The product of the eigenvalues may first be defined up to a leveln≤N,

∏N

n=1

λn=

∏N

n=1

[

mπ^2 n^2

T^2

(

1 −

ω^2 T^2

π^2 n^2

)]

(15.9)

Clearly, the first factor in the product is independent ofω, so we may write

∏N

n=1

λn=NN

∏N

n=1

(

1 −

ω^2 T^2

π^2 n^2

)

NN=

∏N

n=1

(

mπ^2 n^2

T^2

)

(15.10)

TheN→ ∞limit of the infinite product involvingωis convergent, and is related to the infinite
product representation of the sinxfunction, which is given by

sinx=x

∏∞

n=1

(

1 −

π^2 x^2

n^2

)

(15.11)

Thus we have

∏N

n=1

(

1 −

ω^2 T^2

π^2 n^2

)

=

sinωT

ωT

(15.12)

TheN→∞limit of the productNN, however, does not converge. This divergence results from the
fact that a precise calculation of the absolute overall normalization of the path integral would have
required more care than we have applied here. But the most important property ofNNis that it is
independent ofωfor all values ofN. Instead of attempting to calculate this proper normalization
from first principles, we shall leave the overall normalization undetermined, and represent it by
a multiplicative factorN which is independent ofω. We shall then fixNby matching with the
known case of the free particle corresponding toω= 0. Putting all together, we have,

∏∞

n=1

λn=N

sinωT

ωT

(15.13)

whereN is independent ofω. Assembling the entire path integral representation of thematrix
elements of the evolution operator, and matching the overall normalization with that of the free
particle forω= 0, given in (14.17), we findN= 2πi ̄h/m, and thus,

〈qb|U(tb−ta)|qa〉=

(

mω

2 πi ̄hsinωT

) (^12)
exp
{
imω
2 ̄hsinωT
[
(q^2 a+q^2 b) cosωT− 2 qaqb
]}
(15.14)
It is good practice to double check this result in the following way. We compute the trace of the
evolution operators by integrating overq=qa=qb,
Tr
(
U(T)
)

∫
dq〈q|U(T)|q〉

=

∫

dq

(

mω

2 πi ̄hsinωT

) (^12)
exp
{
imωq^2
̄hsinωT
(cosωT−1)
}

=

−i

2 sinωT/ 2

=

e−iωT/^2

1 −e−iωT

(15.15)