# Asymptotic Behavior of z->x and z->y Rotation Matrices

This appendix derives the asymptotic form of the rotation matrices used in [Section XIV] to define rotated position operators, and contains the proof details of theorem 14.5.
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For the rotation of the z-axis into the x-axis,

D^(j)(0, - pi /2, 0)_ab  =

2^(-j) SIGMA_k (-1)^k/k! c(j, a, b; k)

and for the rotation of the z-axis into the y-axis,

D^(j)( pi /2, - pi /2, 0)_ab  =

exp(+ia pi/2) exp(-ib pi) 2^(-j) SIGMA_k (-1)^k/k! c(j, a, b; k)

where

sqrt[(j+b)! (j-b)! (j+a)! (j-a)!]
c(j, a, b; k)  :=  ---------------------------------
(j+b-k)! (j-a-k)! (k+a-b)!

The number of terms in the summation, as a function of j, a, b:
for a = j, hence also for b = j that there is only one term;
generally, the number of terms in the sum is equal to

1 + Min{ j(+|-)a, j(+|-)b }

E.g., [Messiah 1965] v. II p. 1073.

We want the asymptotic form of these matrix elements for very large j,
and moderate a and b.  First address the behavior of the coefficient
c(j, a, b; k).  This can be rewritten as,

(j-b)! (j+a)!              1
[ B(j+b k)k! B(j-a k)k! ---------------- ]^(1/2) --------
(j+b-k)! j-a-k)!         (k+a-b)!

Using Stirling's approximation for large A,

A!  ->  sqrt(2 pi A) A^A e^(-A)

the binomial coefficient is approximated by

B(A B) ->  A^B/B!
so
B(j+b k)k!  ->  (j+b)^k
B(j-a k)k!  ->  (j-a)^k

and

(j-b)!                          (j+b-k)^k
--------    ->  e^(-2b) -----------------
(j+b-k)!                      (j-b)^b (j+b-k)^b

(j+a)!                              (j-a-k)^k
--------    ->  e^(+2a) -----------------------
(j-a-k)!                      (j+a)^(-a) (j-a-k)^(-a)

then c(j, a, b; k) is approximated by,

e^(a-b)   (j+b)^k (j-a)^k (j+a)^a
-------- [ ----------------------- (j+b-k)^(k-b) (j-a-k)^(k+a) ]^(1/2)
(k+a-b)!          (j-b)^b

Considering that j is very much larger than a and b, approximate
c(j, a, b; k) further by,

e^(a-b)
-------- [ j^(2k+a-b) (j+b-k)^(k-b) (j-a-k)^(k+a) ]^(1/2
(k+a-b)!

On the same basis

(j+b-k)^(k-b) (j-a-k)^(k+a)  ->   (j-k)^(2k+a-b)

So
e^(a-b)
c(j, a, b; k)  ->  --------- [ j^(2k+a-b) (j-k)^(2k+a-b) ]^(1/2)
(k+a-b)!

e^(a-b)
->  -------- j^(2k+a-b) [ (1 - k/j)^(2k+a-b) ]^(1/2)
(k+a-b)!

e^(a-b)
->  -------- j^(2k+a-b)
(k+a-b)!

Then the rotation matrix taking the z axis to the x axis has the
asymptotic elements,

D^(j)(0, - pi /2, 0)_ab  ->

2^(-j) SIGMA_k (-1)^k/k! e^(a-b) j^(2k+a-b)/(k+a-b)!

The m-th Bessel function of integral order is defined by the series,

infinity   (-1)^k
J_m(z)  :=  2^(-m) z^m  SIGMA   --------- (z/2)^(2k)
k=0    k! (k+m)!

where

J_(-m)(z)  =  (-1)^m J_(+m)(z)

E.g. [Bronshtein 1985], p. 410

Taking the summation for D^(j)(0, - pi /2, 0)_ab over an infinite range
of positive integral k, gives the asymptotic approximation

D^(j)(0, - pi /2, 0)_ab  ->  e^(a-b) 2^(-j) J_(a-b)(2j)

Since j = 0, 1/2, 1, 3/2, ..., 2j is integral, and for 2j odd or even,
the quantity (a-b) is integral, so the approximation is always in terms
of Bessel functions of integral order.  Furthermore, we are approximating
for large values of the argument of the Bessel function, for which the
the Bessel function is asymptotically approximated by
[Bronshtein 1985], p. 413

J_m(z)  ->  (2/[ pi z])^(1/2) cos(z -  pi m/2 -  pi /4)

Finally the asymptotic approximations are:

D^(j)(0, - pi /2, 0)_ab  ->

( pi j)^(-1/2) 2^(-j) e^(a-b) cos( 2j -  pi (a-b)/2 -  pi /4 )

and

D^(j)( pi /2, - pi /2, 0)_ab  ->

i^a (-1)^b ( pi j)^(-1/2) 2^(-j) e^(a-b) cos( 2j -  pi (a-b)/2 -  pi /4 )

QED

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