(1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) 2 is the probability to find the first particle

Transcription

1 Chapter 1 Identical particles 1.1 Distinguishable particles The Hilbert space of N has to be a subspace H = N n=1h n. Observables Ân of the n-th particle are self-adjoint operators of the form n 1 Â 1 N. We call the N particles distinguishable if e.g. the position and the spin of each particle can be determined separately. In this case the set of pure states is given by the set of normalized wave functions ψ( q 1, m 1 ; ; q N, m N ), modulo a phase with inner produvt < ψ φ >= m 1 =j 1 m 1 = j 1 m N =j N m N = j N d 3N qψ ( q 1, m 1 ; ; q N, m N )φ( q 1, m 1 ; ; q N, m N ) (1.1) In particular, ψ( q 1, m 1 ; ; q N, m N ) is the probability to find the first particle at q 1 with spin z = m 1, the second at q with spin z = m etc. The time evolution of the many particle wave function is then governed by a Hamiltonian of the form { N ( 1 H M n=1 n i q n e ) n A( q c n, t) + e n Φ( q n, t) + µ nsn B n ( q n, t)} + V nm ( q n q m ) (1.) 1 n<m N Let us now consider permutations. These are 1 1 maps Π : (1,, N) (Π(1),, Π(N)) (1.3) 1

2 CHAPTER 1. IDENTICAL PARTICLES If all N particles have the same spin and the same domain for their wave function then there is a unitary representation of the group γ(n) of permutations of N objects Û(Π) : ψ( q 1, m 1 ; ; q N, m N ) ψ( q Π(1), m Π(1) ; ; q Π(N), m Π(N) ) (1.4) 1. Indistinguishable particles The N particles are indistinguishable if all observables of the system are invariant under all permutations, for example the total angular momentum Ĵ = N Ĵ N and the Hamiltonian, provided M n = M, e n = e, µ n = µ for all n n=1 and V nm ( q n q m ) = V ( q n q m ). Subspaces of H with simple transformation behaviour are those for which the wave functions are totally symmetric: H S ψ Û(Π) ψ, Π γ(n) (1.5) or totally anti-symmetric: H A ψ Û(Π) σ(π)ψ, Π γ(n) (1.6) where σ(π) = 1 if Π is a composition of an even number of transpositions and σ(π) = 1 if Π is a composition of an odd number of transpositions. The projectors on H S and H A are given by Ŝ = Ŝ = Ŝ = 1 N! Π γ(n) Û(Π) (1.7) and  =  =  = 1 N! Π γ(n) σ(π)û(π) (1.8) In particular, ÂŜ = Ŝ = 0. The orthogonal complement Hc := (1 Ŝ Â)H has a complicated transformation behaviour under γ(n). Luckily it can be ignored due to the Pauli Exclusion Principle: The physically realizable quantum mechanical state vectors for identical particles are totally symmetric for particles with integer spin, bosons and totally anti-symmetric for particles with half- integer spin, fermions.

3 1.3. YOUNG TABLEAUX 3 Since the hamiltonian for identical particles is invariant under permutations, H A and H S are invariant under time-evolution. The Helium atom (see QMI or my german notes) is the simplest an nicest example where the Pauli exclusion principle explains the spectrum of bound states of many electrons. Taking the basis ψ( q 1, m 1 ; q, m ) = ij χ ij ( q 1, q )φ ij (m 1, m ) (1.9) for the -electron bound states where φ 00 denote the spin singlet- and φ 11, φ 10, φ 1 1 the spin triplet respectively. We then have H = H 0 H 1 where the symmetric orbital wave functions are in H 0 while the anti-symmetric orbital wave functions are in H 1. In particular the symmetry type of the orbital wave functions is opposite (or dual) to that of the spin states. This duality will extend to higher N as well. The spin-independent Hamiltonian preserves H 0 and H 1 and can thus be diagonalized in each subspace separately. This leads to two families of eigenfunctions, denoted by para-helium in H 0 and ortho- Helium in H 1. If we neglect the Coulomb interaction between the electrons the ground state energy of ortho- Helium is then degenerate with the first excited state in para-helium. Time-independent perturbation theory leads to the correct qualitative picture for the correction of the ground state energies. However, better agreement with experiment is obtained though the Rayleigh-Ritz Variational method (see QMI or my german notes) with variational Ansatz χ( q 1, q ) := χ Z ( q 1 )χ Z ( q ) (1.10) where χ Z ( q i ) is the single electron wave function for a continuous atomic number, Z. 1.3 Young tableaux In the process of diagonalizing the hamiltonian of systems with many identical particles it will be important to find subspaces of the hilbert space which carry at the same time irreducible representations of the quantum mechanical rotation group, SU() as well as the permutation group, γ(n). It turns out that representations which are simultaneously irreducible with respect to both groups come about rather naturally as we shall now see. In fact

