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Quantum Mechanical Operators in
Multiresolution Hilbert Spaces
Cite as: AIP Conference Proceedings 963, 536 (2007); https://doi.org/10.1063/1.2836134
Published Online: 08 January 2008
János Pipek
AIP Conference Proceedings 963, 536 (2007); https://doi.org/10.1063/1.2836134
© 2007 American Institute of Physics.
963, 536
Quantum Mechanical Operators in Multiresolution
Hilbert Spaces
János Pipek
Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, H–1521 Budapest, Hungary
Abstract. Wavelet analysis, which is a shorthand notation for the concept of multiresolution analysis (MRA), becomes
increasingly popular in high efficiency storage algorithms of complex spatial distributions. This approach is applied for
describing wave functions of quantum systems. At any resolution level of MRA expansions a physical observable is
represented by an infinite matrix which is “canonically” chosen as the projection of its operator in the Schrödinger picture
onto the subspace of the given resolution. It is shown that this canonical choice is only a particular member of possible
operator representations. Among these, there exits an optimal choice, usually different from the canonical one, which
gives the best numerical values in eigenvalue problems. This construction works even in those cases, where the canonical
definition is unusable. The commutation relation of physical operators is also studied in MRA subspaces. It is shown that
the required commutation rules are satisfied in the fine resolution limit, whereas in coarse grained spaces a correction
appears depending only on the representation of the momentum operator.
Keywords: wavelet, multiresolution, quantum mechanical operators
PACS: 31.15.-p, 71.15.Ap, 71.15.Dx, 71.23.An
THE CONCEPT AND APPLICATIONS OF MULTIRESOLUTION ANALYSIS
Wavelet’s theory is a remarkable discovery in the theory of square integrable function spaces L2 (R n ) and in
applied mathematics. In the past two decades tremendous success has been achieved in the fields of Hilbert-space
basis sets and frames, localized window-Fourier transforms, etc. Wavelets are often used in various fields of science,
especially in data compression, storage of pictures and sound, as well as noise reduction. Among many others, one
of the mostly known applications is the compression standard JPEG2000 [1] approved by the International Standards
Organization's committee in 2001. The last few years have likely been the age of explosive development of high
definition cinema and high definition television (HDTV). In 2006 the first HDTV digital cinema camera with a
hardware realization of a wavelet codec has been announced.
Owing to the extremely fast algorithms developed, multiresolution analysis has drawn a considerable interest in
other fields of science, as well. Goedecker and Ivanov [2] solved the Poisson equation, Cho et al. employed wavelets
in solving the Schrödinger equation for Hydrogen-like atoms [3]. All electron calculations have also been performed
within the framework of the local density approximation applying multiresolution basis sets [4]. Arias and his
coworkers developed a Kohn–Sham equations based wavelet method [5], which was tested for various systems
[6,7]. Besides these practical applications, wavelet theory can be considered as a manifestation of physical concepts
in mathematics. Coherent states, scaling and renormalization group transformation are closely connected to this
newly developed field.
Multiresolution analysis (MRA, popularly named “wavelet analysis”) covers a systematically refined basis
function set of Hilbert spaces. We refer to basic textbooks (see e.g., [8,9]) for the details. Conceptually, MRA
belongs to the class of lattice approximations, which, however, can be systematically expanded for finer grid
resolutions and in the infinite resolution limit the ATTACHMENT
method exactly describes the Hilbert space L2 (R n ) . For the sake
2 INSERTED ON THE FIRST PAGE OF EACH PAPER OF
CREDIT LINE
BE
of better visualization
we (BELOW)
will stay TO
at L
(R) , but the extension to higher dimensionalities does not lead to
VOLUME
2, approximation
PART A
unexpected complications [10]. The zeroth resolution
level
of the Hilbert space is expanded by the
CP963, Vol. 2 Part A, Computation in Modern Science and Engineering, Proceedings of the International Conference on
Computational Methods in Science and Engineering 2007, edited by T. E. Simos and G. Maroulis
© 2007 American Institute of Physics 978-0-7354-0478-6/07/$23.00
536
integer translates s0l ( x) = s ( x − l) of a properly chosen scaling function s (x) . The set
{s0l
l ∈ Z} is an
orthonormal system, if the scaling function satisfies some rather non-trivial conditions [8,9]. Nevertheless, there are
numerous such choices, moreover, additional requirements like smoothness, compact support etc. can be also
satisfied. Daubechies [8] defined basis sets determined by N parameters, where the scaling function is zero outside
of the interval (0,N-1). In the following considerations we will apply such compactly supported basis sets in order to
avoid unnecessary complications regarding convergence questions. The elements of the basis set {s 0l } are “sitting”
on an equidistant lattice with grid spacing 1. The basis functions span the infinite dimensional Hilbert space
H [0] = span{s 0l } . The next step in MRA is scaling the basis as s1l ( x) = 21 / 2 s (2 x − l ) , where the basis functions are
sitting on a grid with grid distance 2-1. After m-fold scaling the elements of the basis set are determined as
s ml ( x) = 2 m / 2 s (2 m x − l) , sitting on a fine grid of spacing 2 − m . In the case of Daubechies-N scaling functions the
length of support for each s ml ( x ) is 2 − m ( N − 1) . The orthonormal set spans the subspace H [ m] = span{s ml } . The
key concept of multiresolution analysis is that the fine approximations should contain all previous course grain
spaces, and in the infinite resolution limit (m → ∞) H [m ] → L2 (R) exactly. This can be summarized by the
following ladder of subspaces
H [ 0] ⊂ H [1] ⊂ K ⊂ H [m] ⊂ K ⊂ L2 (R) .
