From unitary operator to hamiltonian

[KFC]

Hi there,
If the evolution operator is given as follows
$$U(t) = \exp[-i (f(p, t) + g(x))/\hbar]$$

where p is momentum, t is time. Can I conclude that the Hamiltonian is

$$H(t) = f(p, t) + g(x)$$

if no, why?

 

[jfy4]

Well, since you have the hbar in there, that superposition of functions has units of action, so it can't be the hamiltonian, right. Maybe an idea is to taylor expand to linear order in $t$.
$$U=\exp (-i (f(p,t)+g(x))/\hbar )\approx \exp (-i(f(p,0)+\dot{f}(p,0)t+g(x))/\hbar )$$
This kind of looks like the normal form
$$U=e^{-iHt/\hbar}$$
Now you have something next to $t$ and divided by $\hbar$, which at least has to have units of energy. What do you think?

 

[KFC]

Thanks jfy4. So based on your math, $$H=\dot{f}(p, 0)$$, right? Well, but if the Hamiltonian is time-dependent, should the corresponding evolution operator be
$$U=e^{-i\int Hdt/\hbar}$$

If that's true, how do we find the Hamiltonian if there should be integral in the exponential ?

Well, since you have the hbar in there, that superposition of functions has units of action, so it can't be the hamiltonian, right. Maybe an idea is to taylor expand to linear order in $t$.
$$U=\exp (-i (f(p,t)+g(x))/\hbar )\approx \exp (-i(f(p,0)+\dot{f}(p,0)t+g(x))/\hbar )$$
This kind of looks like the normal form
$$U=e^{-iHt/\hbar}$$
Now you have something next to $t$ and divided by $\hbar$, which at least has to have units of energy. What do you think?

 

[jfy4]

I think it is usually quite difficult to set up a $U$ with a time-dependent Hamiltonian, and I agree with the expression you wrote for a time-dependent Hamiltonian, however, assuming what I wrote above in correct ( and that's a big assumption), the hamiltonian is time-independent, since it is
$$H=\frac{\partial f(p,t)}{\partial t}\bigg|_{t=0}$$
thus no integral expression is required.

 

[Jano L.]

Hello KFC,

I'll try to explain the relation between U and H as I see it.

The first notion is the Hamiltonian operator $\hat H$, which plays role in the equation

$$\partial_t \psi = -\frac{i}{\hbar} {\hat H}(t) \psi.$$


There is an alternative description in terms of an evolution operator. The operator $\hat U(t,t_0)$ is called an evolution operator, if it changes the function at time $t_0$ to a function at later time t:

$$\psi(t) = \hat U(t,t_0) \psi(t_0).$$

The evolution operator obeys the equation

$$\frac{\partial \hat U}{\partial t}(t) = \hat H(t) \hat U(t) ~~~(*)$$

In case the Hamiltonian is time-independent, Schroedinger's equation gives $\hat U(t,t_0) = e^{-i\hat H (t-t_0)/\hbar}$.

In case the Hamiltonian is time-dependent, there is no simple formula but there is perturbative series called Dyson series: http://en.wikipedia.org/wiki/Dyson_series


In your case, you seek Hamiltonian from known U. There is, as far as I know, no simple way to proceed. You can try to guess correct form of H that will recover the equation (*); if you succeed, the expression in front of U is your Hamiltonian.

Warning: the derivative

$$\frac{\partial U}{\partial t} \neq -i/\hbar (\dot f(p,t)) \hat U(t),$$

except for case when operators f(p,t) + g(x) at different times commute.

 

[KFC]

Thanks Jano. It clears my doubt about the Hamiltonian and evolution operator. Actually, I found that evolution in some online materials, where they give the evolution operator first and then directly write out the corresponding Hamiltonian. But I didn't see any way to write the Hamiltonian from that evolution operator mathematically. I think they gave it, as you says, by guessing.

Hello KFC,

I'll try to explain the relation between U and H as I see it.

The first notion is the Hamiltonian operator $\hat H$, which plays role in the equation

$$\partial_t \psi = -\frac{i}{\hbar} {\hat H}(t) \psi.$$There is an alternative description in terms of an evolution operator. The operator $\hat U(t,t_0)$ is called an evolution operator, if it changes the function at time $t_0$ to a function at later time t:

$$\psi(t) = \hat U(t,t_0) \psi(t_0).$$

The evolution operator obeys the equation

$$\frac{\partial \hat U}{\partial t}(t) = \hat H(t) \hat U(t) ~~~(*)$$

In case the Hamiltonian is time-independent, Schroedinger's equation gives $\hat U(t,t_0) = e^{-i\hat H (t-t_0)/\hbar}$.

In case the Hamiltonian is time-dependent, there is no simple formula but there is perturbative series called Dyson series: http://en.wikipedia.org/wiki/Dyson_seriesIn your case, you seek Hamiltonian from known U. There is, as far as I know, no simple way to proceed. You can try to guess correct form of H that will recover the equation (*); if you succeed, the expression in front of U is your Hamiltonian.

Warning: the derivative

$$\frac{\partial U}{\partial t} \neq -i/\hbar (\dot f(p,t)) \hat U(t),$$

except for case when operators f(p,t) + g(x) at different times commute.

 

[Jano L.]

No problem. Can you post a link? I am curious what result they found for H.

 

[KFC]

No problem. Can you post a link? I am curious what result they found for H.

Yes, I send you the link via message. Check it out :)