# Hydrogen in Magnetic Field, Interaction Representation

• Muh. Fauzi M.
In summary, the conversation discusses the use of the relation between the external magnetic field and the spin state, and the resulting differential equations for the spin states. The question arises whether the term for the z-component of the magnetic field should be included in the expression for the Hamiltonian and the importance of considering all time-dependent terms in the equations.
Muh. Fauzi M.
The hydrogen is placed in the external magnetic field:
$$\textbf{B}=\hat{i}B_1 cos(\omega t) + \hat{j} B_2 sin(\omega t) + \hat{k} B_z ,$$

Using the relation ## H = - \frac{e\hbar}{2mc} \mathbf \sigma \cdot \mathbf B ##, then I got the form

$$H = H_0 + H' ,$$

where

$$H'= - \frac{e \hbar}{2 m c} (B_1 cos(\omega t) \sigma_1 + B_2 sin(\omega t) \sigma_2 ) .$$

To find the spin state in the interaction representation, I substitute the ## H' ## in

$$i\hbar\frac{\partial}{\partial t} | \psi \rangle = H' | \psi \rangle ,$$

with ## | \psi \rangle = \begin{pmatrix} a \\ b \end{pmatrix} .##

Next, I tried to solving it, e.g. for ## a ##

$$\frac{\partial^2 a}{{\partial t}^2} + \lambda^2 a = 0 ,$$

with

$$\lambda^2 = \big(\frac{e}{2mc}\big)^2[B_1^2 cos^2(\omega t)+B_2^2(sin^2(\omega t))] .$$

Then I'm heading for the boundary condition.

But then I realized that the ## \lambda^2 ## is depend on ## t ##.

So, my question, is it "ok" to continue to the boundary condition, or there is something wrong with my attempt?

How did you get the differential equation for a? It seems to me you should get two coupled differential equations involving a and b. Is what you show the decoupled result? Also, should you not include the term ##B_z \sigma_3## in your expression for ##H'##?

kuruman said:
How did you get the differential equation for a? It seems to me you should get two coupled differential equations involving a and b. Is what you show the decoupled result? Also, should you not include the term ##B_z \sigma_3## in your expression for ##H'##?
I don't include the ## B_z \sigma_3 ## because it time independent.

I realized that I don't differentiate with time the ##{B_1 cos (\omega t) \mp i B_2 sin (\omega t)}## term in the ##\partial a/\partial t## and ##\partial b/\partial t##.

Here's my result after differentiating it (for ## \partial a/\partial t ##),

$$\frac{\partial^2 a}{{\partial t}^2} = - \frac{e\hbar}{2mc}\Big[ \frac{ie}{2mc} [B_1^2 cos^2{\omega t}+B_2^2 sin^2{\omega t}] a + \frac{2mc}{e\hbar} \omega i \frac{B_1 cos \omega t + i B_2 sin \omega t}{B_1 cos \omega t - i B_2 sin \omega t} \frac{\partial a}{\partial t} \Big] .$$

Muh. Fauzi M. said:
I don't include the Bzσ3 B_z \sigma_3 because it time independent.
So what? Is it not a perturbation to ##H_0##? As long as ##B_z \neq 0##, you cannot ignore it.

kuruman said:
So what? Is it not a perturbation to ##H_0##? As long as ##B_z \neq 0##, you cannot ignore it.

I am using Schwinger-Tomonaga equation, where the Hamiltonian that evolve with time is only the time dependent. CMIIW

update, I have finish this work.

## 1. What is the significance of studying the interaction between hydrogen and magnetic fields?

The interaction between hydrogen and magnetic fields plays a crucial role in various fields such as astrophysics, chemistry, and material science. It allows us to understand the structure and behavior of atoms, molecules, and materials, and can also help in the development of new technologies.

## 2. How does hydrogen behave in the presence of a magnetic field?

In the presence of a magnetic field, hydrogen atoms experience a force due to the interaction between the magnetic field and the magnetic moment of the atom. This force causes the atom to align its magnetic moment with the direction of the magnetic field, resulting in different energy levels for the hydrogen atom.

## 3. What is the concept of interaction representation in the study of hydrogen in magnetic fields?

Interaction representation is a mathematical tool used to study the behavior of a system under the influence of an external field. In the case of hydrogen in a magnetic field, it allows us to separate the interactions between the atom and the magnetic field from the internal dynamics of the atom.

## 4. What are some applications of the study of hydrogen in magnetic fields?

The study of hydrogen in magnetic fields has numerous applications, including the development of magnetic resonance imaging (MRI) techniques in medical imaging, understanding the behavior of atoms in magnetic confinement for nuclear fusion, and studying the magnetic properties of materials for technological advancements.

## 5. What are some challenges in studying the interaction between hydrogen and magnetic fields?

One of the main challenges in studying the interaction between hydrogen and magnetic fields is the complexity of the mathematical models involved. This requires advanced computational techniques and high-level mathematical skills. Additionally, the behavior of hydrogen in magnetic fields can also be affected by external factors such as temperature and pressure, making the study more challenging.

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