About the Schrodinger equation, the Hamiltonian, time evolution?

[jeebs]

Forgive me if this is a poorly asked question but I am not yet completely fluent in quantum mechanics and was just looking at the energy eigenvalue equation $$H|\Psi\rangle = i\hbar \frac{\partial}{\partial t}|\Psi\rangle = E|\Psi\rangle$$.
We've got the Hamiltonian operator H acting on the state vector $$|\Psi\rangle$$ and it gives you the eigenvalue E which is what we can measure in experiments. That much is fair enough.
But, it also says that when you apply the operator, you are doing a time evolution on the state $$|\Psi\rangle$$. I am struggling to see what the link is between energy and time evolution. I know I'm not asking a very precise question here, but what is the significance of this?
What does it mean to equate energy with time evolution? For that matter, what does it mean to equation momentum p with $$i\hbar\frac{\partial}{\partial x}$$,
("space evolution" I suppose you might call that?)
This feels like something basic I should be aware of by now.

 

[Fredrik]

If f is the state vector you use to represent the state of the system right now, exp(-iHt)f is the state vector you have to use to represent the state of the system a time t later. Alternatively, you can think of exp(-iHt)f as the state vector you would had to use right now if the system had been put through the same preparation procedure a time t earlier.

Similarly, exp(ipb)f is the state vector you would have had to use if the system had been put through the same preparation procedure a distance b from where it was actually prepared.

exp(ipb) is called a translation operator, and exp(-iHt) a time translation operator. H (the Hamiltonian) is said to be the generator of translations in stime, and p (momentum) is said to be the generator of translations in space.

A good exercise for you is to write the wavefunction that appears in the Schrödinger equation in the form $\psi(x,t)=u(x)T(t)$, and solve for u and T separately. (You should get one equation for u and one for T). Then you can show that

$$\psi(x,t)=e^{-iHt}\psi(x,0)$$

 

[The_Duck]

This is some cool stuff that I don't think is usually covered in an intro QM course. Guess what dynamical variable is associated with "angular evolution" (rotation)?

Ballentine's book on QM discusses this early on. I think Shankar also has some stuff on it, including the analogous relations in classical mechanics.