers to second
the Hamiltonian when expressed in quantum fields as ”second quantized”, and quantization
the method of using annihilation and creation operators acting on a Fock space
as ”second quantization”. As should be clear from the above, this terminology
is misleading in the sense that ψ̂ is not a once more quantized version of the
wave function, but an object which is directly (or via a Fourier transform)
related to particle annihilation. Including interaction terms, the equation for
the quantum field ψ̂ becomes non-linear, while the linearity of the Schrödinger
equation is necessary for the superposition principle and the probability interpretation of quantum mechanics.
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4
Coherent states
If there are physical processes which changes a particle number, one can consequently have quantum mechanical superpositions of stats with different number of particles. A particularly important class of such states are the so called
coherent states which can be defined as the eigenstates of the bosonic annihilation operator,2
aα |φi = φα |φi ,
(14)
where φα is a complex number. (Make sure you understand why there cannot
be any eigenstates of the creation operator!) By expanding |φi in the n-states,
(6), one can derive the following explicit formula,
P
|φi = e
α
φα a†α
|0i
(15)
In the text you will learn about the properties of these states, and here I just
stress two very important relations. Using the commutation relations (5) one
can calculate the overlap between two coherent states
P
hφ|φ0 i = e
α
φ?α φ0α
,
and derive the following resolution of unity,
Z Y ?
φα φα − P φ?α φ0α
α
e
|φihφ| .
1=
α 2πi
(16)
(17)
The second of these formulas show that the coherent states form a basis in
the Fock space, and the first that this basis in not orthogonal and thus overcomplete. In spite of this the coherent states are very useful in particular for
deriving path integrals. In this course they will be important as an example
of states where the creation and annihilation operators have non-vanishing
expectation values. This will be important in the discussion of Bose-Einstein
condensates.
So far we only considered bosons. As mentioned in the text, there is formally very similar construction for fermions. Since fermions anticommute,
the corresponding eigenvalues of the fermionic annihilation operator must also
anticommute (show this!). Such numbers are known to mathematicians as
Grassmann numbers. In this course we shall not use fermionic coherent state
and will thus not need Grassmann numbers. We shall return to this subject
in the discussion of fermonic path integrals later in this course.
2
There is a more general group theoretic definition of coherent states which is very
important in the discussion of e.g. spin systems. This very interesting topic, which is
beyond the scope of this course is discussed in detail in the book A, Perelomov, Generalized
Coherent States and Their Applications Springer Verlag, 1986.
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