General formula for finding Mexican hat wavelets by virtue of Dirac s representation theory and coherent state

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1 arxiv:quant-ph/ v 8 Aug 005 General formula for finding Meican hat wavelets by virtue of Dirac s representation theory and coherent state, Hong-Yi Fan and Hai-Liang Lu Department of Physics, Shanghai Jiao Tong University, Shanghai 00030, China Department of Material Science and Engineering, University of Science and Technology of China, Hefei, Anhui,3006, China February, 008 Abstract The admissibility condition of a mother wavelet is eplored in the contet of quantum optics theory. By virtue of Dirac s representation theory and the coherent state property we derive a general formula for finding Meican hat wavelets. Introduction Wavelet is a small wave which is localized in both time and frequency space[,, 3]. It is this unique characteristic that makes wavelets analysis in some ways superior to Fourier analysis which employs big waves sinusoid or cosine), e.g., wavelets are particularly useful when processing data with sharp discontinuities or compressing image data. Mathematically, a wavelet ψ of the real variable must satisfy the following admissibility condition ψ ) d = 0, ) which suggests that ψ ) behaves like a wave, and in contrast to sinusoid, it decreases rapidly to zero as tends to infinity. The theory of wavelets is concerned with the representation of a function in terms of a two-parameter family of dilates and translates of a fied function, which is usually known as the mother wavelet. A family of wavelets ψ µ,s) µ > 0 is a

2 fig. : scaling parameter, s is a translation parameter, s R) are constructed from the mother wavelet ψ, and the dilated-translated functions are defined as ψ µ,s) ) = µ ψ ) s, ) µ and the wavelet integral transform of a signal function f ) L R) by ψ µ,s) is defined by W ψ f µ, s) = µ ) s f ) ψ d. 3) µ The admissibility condition ) ensures that the inverse transform and Parseval formula are applicable. When ψ ) is an odd function of, it satisfies ) obviously. A typical ψ ), which is an even function of, is the Meican hat wavelet see fig. ) ψ M ) = π / e / ), ) satisfying e / ) d = 0. 5) An important question thus naturally arises: how to find more even functions which also satisfy ), i.e. are there a series of even functions which can be considered as generalized Meican hat wavelets? To our knowledge, this question has not been posed and solved in the literature before. In this letter we shall derive a general formula for finding Meican hat wavelets by taking the advantage of Dirac s representation theory and the coherent state theory in quantum optics, so that more mother wavelets and correspondingly more wavelettransformations can be introduced. Dirac s representation theory is not just a foundation of

3 quantum mechanics theory[]; it has its own special features which allow it to be etended to new theoretical problems. In this Letter we shall show how this theory can help us to directly derive general formula for finding Meican hat wavelets. According to Dirac s representation theory, we can epress Eq. 3) as W ψ f µ, s) = ψ U µ, s) f, 6) where ψ is the state vector corresponding to the given mother wavelet, f is the state to be transformed, and s d U µ, s) 7) µ µ is the squeezing-translating operator[5, 6], is the coordinate eigen-vector of X, X =, which in the Fock space is epressed as = π / ep + a a ) 0, 8) here 0 is the vacuum state annihilated by the bosonic operator a, a 0 = 0, [ a, a ] =, and X = a + a ) /. In order to combine wavelet transforms with transforms of quantum states more tightly and clearly, using the technique of integration within an ordered product IWOP)[5] for a review see[6, 7]) of operators we can directly perform the integral in 7) U µ, s) = = πµ d : ep [ µ + µ : ep + µ ) + µ ) + s µ + s µ a + ] a s µ X : s µ + a + ) µa s µ a s µ X :, 9) where : : denotes normal ordering. Let µ = e λ, so sechλ = µ, tanhλ = µ, using the +µ µ + operator identity e ga a =: ep [ e g )a a ] :, Eq. 9) becomes [ s ] [ U µ, s) = ep + µ ) a tanh λ a s sechλ ep a a + ) ] ln sechλ [ ] a sa ep tanhλ + sechλ. 0) In particular, when s = 0, it reduces to the well-known squeezing operator, U µ, 0) = d = ep[ λ a a ). ) µ µ 3

