Discrete Variable Representation

Transcription

1 Discrete Variable Representation Rocco Martinazzo Contents Denition and properties 2 Finite Basis Representations 3 3 Simple examples 4 Introduction Discrete Variable Representation (DVR) methods were introduced long ago [3, 2, 6, 5, 7,, 4], but only recently have been put on a rm theoretical basis [2, 0, 9, 8, ]. Here, we give a brief (and simplied) account of the recent work by Littlejohn and Cargo. Denition and properties Let H be the Hilbert space of a quantum mechanical system and P the projector onto a subspace S = P H in which we are interested. Let M be the conguration space of the system upon which the coordinate representation of H is based; for example, for a single particle M = R 3 and H x = L 2 (R 3 ). Finally, let {x α } be a set of grid points in M. Then, we say that the combination of the projector P and the grid {x α } forms a DVR set if the set of vectors is orthogonal α = P x α α β = N α δ αβ N α > 0 and complete in S. In this case, the complete, orthonormalized set of vectors F α = α is the DVR set of the space S on the grid {x α }. The above denition makes clear that each DVR state F α is, in some sense, localized around the grid point x α, thereby establishing a natural correspondence between points in conguration space and state vectors. The following,

2 schematic interpretation may be useful. Close to H there are two simple spaces, the Schwartz space of `rapid decreasing functions' and the temperated distribution space of continuous linear functionals on the Schwartz space. The rst space is dense in H; the latter is somewhat larger than H and contains `exotic' elements such as improper vectors. A simple map between M and the distribution space is given by the (improper) eigenvectors of the coordinate operator M x x Then, the DVR maps a set of point of M directly on H and almost preserves the metric properties, in the sense x α x β = δ(x α x β) F α F β = δ αβ By denition, using P = P 2 = P, it follows or equivalently α β = x α P x β = β (x α ) = α(x β ) = N α δ αβ F β (x α ) = δ αβ that is, the coordinate representation of a DVR set is made up of functions which vanish at each other grid points, but at their own. In other words, DVR functions satisfy simultaneously two properties: orthogonality and interpolation property. These properties can be used to get two dierent representations of vectors. Let us rst consider the case ψ S. Then, ψ can exactly be represented by the DVR set ψ = α F α F α ψ = α N α ψ(x α ) F α where the property F α ψ = P x α ψ = x α ψ has been used. The DVR representation of the state vector ψ, ψα DV R = F α ψ, is therefore simply related to the values of the wavefunction at grid points; in this case, one may indierently use the `scalar product denition' ψ DV R α = F α ψ or the `grid denition' ψ DV R α = ψ(x α ) When ψ / S the two expressions above are dierent and only the projection of ψ onto S can exactly be represented by the DVR. That is, two dierent approximation to the state vector are possible, one based on its projection on S ψ ψ S = P ψ = α F α F α ψ and one based on the DVR formula ψ ψ DV R = α ψ(x α ) F α () 2

3 The two vectors ψ S and ψ DV R lie in the same space S but they are only approximately equal, since in this case x α P ψ = ψ S (x α ) ψ(x α ). Both vectors are approximate representation of ψ and, in some sense, they are exponentially close to ψ in the measure that ψ is close to S [2]. Equation denes the (approximate) Discrete Variable representation of the state vector, ψα DV R = F α ψ DV R = ψ(x α ) /2. Now, let ψ, φ S and let us consider the scalar product ψ φ = α φ F α F α ψ = α N α φ (x α )ψ(x α ) In other words, the DVR introduces an exact quadrature rule for the scalar product, ψ φ = ω α φ (x α )ψ(x α ) with ω α = (2) α When ψ, φ / S this formula is the DVR approximation to the scalar product, according to ψ φ ψ DV R φ DV R where vectors on r.h.s. are dened by equation. Note that once a DVR set has been dened in a space S = P H with the grid points {x α } one can obtain further DVR sets by simply throwing out grid points. This is useful if the wavefunction we are interested in is known to be vanishing small at some grid points. The proof follows directly from the denition using the projector P = α F α F α in which the sum is restricted to a subset of the original grid. 2 Finite Basis Representations Let { φ n } be a complete, orthonormal set in S. This set can be used to obtain an alternative representation of any vector in S which we call the Finite Basis Representation, ψ = φ n φ n ψ ψn F BR = φ n ψ n A unitary transformation exists such that F α = n φ n U nα and this can be used to transform FB representations to DV representations, ψ DV R = n (U ) αn ψ F BR n (3) where U nα = φ α F α = ω α φ n(x α ) (4) It is clear, however, from the discussion of the previous section, that eq. 3 is exact only for vectors of S; for a generic vector ψ, eq. 3 relates dierent representations of its projection on S, ψ S. The assumption is always made that ψ S ψ DV R. 3

