Undergrad What does U\dagger H U represent?

[Haorong Wu]

TL;DR

What does $H_I^{'} = U_0^{\dagger} H_I U_0$ represent?

Hi, there. I'm reading a paper about ion trap. There are several Hamiltonians mentioned in the paper.

First, assume $H_0=H_{\text{internal}} + H_{\text{osc}}$, where $H_{\text{internal}}$ is the Hamiltonian for internal states for a ion, and $H_{\text{osc}}$ is the Hamiltonian describing motion of the ion; also, $H_I=V_{\text{interaction}}$ is the interaction Hamiltonian between the ion and a field.

Then, the paper reads, " we have $H \rightarrow H_I^{'} = U_0^{\dagger} H_I U_0$ where $U_0 = exp \left ( -i H_0 t / \hbar \right )$".

I should have read some equations similar to $U_0^{\dagger} H_I U_0$, but I can not rememer what they mean and I can not remember where have I read them.

I think, $U_0$ is a time evolution for the ion, then the ion is transformed by $H_I$ due to a field, and finally it experienced a inverse time evolution because of $U_0^{\dagger}$. But I can not imagine the result for the whole operations.

The paper is “Experimental Issues in Coherent Quantum-State Manipulation of Trapped Atomic Ions“, D. J. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, Journal of Research of the National Institute of Standards and Technology 103, 259 (1998).

Thanks.

 

[Orodruin]

It is a change of picture from the Schrödinger picture to the interaction picture, where both states and the operators of observables depend on time. If the interaction Hamiltonian is zero, then the interaction picture coincides with the Heisenberg picture, where the time evolution is contained in the operators rather than in the states.

See https://en.wikipedia.org/wiki/Dynamical_pictures

 

[Haorong Wu]

It is a change of picture from the Schrödinger picture to the interaction picture, where both states and the operators of observables depend on time. If the interaction Hamiltonian is zero, then the interaction picture coincides with the Heisenberg picture, where the time evolution is contained in the operators rather than in the states.

See https://en.wikipedia.org/wiki/Dynamical_pictures

Thanks, Orodruin. No wonder the paper reads interaction pictures for several times.