1000 Solved Problems in Modern Physics

78 1 Mathematical Physics

(c)<x^2 >=

∑

x^2

e−mmx

x!

=

∑

[x(x−1)+x]

e−mmx

x!

=

∑∞

x= 0

e−mmx

(x−2)!

+

∑∞

x= 0 xe

−mm

x

x!

=e−m

(

m^2 +

m^3

1!

+

m^4

2!

+···

)

+m

=m^2 e−mem+m=m^2 +m

σ^2 =<(x−x ̄)^2 >=<x^2 >− 2 <x>x ̄+< ̄x>^2 =<x^2 >−m^2

σ^2 =morσ=

√

m

(d)Pm− 1 =

e−mmm−^1

(m−1)!

=

e−mmm

(m−1)!m

=

e−mmm

m!

=Pm

That is the probability for the occurrence of the event atx=m−1is

equal to that atx=m

(e)Px− 1 =

e−mmx−^1

(x−1)!

=

e−mmx

x!

x

m

=

x

m

Px

Px+ 1 =

e−mmx+^1

(x+1)!

=

me−mmx

x!(x+1)

=

m

x+ 1

Px

1.94 (a)(q+p)N=qN+NqN−^1 P+

N(N−1)qN−^2

2!

P^2

+···

N!

x!(N−x)!

PxqN−x+···PN

=

∑N

x= 0

N!

x!(N−x)!

PxqN−x=1(∵q+p=1)

(b) We can use the moment generating functionMx(t) about the meanμ

which is given as

Mx(t)=Ee(x−μ)t

=E

[

1 +(x−μ)t+(x−μ)^2

t^2

2!

+···

]

= 1 + 0 +μ 2

t^2

2!

+μ 3

t^3

3!

+···

So thatμnis the coefficient oft

n

n!