Operator in exponential function and generating function

[ChrisLM]

J.J. Sakurai Modern Quantum Mechanics p. 74
It says,
[A,H] = 0;
H|a'> = E_a' |a'>
where H is the Hamiltonian A is any observable |a'> is eigenket of A

then,

exp ( -iHt/h)|a'> = exp (-iE_a't/h)|a'>

where h is the reduced Planck's constant.
I want to know WHY ?
and besides, I would like to ask what is generating function means?
I have encountered
x_new= x + dx
p_new = p
then generating function for infinitesimal translation is
F(x,P) = x · P + p · dx
How can I understand it ?

Thank You ! I wish this time is readily to read !

 

[CompuChip]

J.J. Sakurai Modern Quantum Mechanics p. 74
It says,
[A,H] = 0;
H|a'> = E_a' |a'>
where H is the Hamiltonian A is any observable |a'> is eigenket of A

then,

exp ( -iHt/h)|a'> = exp (-iE_a't/h)|a'>

where h is the reduced Planck's constant.
I want to know WHY ?

You can use the definition:

$$e^{-iHt/h} = \sum_{n = 0}^\infty \frac{1}{n!} \left( -\frac{i}{h} H t \right)^n$$
Now t and i/h are numbers, so
$$\left( -\frac{i}{h} H t \right)^n = \left( -\frac{i}{h} t \right)^n H^n$$
and you can easily show that
$$H^n |a'\rangle = E_{a'}^n |a'\rangle$$
to get
$$\sum_{n = 0}^\infty \frac{1}{n!} \left( -\frac{i}{h} E_{a'} t \right)^n = e^{-i E_{a'} t / h}$$