[seqfan] Re: A000217, A002629, and Quantum Mechanics

[israel at math.ubc.ca]

Your Subject has A002629, but the body of your message mentions only 
A002620. Your statement

"The ratio a000217( n ) / a002620( n + 1 ) is already small for n = 5, and
 approaches zero for higher order perturbations, viz. as n -----> infinity"

is wrong as stated. That ratio is 2*(n+1)/(n+2) for even n and 2*n/(n+1) 
for odd n, and thus approaches 2 as n -> infinity. On the other hand, if 
you meant a002629 instead of a002620, then certainly the ratio goes to 0, 
as we see from the asymptotics: a000217(n) ~ n^2/2, a002629(n) ~ (n-1)!.

Cheers,
Robert

On Apr 28 2015, Brad Klee wrote:

>Hi Seqfans,
>
>I am writing you all today to describe a ( slightly unproven ) symmetry
>transformation I discovered while studying a famous problem from quantum
>mechanics ( can you guess which one ? ).
>
>The alternative definition of A000217:
>
>"Number of legal ways to insert a pair of parentheses in a string of n
>letters. E.G., there are 6 ways for three letters."
>
>implies that this sequence also counts the number of unique triples { i, j,
>k } that satisfy the equation
>
>i+j+k = n .
>
>Exactly a000217(n+1) unique differential operators comprise the total n_th
>order term of the Taylor expansion of a potential V which is a function of
>three orthogonal, generalized coordinates.
>
>In order to avoid ever using any Clebsch-Gordon coefficients ( I still
>can't understand Wigner's Eckart theorem ), I am interested in reducing the
>elements of a three-by-three perturbation matrix to dependency upon a
>minimal number of symmetry-allowed parameters. Reduction follows
>immediately from the indistinguishable assumption applied to the set of
>three orthogonal, generalized coordinates, and the set of three orthogonal,
>spanning wavefunctions.
>
>At most the perturbation matrix could depend upon 9 * a000217(n+1)
>parameters of the form ( in the Dirac notation ),
>
>< i | D[ n , V(q1,q2,q3) ]  | j >,
>
>where i and j are wavefunctions from the spanning set, and D[ n ,
>V(q1,q2,q3) ] is one of the summands of the n_th order term of the
>expansion of potential V about the origin of coordinates.
>
>Fortunately, many of these elements can be eliminated by applying the
>indistinguishable assumption and solving a system of linear equations, thus
>reducing the dependency to 2*a002620( n + 2 ) expectation values (
>apparently ). For some but not all values of the parameters, the
>perturbation matrix will have a degenerate eigenspace.
>
>Finally I reach an abstract formulation of the indistinguishable
>assumption, which appears to be a relation between a000217( n ) and
>a002620( n + 1 ). Notice that each triple {i,j,k} is invariant under a
>permutation from group S3 ( symmetric ), P ( parity ), or I ( identity ). I
>evaluate a linear functional of the triple, weighting each term according
>to its symmetry,
>
>S3 -----> 1,
>P   -----> 2 / 3,
>I    ------> 1 / 2 ,
>
>and summing the weights. For example, just in case you need to take a look
>at the fourth order terms, a000217( 5 ) = 15. Values of the 15 triples
>{i,j,k} are
>
>Triple  : occurrence : weight
>---
>{4,0,0} : 3 : 2 / 3
>{3,1,0} : 6 : 1 / 2
>{2,2,0} : 3 : 2 / 3
>{2,1,1} : 3 : 2 / 3
>
>Evaluating the functional,
>
>3 * 2 / 3 + 6 * 1 / 2 + 3 * 2 / 3 + 3 * 2 / 3
>= 2 + 3 + 2 + 2 = 9
>= a002620( 6 ) .
>
>At fourth order, expect to find not 135 free parameters, but rather 18. You
>see, the indistinguishable assumption enacts a very strong constraint. The
>ratio a000217( n ) / a002620( n + 1 ) is already small for n = 5, and
>approaches zero for higher order perturbations, viz. as n -----> infinity.
>
>
>Empirically this procedure is verified for many terms and I have a cool
>picture drawn up that explains everything, but proving it could be
>difficult, especially for me. I expect relations such as these to continue
>to arise as I investigate expansions with larger numbers of generalized
>coordinates and higher dimensional matrices.
>
>I continue to work under the assumption that publishers will not want to
>hear from me, but I will try to submit something sometime anyways.
>
>Please send questions or comments.
>
>Cheers,
>
>Brad
>
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