(1)
OPERATORS OF PHYSICAL QUANTITIES IN GRANULAR HILBERT SPACES
Since the early works of von Neumann [11] it is clear that all quantum mechanical theories should work in
specific representations of the abstract Hilbert space. It is worth emphasizing that the infinite dimension of the
Hilbert space is an essential ingredient of the description. The coarse grained (or “granular”) Hilbert spaces H [m ]
are good candidates for a correct quantum mechanical description in this respect, as they are infinite dimensional for
any m, opposed to the usual linear combination of atomic orbital (LCAO) schemes. Moreover, all these spaces are
equivalent to the separable abstract Hilbert space, and as such, they can serve for a complete description of any
quantum mechanical system, on their own right. It is not clear, however, how the operators belonging to the various
physical quantities have to be represented in these granular spaces. Among others mentioned previously [2-7], we
have also studied some specific questions of quantum systems in the framework of MRA [12-15], taking it for
granted that the matrix elements of the infinite matrix representing the physical quantity Q in the space H [m ] should
be calculated as
Q = s Qˆ s
(2)
kl
mk
ml
where Q̂ is the Schrödinger operator representation of Q in L2 (R) . (For example, Q̂ = x ⋅ for the position operator,
whereas Qˆ = −i d dx for the momentum.) We have just recently realized [16] that the “canonical” choice (2) leads to
a systematic error at finite resolutions even in the case of the simplest system of a free particle. The regular grid of
the basis set introduces an artificial consequence of periodicity, causing an unphysical increase of the kinetic energy.
Although this error disappears in the m → ∞ limit, in a practical calculation, however, one should stay at a relatively
low resolution m. We have presented an explicit method of preparing a near optimal kinetic energy matrix which
leads to more appropriate results in numerical MRA based calculations. This construction works even in those cases,
where the canonical definition is unusable (i.e., the derivative of the basis functions does not exist). The strange
experience with the kinetic energy operator Q̂ = − d 2 2dx 2 poses the questions, what is the proper representation of
other physical quantities, how commutation rules are satisfied, how the eigenfunctions of Q in H [ m ] are related to
those in L2 (R) . These problems are discussed in this contribution.
As the granular subspaces are defined through a specific basis set, the application of matrix mechanics is a
natural consequence. In its original Born–Heisenberg–Jordan form [17,18] any physical quantity Q is represented by
an infinite matrix Qkl for which no specific prescriptions are necessary, except hermiticity and that the representing
matrices of any two physical quantities P and Q should satisfy commutation rules corresponding to their Poisson’s
bracket expression in classical mechanics [19]. On the other hand, in the space of square integrable functions the
537
form of physical operators is uniquely defined by the usual Schrödinger’s prescription. The canonical definition (2)
of the matrix Qkl appears in showing the equivalence of L2 and ℓ2 (i.e. the equivalence of wave- and matrix
mechanics). The granular space H [ m ] , however, essentially differs from L2 (R) and there is no need that the matrix
elements Qkl in H [ m ] coincide with the matrix elements of an operator Q̂ in an other Hilbert space L2 (R) . In the
following section we will investigate the question what properties are satisfied by the representing matrices of
simple physical quantities in MRA subspaces.
