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- From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
- Newsgroups: sci.physics
- Subject: Re: Wavelets & Coherent States ?
- Message-ID: <1993Jan7.021947.25511@murdoch.acc.Virginia.EDU>
- Date: 7 Jan 93 02:19:47 GMT
- References: <qg62r5g@rpi.edu> <1993Jan3.211626.3723@EE.Stanford.EDU> <1993Jan6.230830.6733@galois.mit.edu>
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- In article <1993Jan6.230830.6733@galois.mit.edu> jbaez@riesz.mit.edu (John C. Baez) writes:
- >In article <1993Jan3.211626.3723@EE.Stanford.EDU> siegman@EE.Stanford.EDU (Anthony E. Siegman) writes:
- >>In article <qg62r5g@rpi.edu> sassoj@aix.rpi.edu (John J. Sasso Jr.) writes:
- >>
- >>>quantum mechanics. Although I have has a basic course in Q.M., can anyone
- >>>explain to me what coherent states are? Do they have anything to do with
- >>>phase-space localization?
- >>
- >> I also think you are perceptive in noting a possible connection
- >>between wavelets and coherent states. Coherent states provide an
- >>"over-complete" basis for the SHO, which means that the expansion of a
- >>given arbitrary SHO state in coherent states is not unique (although
- >>there are preferred ways to do the expansion), and I suspect the same
- >>may be true of wavelets.
- >
- [stuff deleted]
- >
- >Coherent states are rather different. They don't form a basis; they are
- >"overcomplete". Basically, they are just functions of the following
- >form: a Gaussian translated by a certain amount and then multiplied by a
- >complex exponential. If we make the Gaussian tall and skinny, we have a
- >function that's fairly well localized in position space; if we make it
- >short and wide, it's fairly well localized in momentum space. If one is
- >looking for a quantum state that approximates a given classical state,
- >one is probably looking for a coherent state... they are often used for
- >that in quantum optics.
-
- And the wavelet people call these 'frames'. They are not required
- to be orthogonal and can be thought of as a generalization of
- a wavelet basis. The only reference that seems to be handy
- is Heil and Walnut's article in SIAM Review (31:628 (1989))
- which is right after Strang's introductory article on wavelets
- and dilation equations. I believe that 'Wavelets and Their
- Applications' has some relevant articles explicitly pointing
- out the connections between coherent states representations and
- wavelets, but I cannot seem to find the book in the mess.
-
- dale bass
-
- --
- C. R. Bass crb7q@virginia.edu
- Department of Wildebeest
- Transvaal (804) 924-7926
-