4 4 CHAPTER 1. IDENTICAL PARTICLES we have encountered an example of this already when discussing product representations in section 3.: the product of two spin 1 representations is a direct sum of two irreducible representations, the spin 0 which is totally anti-symmetric and the spin 1 representation which is totally symmetric. A convenient short hand notation representing this fact is in terms of the so-called Young-tableaux. We represent a irreducible fundamental representation D 1 of SU() by a box. More precisely + and - stand for spin z = and spin z = respectively. A product representation D 1 D 1 then corresponds to and (1.11) To continue we will assume that for boxes which are horizontally aligned symmetrization of its entries is assumed whereas for vertically aligned antisymmetrization of its entries is assumed. We will call general young tableaux with the above symmetry properties irreducible symmetrizers, e λ, for reasons that will become clear shortly. In particular, for the above example { } e 1 := = + +, + -, - - (1.1) is a triplet whereas e := = + - (1.13) is a singlet. The index λ (λ = 1,, in this example) specifies the symmetry type. It turns out that the compatibility between irreducible symmetrizers and irreducible representations of the rotation group is valid for higher tensor product representations as well. If we consider as an example the case of 3- identical spin 1 particles we have the following young tableaux The totally ant-symmetric representation (1.14) (1.15)

5 1.3. YOUNG TABLEAUX 5 of γ(3) does not appear for spin 1 particles. Filling in the numbers we have (1.16) representing the quartet with spin 3 of SU() where Ĵ± act horizontally. In addition we have (1.17) representing a doublet of SU(). Since D 1 D 1 D 1 = D 3 D 1 D 1 the complete decomposition of the tensor representation requires two doublets. These two doublets arise by identifying the boxes in (1.18) with the particles 1,, 3 and 1, 3, respectively. More generally we define the irreducible symmetrizers as follows: Take a Young tableau of a certain symmetry type, λ and fill it with the numbers 1,, N in such a way that the numbers strictly increase from the left to the right and strictly increase from top down, eg for N = 6, , , (1.19) for N = 3 we then have (1.0) Such diagrams are called standard young tableaux. We see that for N = 3 there are two young tableaux of the mixed symmetry type related to each other by the permutation Π = (, 3). In general there may be several standard young tableaux of the same symmetry type related to each other by a permutation Π of its entries. The irreducible symmetrizer corresponding to a given standard young tableau is defined by ê Π λ := v λ,h λ σ(v λ )v λ h λ (1.1)

6 6 CHAPTER 1. IDENTICAL PARTICLES where h λ is an arbitrary horizontal permutation of the entries of e λ and v λ is an arbitrary vertical permutation of the entries of e λ. For instance ê = (e (13)) (e + (1)) = e + (1) (13) (13) ê (3) = (e (1)) (e + (13)) = e + (13) (1) (31) (1.) with the convention 1 (13) : 3 (1.3) 3 1 For illustration ê (3) + ++ > = 0 (1.4) ê (3) ++ > = ++ > > + + > + + >, etc. We are now ready to formulate the general result. This is a consequence of the following Theorem (e.g. Wu Ki-Tung): (a) Let V be a vector space, V N = V V its N-fold tensor product and L{ λ = {Π e λ Π γ(n)} the left ideal generated by e λ then r α > r Lλ, α > V N, fixed } is an irreducible representation of γ(n). (b) If V m is an m-dimensional vector space carrying a representation of a subgroup of GL(m, C) and Vm N = V m V m is the N-fold tensor product, then [Û(Π), Û(g)] = 0 for all Π γ(n) and g GL(m, C). As a consequence of part (b) we have { } eλ α > α > V N (1.5) defines an irreducible representation of SU() (see e.g. Wu Ki-Tung for a proof). Furthermore V N can be decomposed into irreducible representations of γ(n) and SU() with standard basis { λ, α, a >} (1.6) where λ specifies the symmetry type, α labels different irreducible representations of γ(n) and a labels different irreducible representations of SU(). In particular, Û(g) λ, α, a >= β (U(g)) αβ λ, β, a > (1.7)

