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

[Brad Klee]

Hi Robert,

You caught me going a little too fast on the equations there!  Thanks for
the help!

Actually the subject line is a typo, and the small ratio should actually
be  a002620( n + 1 ) / a000217( n ) . According to your correct analysis,
this would approach 1 / 2. The asymptotic ratio of reduced parameters to
unreduced parameters should then be 1 / 2 * 2 / 9 = 1 / 9.

In retrospect, finite limit should really be an obvious consequence of the
fact that both sequences are essentially quadratic. Now that I am thinking
straight again, there is another way to take the limit of a002620( n + 1 )
/ a000217( n ). The function can be written as

floor(n^2/4) = ( n^2 - i ) / 4,

with ( possibly ) values of i = 0,1,2,3. In the limit these values are
inconsequential, and the ratio proves to be 1 / 2.

1:30 a.m. in someways is the best and worst time to finally get some work
done. Dangers of working late are compounded when you turn twenty-eight and
spend the entire day writing a soliloquy:

To work or not to work, that is the question--
Whether 'tis Nobler in the mind to suffer
The outrageous confusion of exile,
Or to turn away from all studies,
And by opposing, end them?

I did some more computations and then found sequence A000292... the
tetrahedral numbers. These apparently are constrained by an exponent
equation something like:

w + x + y + z = n.

The group of maximal symmetry is expanded to S4, and when the quartet takes
on integer values, the subgroup symmetry can be S3, V ( vierergruppe ), P,
or I. Again I'm moving to fast, but the reduction factors ( this time I'm
not dividing by two ) could be, should be, or at least probably are similar
to:

S4 ---> 2
S3 ---> 4 / 4
V   ---> 5 / 6
P   ---> 7 / 12
I   ---> 10 / 24

In general it is not possible to divide one of these series by 2 and retain
all integer terms. Without diving by 2, the origin of the factors becomes
more clear, but probably still confusing. The denominator is determined by
the index of the group G and the subgroup SG according to

d = |G| / |SG|.

The numerators are determined by a graphical procedure that seems well
motivated, but I will need to double check using the formulation of the
indistinguishable assumption in terms of matrix transformation. If I got
the numerators correct, the corresponding series counting the reduced
parameters should start with the following terms:

2, 4, 9, 15, 25, 36, 53, 71, 96, 123, 158, 195, 242, 291, 351, 414,
489, 567, 659, 754, 864, 978, 1108, 1242, 1394, 1550, 1725, 1905,
2105, 2310, 2537, 2769, 3024, 3285, 3570, 3861, 4178, 4501, 4851,
5208, 5593, 5985, 6407, 6836, 7296, 7764, 8264, 8772, 9314, 9864,
10449, 11043, 11673, 12312, 12989, 13675, 14400, 15135, 15910, 16695,
17522, 18359, 19239, 20130, 21065, 22011, 23003, 24006, 25056, 26118,
27228

Following from our discussion here, I think it's reasonable to first assume
that the fit will be a cubic function like k x^3 such that the asymptotic
ratio is a fixed constant. Up to these 70 terms, I get something like k =
.07715 .

This sequence is not in OEIS, but I am not suggesting to add it until it
has been properly reviewed, which will probably require a more formal
presentation ( some of this summer's "work" ).

Thanks,


Brad








On Tue, Apr 28, 2015 at 11:19 AM, <israel at math.ubc.ca> wrote:

> 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
>>
>> _______________________________________________
>>
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
>>
>>
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