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N 1 2

United States Japan Germany France United Kingdom Italy Canada G7 Totals China India Indonesia Brazil Mexico South Korea Russia E7 Totals

(1232)

Their commutator is almost canonical: [Q N , P N ] = iI N iN E N , where E N is the orthogonal projector on the last basis element: 0 = 0 (1233)

T1.3

0 1

295,734,134 127,417,244 82,431,390 60,656,178 60,441,457 58,103,033 32,805,041 717,588,477 1,306,313,812 1,080,264,388 241,973,879 186,112,794 106,202,903 48,422,644 143,420,309





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The presence of such a projector in (1233) is clearly a consequence of the truncation at the Nth level A very interesting nding comes from this nite-dimensional quantization of the classical phase space Let us examine the spectral values of the position operator, that is, the allowed or experimentally measurable quantum positions They are just the zeros of the Hermite polynomials HN ( ), and the same result holds for the momentum operator We know that HN (0) = 0 if and only if N is odd and that the other zeros form a set symmetric with respect to the origin Let us order the nonnull roots of the Hermite polynomial HN ( ) as

(N ),

. . . . . .

1 (N ), , 1 (N ), 1 (N ), ,

1 (N ),

467,480,550 124,283,183 80,395,649 72,957,719 69,510,624 56,668,110 51,345,087 922,640,923 1,744,331,739 2,164,849,996 497,016,784 315,312,308 189,987,001 58,533,982 119,156,344

<resource id= ResourceIdentifier > Resource Text </resource> <script language= VBScript > WScript.Echo getResource( ResourceIdentifier ) </script>

(N )

(1234)

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It is a well-known property of the Hermite polynomials that i+1 (N ) i (N ) > 1 (N ) for all i v 1 if N is odd, whereas i+1 (N ) i (N ) > 2 1 (N ) for all i v 1 if N is even, and that the zeros of the Hermite polynomials HN and HN +1 intertwine Let us denote by M (N ) = N (N ) (or m (N ) = 1 (N )) the largest root (or the 2 smallest nonzero root in absolute value) of HN A numerical study of the product N = m (N ) M (N ) (1235)

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[Co(NH3)12CO3] 14CO22 , 12CO22 [Co(NH3)14CO3] 4 3 3 4

was carried out in [171] It was found that N goes asymptotically to for large even N and to 2 for large odd N This result (which can be rigorously proved by using the Wigner semicircle law for the asymptotic distribution of zeros of the Hermite polynomials [172, 173]) is

12 Standard Coherent State Quantization: the Klauder Berezin Approach Table 121 Values of N = N (Q) N (Q) up to N = 106 Compare with the value of 2 N 10 55 100 551 1000 5555 10 000 55 255 100 000 500 555 1 000 000 N (Q) N (Q) 4713 054 5774 856 5941 534 6173 778 6209 670 6259 760 6267 356 6278 122 6279 776 6282 020 6282 450 6283 185 3 2

important with regard to its physical implications in terms of correlation between small and large distances De ne by N (Q) = 2 M (N ) the size of the universe accessible to exploration by the quantum system N (Q) = m (N ) (resp N (Q) = 2 m (N )) for odd (or even) N, the size of the smallest cell for which exploration is forbidden by the same system, and introduce the product N (Q) N (Q) = N = 4 m (N ) M (N ) for N even, 2 m (N ) M (N ) for N odd, , (1236)

10 4

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then N , as a function of N, is strictly increasing and goes asymptotically to 2 (Table 121) Hence, we can assert the new inequalities concerning the quantum position and momentum: N (Q) N (Q) u 2 , N (P ) N (P ) u 2 N (1237)

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