QMGreensite_merged

156 CHAPTER10. SYMMETRYANDDEGENERACY

Then

TH ̃[

∂

∂x

,x]ψ(x) = H ̃[

∂

∂x′

,x′]ψ(x′)

= H ̃[

∂

∂x

,x]ψ(x′)

= H ̃[

∂

∂x

,x]Tψ(x) (10.16)

Fromthisequation,weseethattheoperatorTcommuteswiththetheHamilitonian
operator
[T,H ̃]= 0 (10.17)

Nownoticethat

TF(x) = F(x+a)

= F(x)+

∂F

∂x

a+

1

2

∂^2 F

∂x^2

a^2 +...

=

∑∞

n=0

an

n!

∂n

∂xn

F(x)

= exp

[

a

∂

∂x

]

F(x)

= exp[iap/ ̃ ̄h]F(x) (10.18)

whichmeansthat T istheexponentialofthe momentumoperator. SinceT com-
muteswithH ̃foranydisplacementa,itfollowsthatthemomentumoperatorpalso
commuteswithH
[p, ̃H ̃]= 0 (10.19)

Thisis easytocheck forthefreeparticle Hamiltonian,since H ̃ = p ̃^2 / 2 m, andp ̃
commuteswithp ̃^2.
InLecture8,wefoundanequationofmotionforexpectationvalues

d

dt

=

i

̄h

<[Q,H]> (10.20)

ItfollowsthatifanHermitianoperatorcommuteswiththeHamiltonian,thecorre-
spondingobservableisconserved:

d

dt

= 0 (10.21)

Inthespecialcaseofthefreeparticle,wethereforehaveconservationofmomentum

d

dt

<p>= 0 (10.22)