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Commutation Relations, Normal Ordering, and Stirling Numbers
book

Commutation Relations, Normal Ordering, and Stirling Numbers

by Toufik Mansour, Matthias Schork
September 2015
Intermediate to advanced content levelIntermediate to advanced
528 pages
19h 34m
English
Chapman and Hall/CRC
Content preview from Commutation Relations, Normal Ordering, and Stirling Numbers
The Weyl Algebra, Quantum Theory, and Normal Ordering 169
This follows also from the operator relation [756, 904, 905]
n
k
:= ˆnn 1) ···n k +1)=(ˆn)
k
, (5.49)
which we recognize in the representation ˆn → XD = x
d
dx
as Boole’s relation (1.13). Recall
from (1.37) that Katriel showed
ˆn
m
=
m
k=1
S(m, k)(ˆa
)
k
ˆa
k
=
m
k=1
S(m, k):ˆn
k
:, (5.50)
implying due to n|ˆn
m
|n = n
m
and (5.48) that n
m
=
k
S(m, k)(n)
k
, which is precisely
(1.3). See [731] for an early physical application of (5.50). The antinormal ordering analog
of (5.49) is given by
.
.
n
k
.
.
.=(ˆn +1)···n + k)=ˆn +1
k
,
implying as analog of (5.48) the relation n|
.
.
n
k
.
.
.|n = n +1
k
. In [453],
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Publisher Resources

ISBN: 9781466579897