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Explicit form of time evolution operator 
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#1
Dec1610, 08:30 AM

P: 8

The time evolution operaton may be written formally as:
This is an actual solution to: only in the case that [H(t_{1}),H(t_{2})]=0 (that is: the hamiltonian commutes in different instants of time) Of course, this includes the case of a time independent hamiltonian. If this is not the case, the actual U(t) may be obtained with the Dyson series. This is what is stated in Sakurai, for example. But I don't really understand WHY you cannot use the exponential form of U(t) in case the Hamiltonian doesn't commute at different times. I guess it may have something to do with its integrability, but I don't know. Any hints? 


#2
Dec1610, 08:55 AM

Sci Advisor
P: 3,593

Well, you may try it out. Assume that you have H(t)=H_1 for 0<t<t_1 and H(t)=H_2 for t_1< t<t_2. For t>t_2, the time evoltion operator will be a product of two exponentials of the form you gave. Can you write it as a single exponential of an integral if H_1 and H_2 don't commute?



#3
Dec1610, 08:58 AM

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P: 3,593

Btw. there are other representations than the Dyson series, e.g. the Magnus expansion.



#4
Dec1610, 09:17 AM

P: 8

Explicit form of time evolution operator
No, of course you can't because the product of exponentials of two operators can't be written as the exponential of the sum unless the operators commute!
Thank you! 


#5
Dec1610, 10:24 AM

Sci Advisor
P: 1,943

the rsult, but they are in terms of infinite series, and not very useful. 


#6
Dec1610, 10:28 AM

Sci Advisor
P: 3,593

Of course you can write it as a single exponential, but certainly not as the exponential of an integral over H(t).



#7
Dec1610, 10:40 AM

P: 527

The time evolution operator for Hamiltonians which are not constant in time is
[tex] U(t,t_0) = \mathcal{T}\exp\left(\frac{i}{\hbar}\int_{t_0}^t dt' H(t')\right)[/tex] where [tex]\mathcal{T}[/tex] is the time ordering symbol. This exponential is just a shortway of writing the following power series: [tex]U(t,t_0) = 1 + \sum_{n=1}^{\infty} \frac{1}{n!}\left(\frac{i}{\hbar}\right)^n} \int dt_1\cdtos dt_n \mathcal{T}\left[H(t_1)\cdots H(t_n)\right][/tex] This series is what's actually the time evolution operator. You can look it up in e.g. the books by Rammer, Mahan or Fetter and Walecka. Again, the proof relies on the fact that this power series is a unitary solution to the Schrodinger equation. 


#8
Dec1610, 10:43 AM

P: 8

I think DrDu is right, when you integrate you are sort of adding hamiltonians at different t's, and then the exponential form doesn't hold anymore unless they all commute.



#9
Dec1610, 10:57 AM

P: 527




#10
Dec1610, 11:01 AM

P: 8

Right xepma :) Thanks to all of you!



#11
Dec1610, 01:47 PM

Sci Advisor
P: 3,593

xempa, yes, you can write it with time ordering, but this is nothing else than the Dyson series bluesunday mentioned in the first post.



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