4 For a review of the squeezed state theory we refer to[8, 9, 0]. Now we analyze the condition ) for mother wavelet in the contet of quantum optics theory. Due to Dirac s representation transformation π eip d = p, where p is the momentum eigenstate, p = π / ep p + ) ipa + a 0, we have π d = p = 0, ) which can help us to recast the condition ) into Dirac s ket-bra formalism, ψ ) d = 0 p = 0 ψ = 0, 3) which indicates that the probability of a measurement of ψ by the projection operator p p with the value p = 0 is zero. Now we want to find such ψ that obeys p = 0 ψ = 0. By considering a n / n! 0 = n is the orthogonal basis of Fock representation, without loss of generality, we can epand ψ as ψ = G a ) 0 = g n a n 0, ) n=0 where g n are such chosen as to let ψ obey the condition 3). Then using the overcompleteness relation of coherent states useful representation in quantum optics and can describe laser[, ]), d z z z =, 5) π where z = ep za z a ) 0, 6) a z = z z, and we have ) z p = 0 z = π / ep, 7) d z p = 0 ψ = p = 0 π z z g n a n 0 n = π / d z g n n π e z z n z m m! = π / m n m! mg nδ n,m n! = π / n ) m n)! n! n g n = 0. 8)

5 Then condition 9) provides a general formalism to find the qualified wavelet. To illustrate the usage of 9), assuming that in 9) g n = 0 for n > 3, so the coefficients of the survived terms should satisfy g 0 + g + 3g + 5g 6 = 0, 9) and ψ becomes ψ = g 0 + g a + g a + g 6 a 6) 0. 0) Projecting it onto the coordinate representation, we get the qualified wavelets ψ ) = ψ = π / e / [ g 0 + g ) + g + 3 ) +g )], ) where we have used n = n n! π H n ) e /, ) and the Hermite polynomials definition H n ) = ) n dn e. 3) d ne Now we take some eamples and depict them in figures. Case : when we take g 0 =, g = and g = g 6 = 0, we immediately obtain the Meican hat wavelet see fig. ) as in ). Hence ) a 0 is the state vector corresponding to the Meican hat mother wavelet. Case : when g 0 =, g =, g = and g 6 = 0, we obtain see fig. ) ψ ) = π / e / + ), ) which also satisfies the condition π / e / + ) d = 0. 5) Note that when g 0 =, g =, g = and g 6 = 0, we obtain a slightly different wavelet see fig. 3). Therefore, as long as the parameters conforms to condition 9), we can adjust their values to control the shape of the wavelet. Case 3: when g 0 =, g =, g = and g 6 =, we get see fig. ) ψ 3 ) = π / e / ), 6) 5

6 fig. : fig. 3: fig. : 6

7 and π / e / ) d = 0. 7) From these figures it is observed that the number of the crossing points of the curve at the -ais is equal to the highest power of the wavelet function. In summary, by converting wavelets and its admissibility condition into the framework of Dirac s ket-bra formalism and using the coherent state s well-behaved properties we have derived the general formula for composing Meican hat wavelets, based on which more qualified wavelets can be found and more wavelet transformations can be defined. This work again shows the powerfulness of Dirac s representation theory. References [] Daubechies, I. Ten Lectures on Wavelets Philadelphia, PA: Society for Industrial and Applied Mathematics, 99). [] Kaiser, G. A Friendly Guide to Wavelets Cambridge, MA: Birkhäuser, 99). [3] Chui, C. K. An Introduction to Wavelets San Diego, CA: Academic Press, 99). [] Dirac, P. A. M. The Principle of Quantum Mechanics fourth edition, Oford University Press, 958). [5] Fan Hong-yi, H. R. Zaidi and J. R. Klauder, Phys. Rev. D 987) 83; Fan Hongyi and H. R. Zaidi, Phys. Rev. A ) 985; Hongyi Fan and J. R. Klauder, J. Phys. A 988) L75; Hongyi Fan and Hui Zou, Phys. Lett. A 5 999) 8. [6] Hong-yi Fan, J. Opt. B: Quantum & Semiclass. Opt ) R7; Inter. J. Mod. Phys. 8 00) 387. [7] A. Wünsche, J. Opt. B: Quantum & Semiclass. Opt. 999) R. [8] D. F. Walls, Nature 3 986) 0. [9] R. Loudon and P. L. Knight, J. Mod. Opt ) 709. [0] V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 00) R. [] R. J. Glauber, The Quantum Theory of Optical Coherence ) 59; Phys. Rev ) 766. [] J. R. Klauder and B. S. Skargerstam Coherent States World Scientific, Singapore, 985). 7