Many-body Quantum Mechanics

Transcription

1 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 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, Basis for N Fermions or Bosons First recall that a many body wave function Ψ( r 1, r 2,... r N ) is related to the corresponding state vector by, Ψ( r 1, r 2,... r N ) = r 1, r 2,... r N Ψ (1) where r 1, r 2,... r N 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 ˆx rστ where σ is a spin and τ some other internal index, e.g. isospin or a layer or band index. The state vector Ψ is an element in the N-particle Hilbert space H N = H H H... H, where H is the one-particle Hilbert space. If α is an ON basis of H, the states of H N are spanned by the tensor products α 1,... α N ) = α 1 α 2... α N. 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( r 1, r 2,... r, r... r N ) = ±Ψ( r 1, r 2,... r, r... r N ) (2) where the plus and minus signs are for bosons and fermions respectively. 1 The N-particle Hilbert spaces for bosons, B N, and fermions, F N are subspaces of the full Hilbert space H N, obtained by projecting on the subspaces where the corresponding wave functions satisfy (2). The explicit construction B N and F N 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. 1

2 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): α 1... α N = 1 N! α n α! ζ P α P1 α P2... α P2 (3) P 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 corresponding N particle Hilbert spaces, B = n=0b N and F = n=0f N 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 λ 1... λ N n λ + 1 λλ 1... λ N (4) a λ where n λ is the number of particles in λ 1... λ N 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 λ 1 the second in state λ 2 etc.. For bosons there can be any number of particles in the same state, so the same label λ i can occur many times. For fermions each λ i can occur only once. By acting with the creation operator a λ we create a new state with one extra particle in the state a λ 2

3 labeled by λ. Finally, the annihilation operator a λ 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 0 by consecutive action with the creation operators, and also how to derive the very important commutator algebra, [a λ, a µ] ± = δ λµ ; [a λ, a µ ] ± = 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 λ i. 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, n α1, n α2,... = (a α 1 ) nα 1 nα1! (a α 2 ) nα 2 nα2!... 0 (6) An important property of these states is, a λ n β1, n β1,... n λ... = n λ n β1, n β1,... (n λ 1).... (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 a α and a α operators via a basis change, and conventionally denoted by ˆψ and ψ, ˆψ(x) = α ˆψ (x) = α x α a α (8) α x a α. The quantum field operators satisfy the commutation relation, [ ˆψ(x), ˆψ (y)] = δ(x y). (9) 3

4 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 ˆρ( r) = ˆψ ( r) ˆψ( 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, Û + ˆV = d 3 r U( r)ˆρ( r) + 1 d 3 rd 3 r V ( r, r )ˆρ( r)[ˆρ( r ) δ 3 ( r r )] (11) 2 where U( r) is an external potential, and V ( r, r ) 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 T = h2 d 3 r 2m ˆψ ( r) 2 ˆψ( r). (12) These are all examples of so called one- and two-body operators, which are discussed in general in the text. 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 ˆψ takes the form i h d ) dt ˆψ( r) = ( h2 2m 2 + U( r) ˆψ( r) (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 the Hamiltonian when expressed in quantum fields as second quantized, and 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 ˆψ 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 ˆψ becomes non-linear, while the linearity of the Schrödinger equation is necessary for the superposition principle and the probability interpretation of quantum mechanics. second quantization 4

5 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 a α φ = φ α φ, (14) where φ α is a complex number. (Make sure you understand why there cannot be any eigenstates of the creation operator!) By expanding φ in the n-states, (6), one can derive the following explicit formula, φ = e α φαa α 0 (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 φ φ = e α φ αφ α, (16) and derive the following resolution of unity, φ 1 = αφ α 2πi e α φ αφ α φ φ. (17) α 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,