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
$$\sum_{n = 0}^\infty \frac{1}{n!} \left( -\frac{i}{h} E_{a'} t \right)^n = e^{-i E_{a'} t / h}$$

 

[Entropia]

I am happy to provide a response to your question about the operator in exponential function and generating function.

Firstly, let's start with the operator in exponential function. In quantum mechanics, operators are mathematical objects that represent physical observables, such as position, momentum, and energy. The exponential function is a mathematical function that can be used to describe the time evolution of a quantum state.

In the provided equation, [A,H] = 0, we see that the commutator of the operator A and the Hamiltonian H is equal to zero. This means that A and H can be measured simultaneously, without any uncertainty. This leads to the second equation, H|a'> = Ea' |a'>, where |a'> is an eigenket of A and Ea' is the corresponding eigenvalue. This equation tells us that the state |a'> is an eigenstate of the Hamiltonian, with an energy eigenvalue of Ea'.

Now, let's move on to the exponential function. In quantum mechanics, the time evolution of a quantum state is given by the Schrodinger equation, which contains the Hamiltonian operator. By applying the operator in exponential form, exp ( -iHt/h), to the state |a'>, we can describe the time evolution of this state. The h in the denominator is the reduced Planck's constant, and it appears due to the quantization of energy in quantum mechanics.

Regarding the generating function, it is a mathematical tool used to describe the transformation of variables in a system. In the case of infinitesimal translation, the generating function is given by F(x,P) = x · P + p · dx. This means that the new position, xnew, is equal to the old position, x, plus an infinitesimal change, dx. Similarly, the new momentum, pnew, remains unchanged. This generating function is useful in understanding the transformation of variables in a quantum system and can be applied to various physical systems.

I hope this explanation helps you understand the concepts of operator in exponential function and generating function. As always, it is important to continue learning and exploring these concepts to gain a deeper understanding. Happy studying!