4 Operator representations and transformations follow easily from F α A F β = nm F α φ n φ n A φ m φ m F β i.e. A DV R = U A F BR U (5) which is exact only for operators projected onto S, A P = P A and, as before, the assumption is always made that α S = P A F α α DV R. Local operators may be approximated as follows F α A F β = A(x α )δ αβ (6) if one takes into account the localized properties of the DVR vectors. As before, this formula is exact only if A = P A. In practice, one uses the DVR approximation of eq. 6 for operators which are local in system coordinates and, starting from an appropriate FBR for non-local operators, (s)he gets (approximate) DV representations for the latter using eq. 5. Usually, the FB set diagonalizes the operator A and thus (A DV R ) αβ = n ωα ω β φ n (x α )φ n(x β )a n where a n are the eigenvalues of A. A more general result is (f(a) DV R ) αβ = n ωα ω β φ n (x α )φ n(x β )f(a n ) 3 Simple examples Dening a DVR set is generally dicult and not even always possible. Usually, one starts with a set of orthogonal vectors { φ n }, denes the projector P = n φ n φ n and looks for grid points x α such that the vectors P x α = α = n φ n φ n(x α ) are orthogonal, α β = n φ n (x α )φ n(x β ) = N α δ αβ (7) This task may be very dicult or even impossible. However, for several onedimensional problems a convenient choice for { φ n } is a real orthogonal polynomial set times the square root of some weighting function W (x), φ n (x) = W /2 (x)p n (x) φ n φ m = W (x)p n (x)p m (x)dx = δ nm 4

5 In such case, given that P n (x) (n = 0,..N ) are polynomials of degree n one chooses as grid points the zero of P N (x), the rst neglected polynomial P N (x α ) = 0 α =,..N They are known as Gauss-quadrature points. The above DVR condition of eq. 7 may be rewritten as follows α (x β ) = N n=0 W (x α )W (x β )P n (x α )P m (x β ) δ αβ Now, from Gauss' quadrature theory it is known that the following relation holds δ nm = W (x)p n (x)p m (x)dx α w α W (x α )P n (x α )P m (x α ) (8) for n, m = 0,,..N. Here {x α } are Gauss quadrature points and w α are Gauss quadrature weights. This formula tells us that scalar products involving P n (x) n = 0,..N polynomials (and their linear combinations) are exactly computed by quadrature with a careful choice of grid points. From eq. 8 it follows that the vectors X β = {Xm} β N m= = { W (x β )P m (x β )} N m= are linearly independent since they are orthogonal with respect to the weights w β, and holds. In other words N w β α (x β )Xm β = W (x α )P m (x α ) β N w β α (x β )X β = X α β= or, equivalently, w β α (x β ) = δ αβ. Thus, the set F α (x) = N w α n=0 W (x)w (xα )P n (x)p m (x α ) is the DVR set. Another very common, one-dimensional set is the sync set. To introduce it, let us consider M = R and the following projection operator P = +pmax p max dp p p where p are momentum eigenstates. This operator and the grid allow us to dene a DVR set. Indeed, α = {x α = π p max α} α Z +pmax p max 5 dp p e ipxα 2π

6 and α β = +pmax p max dp eip(xα xβ) 2π = δ αβ p max π Therefore, a DVR set can be dened, π sin[p max (x x α )] F α (x) = α (x) = p max πpmax (x x α ) which is known as sync DVR set. Note that in this case the space S = P H is innite dimensional and the DVR set is innite. Its completeness follows from its momentum representation π p F α = p α = e i π pmax pα p max 2pmax which is the known complete Fourier basis set for square-integrable functions in M = [ p max, p max ]. Note also that this DVR set allows one to exactly represent any band-limited function through an innite but discrete set of function evaluations (or measurements) S ψ ψ = π F α ψ(x α ) p α max Actual sync sets used in practice are obtained from that dened above by throwing out points, e.g. retaining points α N. The corresponding space S comprises only discrete momentum values, p α = pmax N α, and is the space of band-limited, square-integrable functions in M =[ L, L] with L = πn/p max. This set underlies the use of Fast Fourier Transforms. References [] D. Baye and P. H. Heenen. Generalized mesh for quantum mechanics. J. Phys. A, 9:204, 986. [2] A. S. Dickinson and P. R. Certain. Calculation of matrix elements for one-dimensional quantum-mechanical problems. J. Chem. Phys., 49:4209, 965. [3] D. O. Harris and D. G. Engerholm. Calculation of matrix elements for one-dimensional quantum-mechanical problems and the application to anharmonic oscillators. J. Chem. Phys., 43:55, 965. [4] J. C. Light and T. Carrington. Dvrs and their utilization. Adv. Chem. Phys., 4:263, [5] J. C. Light, I. P. Hamilton, and J. V. Vill. Generalized discrete variable approximation in quantum mechanics. J. Chem. Phys., 82:400, 985. [6] J. V. Lill, G. A. Parker, and J. C. Light. Chem. Phys. Lett., 89:483, 982. The periodic properties ψ( L) = ψ(l) of the plane-wave set are inessential for L 2 (M). 6

7 [7] J. V. Lill, G. A. Parker, and J. C. Light. The discrete variable nite basis approach to quantum scattering. J. Chem. Phys., 85:900, 986. [8] R. G. Littlejohn and M. Cargo. Airy discrete variable representation bases. J. Chem. Phys., 7:37, [9] R. G. Littlejohn and M. Cargo. Bessel discrete variable representation bases. J. Chem. Phys., 7:27, [0] R. G. Littlejohn and M. Cargo. Multidimensional discrete variable representation bases: Sinc functions and group theory. J. Chem. Phys., 6:7350, [] R. G. Littlejohn and M. Cargo. Tetrahedrally invariant discrete variable representation basis on the sphere. J. Chem. Phys., 7:59, [2] R. G. Littlejohn, M. Cargo, T. Carrington, K. A. Mitchell, and B. Poiter. A general framework for dvr basis sets. J. Chem. Phys., 6:869,