# What does U\dagger H U represent?

• I
• Haorong Wu
In summary: The equation for the time evolution of the ion is ##U_0^{\dagger} H_I U_0##. It 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.
Haorong Wu
TL;DR Summary
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.

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 and vanhees71
Orodruin said:
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.

## 1. What is U\dagger H U?

U\dagger H U represents the transformation of a quantum state by a unitary operator U followed by its Hermitian conjugate U\dagger. It is commonly used in quantum mechanics to describe the evolution of a quantum system.

## 2. What does the symbol U\dagger represent?

The symbol U\dagger, also known as U-adjoint, represents the Hermitian conjugate of a unitary operator U. It is obtained by taking the transpose of the complex conjugate of the matrix representation of U.

## 3. Why is U\dagger H U important in quantum mechanics?

U\dagger H U is important because it represents a unitary transformation, which preserves the inner product and the norm of a quantum state. This is a fundamental principle in quantum mechanics and is essential in understanding the behavior of quantum systems.

## 4. How is U\dagger H U related to the time evolution of a quantum system?

U\dagger H U is related to the time evolution of a quantum system through the Schrödinger equation, which describes how a quantum state changes over time. U\dagger H U is used to represent the unitary transformation of a quantum state at a specific time point.

## 5. Can U\dagger H U be used to measure the energy of a quantum system?

No, U\dagger H U cannot be used to measure the energy of a quantum system. It is a mathematical representation of a unitary transformation and does not have a physical interpretation. The energy of a quantum system is typically measured using other operators, such as the Hamiltonian operator.

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