The second quantization In this section we discuss why working with wavefunctions is not a good idea in systems with N ∼ 1023 particles, and introduce much more suitable notation for this purpose. 1 Quantum mechanics for one particle - review of notation The dynamics of the particle is described by some Hermitian Hamiltonian H^ , acting on the one- particle Hilbert space H. The Hilbert space is generally a product of Hilbert spaces associated with the translational and the spin degrees of freedom, H = Htr ⊗ Hspin. However, sometimes we will \forget" about the spin degrees of freedom, and be concerned only with the location in space of the so-called \spinless" particle. In this case H^ contains no spin operators. Other times we will \forget" the translational degrees of freedom and concern ourselves only with the dynamics of localized spins { this works, for instance, in insulators with non-trivial magnetic properties. Examples of one-particle Hamiltonians: 2 ^ ^~p H = 2m ! free (possibly spinless) particle; 2 ^ ^~p ^ H = 2m + V (~r; t) ! particle in an external field. h i2 ^ 1 ^ ~ ^ ^ ^~ ~ ^ H = 2m ~p − qA(~r; t) + qΦ(~r; t) − gµBS · B(~r; t) ! spinful particle of charge q in an ~ ~ ~ @A external classical electromagnetic field, B = r × A; E = −∇Φ − @t . ^ H^ = −γS~ · B~ (t) ! localized particle of spin S in external magnetic field B~ (t). The list goes on and on. Note: all quantities with a hat are operators acting on the Hilbert space (sometimes trivially). We can always work in an abstract Hilbert space, with abstract states denoted by j i. Some- times, however, it is convenient to use particular representations, such as the ~r-representation. In this case, we use the complete, orthonormal basis set for the Hilbert space: j~r; Szi = j~ri ⊗ jS; Szi 2 ^ ^ 2 where ~rj~ri = ~rj~ri are eigenstates of the position operator, while S~ jS; Szi = ¯h S(S + 1)jS; Szi, ^2 S^zjS; Szi = ¯hSzjS; Szi are eigenstates of the spin operators S~ ; S^z. It is convenient to introduce the simplified notation x = (~r; Sz) such that jxi = j~r; Szi. In this case, the orthonormation and the completeness relations are written as: 0 0 0 0 hxjx i = h~rj~r ihS; S jS; S i = δ(~r − ~r )δ 0 = δ 0 (1) z z Sz ;Sz x;x and S X Z X 1 = d~rj~r; Szih~r; Szj = dxjxihxj (2) Sz =−S The last notation should hopefully remind us to sum over all discrete variables, and integrate over all continuous ones. In this basis, any state can be decomposed as: S X X Z j i = dxjxihxj i = d~r (~r; Sz)j~r; Szi Sz =−S 1 where (x) = (~r; Sz) = h~r; Szj i, also known as the wavefunction, is the amplitude of probability that the particle in state j i is at position ~r with spin Sz. This amplitude is generally a complex number. Note: if the particle has a spin S > 0, then h~rj i is a spinor with 2S + 1 entries, 0 (~r; S) 1 B (~r; S − 1) C h~rj i = (~r) = B C @ ::: A (~r; −S) Then, for instance, the probability to find the particle at ~r is y(~r) (~r) = P j (~r; S )j2, as Sz z expected. Clearly, if we know the wavefunctions (x) 8x, we know the state of the system. Any equation for j i can be turned into an equation for (x) by simply acting on it with hxj, for example: j 1i = j 2i ! 1(x) = 2(x). However, we also have to deal with expression of the form hxjA^j i, where A^ is some abstract operator. In fact, the only operators that appear in the Hamiltonian are (combinations of): ^ hxj~rj i = ~r (~r; Sz) ^ hxj~pj i = −i¯hr (~r; Sz) p hxjS^zj i = ¯hSz (~r; Sz); hxjS^±j i =h ¯ (S ∓ Sz)(S ± Sz + 1) (~r; Sz ± 1) P 0 0 0 where S^± = S^x ± iS^y. In general, hxjA^j i = dx hxjA^jx i (x ). As long as we can compute the matrix elements of the operator in the basis of interest, we have its representation in that basis. Again, note that if we project on h~rj only, these are equations for spinors. In this case, the last equation changes to a matrix equation 0 ¯hS 0 ::: 0 1 0 (~r; S) 1 ^ ~ B 0h ¯(S − 1) ::: 0 C B (~r; S − 1) C h~rjSzj i = Sz (~r) = B C B C @ ::::::::: 0 A @ ::: A 0 0 0 −¯hS (~r; −S) i.e. each spin-component gets multiplied by its particular spin projection. One can also find the matrix representations for the operators S^x; S^y (exercise - do it!). 