
174 Commutation Relations, Normal Ordering, and Stirling Numbers
5.3.3 Expressions Involving Series
In this section, we consider some expressions which involve infinite series. In a slightly
implicit fashion, the coherent states considered above contained some “infinite” aspects;
see Section 5.2.9. One of the most important operators is the exponential function of the
number operator, that is, exp(λˆn)=exp(λˆa
†
ˆa). It is defined by the exponential series,
exp(λˆa
†
ˆa)=
∞
n=0
λ
n
n!
(ˆa
†
ˆa)
n
.
We start with a famous result, which is often attributed to Schwinger [985] (see also [739,
755, 756, 795,904, 905, 1142]), but was derived by McCoy
22
in a different form ...