QuantumPhysics.dvi

where the Bernoulli numbers are defined by

x

ex− 1

=

∑∞

n=0

Bn

xn

n!

B 2 =−

1

6

, B 4 =−

1

30

,··· (19.99)

Assuming a sharp cutoff, so thatθis a step function, we easily computef(n),

f(n) =

1

6 π

(

ω^3 c−

π^3 n^3

a^3

)

(19.100)

Thus,f(2p−1)(0) = 0 as soon asp >2, whilef(3)(0) =−π^2 /a^3 , and thus we have

E(a)−E 0

L^2

=

π^2

a^3

B 4

4!

=−

π^2

720

̄hc

L^2

a^3

(19.101)

This represents a universal, attractive force proportional to 1/a^4. To make their dependence ex-
plicit, we have restored the factors of ̄handc.