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Evolution Operator

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JK423
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[SOLVED] Evolution Operator

If U=Exp[-iHt/h] is the evolution operator, and H the hamiltonian one,
i cant 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>.

??
 

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If U=Exp[-iHt/h] is the evolution operator, and H the hamiltonian one,
i cant 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

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

then you can get that equation. I think that "*" is just a multiplication instead of a cc. :smile:
 
JK423
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Ohh it`s just a multiplication!! Sorry for that.. :rolleyes:
Thanks a lot... I would never have thought of expanding [tex]e^{x}[/tex]!
 
31
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Ohh it`s just a multiplication!! Sorry for that.. :rolleyes:
Thanks a lot... I would never have thought of expanding [tex]e^{x}[/tex]!
It's a very useful trick~try to make good use of it~
 
nrqed
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Ohh it`s just a multiplication!! Sorry for that.. :rolleyes:
Thanks a lot... I would never have thought of expanding [tex]e^{x}[/tex]!
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
Gold Member
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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
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Hello again!
A minor question came up, and i really dont 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.
 
Last edited:

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