7 1.3. YOUNG TABLEAUX 7 Similarly Û(Π) λ, α, a >= b (U(Π)) ab λ, α, b > (1.8) The theorem stated applies analogously to higher rank groups such SU(m). For instance, for SU(3) (whose fundamental representation is 3- dimensional) the possible young tableaux corresponding to irreducible components of tensor representations are of the form (1.9) In order to determine the dimension of the representation of SU(m) corresponding to a given young tableau one proceeds as follows: Insert the number m in the upper-left-hand box, and increase by 1 as you go to the right, and decrease by one as you go down. For SU(3), then, the last diagram above would look like (1.30) If we denote by F the product of these numbers (here F = 360) then the dimension of the representation is given by D λ = F x e λ hook(x) (1.31) where the hook length, hook(x), of a box x in Young diagram e λ of symmetry type λ is the number of boxes that are in the same row to the right of it plus those boxes in the same column below it, plus one (for the box itself). In our example we get D λ = = 8 (1.3) There is a similar formula for he dimension of the irreducible representation of γ(n) with symmetry type λ dim(λ) = N! x e λ hook(x) (1.33)

8 8 CHAPTER 1. IDENTICAL PARTICLES These formulas are particularly useful in nuclear physics. Let us now consider N identical spin 1 particles. The possible young tableaux have at most two rows. So they are of the form s 1 s s n s 1 s s n s 1 s s n etc. (1.34) Note that the spin of the representation decreases by one as we move to the next young tableau to the right. From this we infer the important result that The symmetry type and the spin of the irreducible components of the N- fold tensor product representations of identical spin 1 particles determine each other uniquely. 1.4 N-electron atoms N-electron atoms with small to intermediate atomic number can be described, approximatively, by the Hamiltonian ( N ( ) ) 1 H M i q n Ze + e 1 (1.35) q n q n q m n=1 1 n<m N with Hilbert space ( H orbit H spin) A. Both, H orbit and H spin carry unitary representations Û orbit (Π) and Û spin (Π) of the permutation group γ(n) and Û(Π) = Û orbit (Π) Û spin (Π). The projector (8.6) clearly commutes with the angular momentum ˆ L and parity. Therefore H orbit can be decomposed into a direct sum of subspaces with definite angular momentum L, parity, P, and symmetry type, β. The spin-independent Hamiltonian H = β L P =± (h βlp 1 β 1 L 1 P ) (1.36) can be diagonalized separately in each subspace. If we furthermore ignore the Coulomb interaction between the electrons then h βlp is a generalization of the Hamiltonian for a central potential with Ansatz χ( q) = f l ( q )Y lm (θ, φ), P l = ( 1) l (1.37)

9 1.4. N-ELECTRON ATOMS 9 The minimal degeneracy of the energy eigenvalues in H B is dim(β) (l + 1) where dim(β) = N! x e β hook(x) (1.38) is the dimension of the irreducible representation of γ(n) with symmetry type β. The eigenvalues of h βlp form the spectroscopic term series, denoted by n β L P, n = 1,,. Including the spin we have mutually commuting unitary representations Û(Π), of γ(n) and Û(g), of SU(). We have seen that in HS the spin s determines the symmetry type, α(s). The minimal degeneracy is then dim(β)(l + 1)dim(α(s))(s + 1). Requiring furthermore antisymmetry of the total wave function the symmetry type β is also fixed by s since antisymmetry requires the young tableau ẽ α to be dual to e α. The duality consists of exchanging the rows and columns, eg. e α = = e β = (1.39) As a consequence, including the spin the minimal degeneracy is (s+1)(l+1) since the totally antisymmetric representations of γ(n) on ( H orbit H spin) are one-dimensional. We label the states as the spectroscopic term series s L P. As an illustration we consider the totally anti symmetric combination of the orbit- and spin wave functions of mixed symmetry ψ( q 1, m 1 ; ; q 3, m 3 ) = (χ e φẽ )( q 1, m 1 ; ; q 3, m 3 ) := χ e i ( q 1,, q 3 )φẽ i (m 1,, m 3 ) (1.40) i=1 where χ e i and φ e i are the basis vectors in of the e -representation in H orbit and H spin respectively. Further Reading A concise overview of the applications of young tableaux in quantum mechanics can be found in Sakurai s Modern Quantum Mechanics. A more detailed

10 10 CHAPTER 1. IDENTICAL PARTICLES description can be found in L.E. Ballantine, Quantum Mechanics: A Modern Development. For a more in depth discussion see eg. Lie Algebras in Particle Physics by Howard Georgi. The proofs of the theorems in this section and a detailed discussion of the relation between Young tableaux and permutations can be found in Wu-Ki Tungs s book.