Solve Evolution Operator: <J|U|E>=Exp[-iEt/h]*<J|E>??

[JK423]

[SOLVED] Evolution Operator

If U=Exp[-iHt/h] is the evolution operator, and H the hamiltonian one,
i can't figure out how to derive the following equation.
Let`s call |E> an eigenstate of H, E it`s eigenvalue and |J> a random basis.
Then:
<J|U|E>=Exp[-iEt/h]*<J|E>.

??

 

[tnho]

If U=Exp[-iHt/h] is the evolution operator, and H the hamiltonian one,
i can't figure out how to derive the following equation.
Let`s call |E> an eigenstate of H, E it`s eigenvalue and |J> a random basis.
Then:
<J|U|E>=Exp[-iEt/h]*<J|E>.

??

what is that "*" mean? complex conjugate (cc)?? or just multiplication?

Use

e^{x}=\sum_n x^n/n! and H|E\rangle = E|E\rangle

then you can get that equation. I think that "*" is just a multiplication instead of a cc.

 

[JK423]

Ohh it`s just a multiplication! Sorry for that..
Thanks a lot... I would never have thought of expanding e^{x}!

 

[tnho]

Ohh it`s just a multiplication! Sorry for that..
Thanks a lot... I would never have thought of expanding e^{x}!

It's a very useful trick~try to make good use of it~

 

[nrqed]

Ohh it`s just a multiplication! Sorry for that..
Thanks a lot... I would never have thought of expanding e^{x}!

If you have an operator A and an eigenstate of a with eigenvalue "a" (so A|a> = a |a> ) then any function f(A) which can be Taylor expanded will give

f(A) |a> = f(a) |a>

in other words, applying the function of the operator on an eigenstate gives the function of the eigenvalue.

 

[JK423]

If you have an operator A and an eigenstate of a with eigenvalue "a" (so A|a> = a |a> ) then any function f(A) which can be Taylor expanded will give

f(A) |a> = f(a) |a>

in other words, applying the function of the operator on an eigenstate gives the function of the eigenvalue.

Yeah, i just proved it. Very usefull!
Thanks for ur help!

 

[JK423]

Hello again!
A minor question came up, and i really don't want to make a new thread for it so i post it here.
In quantum mechanics, what`s the definition of the "two-level systems"? I understand that the state vector is in the form |Ψ>=a|1>+b|2>, where |1>,|2> is a basis of the state space.
Then i think of the particle in a box. The energy is quantized (lets say that the possible values are E1 and E2) while position x is continuous. So, in the first case we would have: |Ψ>=a|E1>+b|E2> and in the second one: |Ψ>=Integral(Ψ(x) |x> dx).
So if we use as a basis the eigenstates of the energy, our system would be a "two-level system". However, in {x} representation, we would have an "infinite-level system".

So what`s the definition of a "two-level system" since the number of levels depend on the basis we use?


*EDIT*: If it`s not permitted to ask irrelevant to the "Evolution Operator" questions, pls let me know so that i`ll make a new thread.