1 Example: consider a spin- 2 particle in an external magnetic field, described by the abstract Hamiltonian 2 ^~p ^ H^ = − γS~ · B~ (t) 2m d ^ In the ~r-representation, the Schr¨odingerequation i¯h dt j (t)i = H(t)j (t)i can be written in a compact form, as a spinor equation: d (~r) ¯h2 ^ i¯h = − r2 − γS~ · B~ (t) (~r) dt 2m where (~r; ") (~r) = (~r; #) 1 ^ h¯ and the spin- 2 matrices are Si = 2 σi, where the matrices σ are called the Pauli matrices: 0 1 0 −i 1 0 σ = ; σ = ; σ = x 1 0 y i 0 z 0 −1 2 You should convince yourselves that the two equations for the wavefunctions (~r; Sz) that we obtain by projecting the abstract equation onto h~r; Szj are equivalent to this one spinor equation. All this may seem rather trivial and somewhat of a waste of time. However, it useful to remember that using wavefunctions (x) is a choice and should be done only when convenient. When not convenient, we may use another representation (for instance of momentum states, i.e. ~ projecting on jk; Szi basis, or coherent states, or any other number of options). Or, we may decide to work directly in the abstract space. 2 Why we need 2nd quantization for systems with N 1 particles As we have seen, we can formulate the quantum mechanics for a single particle in an abstract Hilbert space, although in practice we (almost) always end up working in some given representation, in terms of wavefunctions. Can we generalize this approach to many-body systems? Yes, we can quite easily, if the number N of particles is fixed. In this case, the many-body state in the ~r- representation is (we use again x = (~r; σ)): hx1; : : : ; xN jΨ(t)i = Ψ(x1; : : : ; xN ; t) which is the amplitude of probability that at time t, particle 1 is at x1 = (~r1; σ1), etc. Note that if the number of particles is not fixed (as is the case in a grand-canonical ensemble) we're in trouble deciding what hj to use. But let us ignore this for the moment, and see how we get in trouble even if the number of particles is fixed. We almost always decompose wavefunctions in a given basis. For a single particle, we know that any wavefunction Ψ(x; t) can be decomposed in terms of a complete and orthonormal basis P φα(x) = hxjαi as Ψ(x; t) = α cα(t)φα(x) { this reduces the problem to that of working with the time-dependent complex numbers cα(t). If we have a complete basis for a single-particle Hilbert space, we can immediately generate a complete basis for the N-particle Hilbert space (the particles are identical), as the products of one-particle basis states: X Ψ(x1; : : : ; xN ; t) = cα1,...,αN (t)φα1 (x1)φα2 (x2) ··· φαN (xN ) α1,...,αN As usual, cα1,...,αN (t) is the amplitude of probability to find particle 1 in state α1 and located at x1, etc. However, because the particles are identical we know that the wavefunctions must be symmetric (for bosons) or antisymmetric (for fermions) to interchange of any particles: P Ψ(x1; : : : ; xN ; t) = ξ Ψ(xP1 ; : : : ; xPN ; t) where 1 2 :::N P1 P2 :::PN is any permutation, P is its sign (number of transpositions), and from now on we use the notation: −1; for fermions ξ = (3) +1; for bosons 3 It follows immediately that c...,αi,...,αj ;::: = ξc...,αj ,...,αi;::: and so c...,αi,...,αi;::: = 0 for fermions: we cannot have two or more fermions occupying identical states αi { the Pauli principle is automatically enforced through this symmetry. Because of this requirement, the physically meaningful many-body fermionic (bosonic) wave- functions are from the antisymmetric (symmetric) sector of the N-particle Hilbert space, and we only need to keep the properly symmetrized basis states, which we denote as: s 1 X P φα1,...,αN (x1; : : : ; xN ) = Q ξ φα1 (xP1 )φα2 (xP2 ) ··· φαN (xPN ) (4) N! i ni! P 2SN The factor in front is the normalization constant; ni is the total number of particles in the same state αi (only important for bosonic systems; in fermionic systems all ni = 1), and the summation is over all possible permutations. You should check that indeed, these functions are properly normalized. For fermions, such a properly antisymmetrized product of one-particle states is called a Slater determinant. The set of all these properly symmetrized/antisymmetrized basis states φα1,...,αN (x1; : : : ; xN ) can be shown to form a complete basis for the N-particle Hilbert space, when the particles are bosons/fermions. Then, any N-body wavefunction can be written as: X Ψ(x1; : : : ; xN ; t) = cα1,...,αN (t)φα1,...,αN (x1; : : : ; xN ) α1,...,αN This means that even if we are extremely lucky, and only a single combination of one-particle states α1; : : : ; αN is occupied, so that the sum contains a single φα1,...,αN (x1; : : : ; xN ) basis wavefunction, this alone contains on the order of N! terms (the sum over all permutations in Eq.