PROPERTIES OF MRA REPRESENTATIONS OF SIMPLE PHYSICAL OPERATORS
Although we would like to broaden the set of possible matrix representations, in order to avoid any weird form of
Qkl , we will keep some properties of the canonical approach, which are summarized below. For the momentum
operator P̂ = −i d dx we get the canonical matrix element
Pklcan = s mk Pˆ s ml = −i 2 m Rk −l where R j = ∫ s ( x) s ' ( x + j ) dx
(3)
and for the position operator X̂ = x ⋅ we have
can
X kl
= s mk Xˆ s ml = 2 − m (Yk −l + k δ 0 ,( k −l ) ) where Y j = ∫ s ( x ) x s ( x + j ) dx.
(4)
Here, overbar means complex conjugation, prime denotes the derivative by x, and the formulas are consequences of
the definition of s ml , the orthonormality of the basis set and some straightforward integral variable transformations.
The numerical determination of the values of R j and Y j is not simple, but it can be carried out knowing the
parameters which define the scaling function s (x) . Some of their properties are, however, clear. Partial integration
and s (±∞) = 0 (as s is square integrable) result in R − j = − R j , whereas integral variable transformation and
orthonormality leads to Y − j = Y j . These are necessary conditions to ensure hermiticity of Pkcan
and X kcan
l
l . For the
Daubechies-N scaling functions which are compactly supported on the interval (0,N-1), due to the zero overlap
between the shifted versions of the functions R j = Y j = 0 if j > N − 2 .
The canonical matrix elements depend only on the difference k − l of the indices (except for the second term in
the position matrix elements), which is a consequence of the shift invariance of the subspace H [m ] for grid
translations. In the general case this natural symmetry has to be conserved, which leads to the assumption that any
infinite matrix representing the physical quantities P and X should be written as
(5)
Pk[lm ] = −i 2 m Rk −l
and
X k[ ml ] = 2 − m (Yk −l + k δ 0,( k −l ) )
with properly chosen values of R and Y, satisfying R − j = − R j and Y − j = Y j . The requirement R j = Y j = 0 if
j > N − 2 (which is a consequence of the canonical rule (2)) is not necessary, however, it is very practical from the
computational point of view. Let us check now, how definition (5) affects the commutation relation of P and X.
Using the above definitions
∞
[ m] [ m]
[m] [m]
( X [m ] P [ m ] − P [ m ] X [ m] ) kl = ∑ ( X kn
Pnl − Pkn
X nl ) = − i (k − l) Rk −l .
(6)
n = −∞
We can realize that the commutator does not depend on the specific representation of X, only the representation of
the momentum counts. This fact is a simple consequence of the band matrix form (5) of the physical operators.
Result (6), unfortunately, can never be compatible with the required commutation rule (XP-PX) = i1. Clearly, this
would require ( X [ m] P [ m ] − P [ m ] X [ m ] ) kl = iδ kl with zero off-diagonals and non-zero diagonals. Expression (6)
gives, on the other hand, zero diagonals and non-zero off-diagonals. This disappointing result can be released by the
following considerations. Equation (6) can be rewritten in the following form
( X [m ] P [ m ] − P [ m ] X [ m] ) kl = i δ kl + Okl with O kl = −i (δ 0,k −l + (k − l) Rk −l )
(7)
where Okl are elements of a “zero in mean” matrix. The explanation of this concept is as follows. There are only
few non-zero elements Okl around its diagonal, namely, for k − l ≤ N − 2 , acting on a spatial distance of order
538
2 − m (2 N − 3) . For large resolutions ( m → ∞ ), this distance becomes small compared to the characteristic distance
of changes in the wave functions. If cl are the expansion coefficients of this function, cl ≈ c k for those l where
Okl is non-zero. The action ∑l Okl cl ≈ (∑l Okl ) c k = 0 is equivalent with a zero operator if ∑l Okl = 0 . In this
case Okl is called zero in mean. For (7), this property follows from
∑ j j R j = −1
(8)
which holds for the canonical matrix elements. The proof is not elementary and will not be given here. For any other
representations of the momentum operator, (8) is a necessary requirement satisfied by the R j values.
Further constraints for the possible choices of the momentum matrix elements follow from the eigenvalue
problem. It can be shown that the projection of the wave function e ikx with wave number k to the subspace H [m ] is
]
an eigenvector of Pk[m
l defined by (5) with the eigenvalue
− i 2 m ∑ j R j e −ikm j
(9)
where k m = 2 − m k is the scaled wave number. The optimum choice of R j is determined by the requirement that the
2π-periodic function (9) should approximate the linear function k (which is the correct eigenvalue of P) around
k = 0 as well as possible. The reasoning is very similar to that given in [16] for the case of the kinetic energy
operator.
The ideas and methods used here are applicable to other physical quantities like the position operator, as well.
ACKNOWLEDGMENTS
This work was supported by the Országos Tudományos Kutatási Alap (OTKA), Grant No. T046868.
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