# Time Evolution of Hydrogen Atom in a Magnetic Field

• Xyius
In summary, the task at hand involves computing the probability of finding a hydrogen atom in its ground state with spin down after a constant magnetic field is turned on at time t=0. The problem is solved exactly using the Hamiltonian from the Zeeman effect with the terms involving ##\vec{A}^2## or ##\vec{B}^2## neglected. The exact solution for the time evolution of a ket is found using the expansion coefficients ##C_n(t)## and the approximate solution is found using the Dyson series with first order being equal to ##C_n^1##. However, there is a hurdle in the problem as the matrix form of the perturbation term results in uncoupled equations for the expansion
Xyius

## Homework Statement

A hydrogen atom is prepared in its ground state with spin up along the z-direction.
At time t = 0 a constant magnetic field ##\vec{B}##
(pointing in an arbitrary direction determined
by ##\theta## and ##\phi##) is turned on. Neglecting the fine structure and terms proportional to ##\vec{A}^2##
Compute the probability that the atom will be found in the ground state with spin down as
a function of time.
(i) Solve this problem exactly
(ii) Use first order perturbation theory
(iii) Expand your solution in (i) in Taylor series for short times and compare you result
with the one obtained in (ii).
3

## Homework Equations

So to me this seems exactly like the Zeeman effect. So naturally I am using the Hamiltonian from this analysis (with the ##\vec{A}^2## or ##\vec{B}^2## term neglected.)

$$\hat{H}=\frac{\hat{\vec{p}}^2}{2m_e}+V_C(r)-\frac{e}{2m_e c}B(\hat{L}_z+2\hat{S_z})$$

Where ##V_C(r)## is the coulomb potential, and is part of the unperturbed Hamiltonian.

The exact solution for the time evolution of a ket of the form..

$$|\alpha,t_0,t>_I=\sum_n C_n(t)|n>$$

is found by finding the expansion coefficients ##C_n(t)## using..

$$i\hbar\frac{\partial}{\partial t}C_n(t)=\sum_m e^{i(E_n-E_m)/\hbar}V_{nm}(t)C_m(t)$$

And the approximate form is found by the Dyson series, with first order being equal to..
$$C_n^1=-\frac{i}{\hbar}\int_{t_0}^{t} e^{i(E_n-E_i)\\hbar}V_{ni}(t')dt'$$

## The Attempt at a Solution

There is one major hurdle that is stopping me from making any progress on this problem, and that is the fact that if I write the perturbation term in its matrix form, from the Zeeman effect, the elements are equal to..

$$-\frac{e\hbar Bm}{2m_e c}\left[ 1 \pm \frac{1}{(2l+1)} \right]$$

Where this matrix is diagonal in the |n,l,j,m> basis. Thus, when I apply the third equation (the exact solution for the expansion coefficients) from section 2 above, I get uncoupled equations for the expansion coefficients, meaning a diagonal matrix, meaning, when I try to find any transition probabilities, I get zero from orthogonality!

I have realized that this comes from the fact that my hamiltonian is no different than the Zeeman effect and goes not depend on time. I feel like this has to be my source of error.

Where am I going wrong??

Okay so I realized that the magnetic field is not only in the Z direction as the Zeeman effect was. So now I redid the problem and got this as the perturbation.

$$-\frac{eB}{2m_e c}\left[ \hat{L}_y-\hat{L}_x-\hat{L}_z-2\hat{S}_z \right]$$

I am still having problems because he doesn't specify what n has to be. I am assuming the electron is bound and can only go from spin up to spin down, making this a two state problem. Is this assumption wrong? Because I still get a diagonal matrix even with the above Hamiltonian. I can scan my work if anyone needs to see it.

## 1. How does a magnetic field affect the time evolution of a hydrogen atom?

A magnetic field can cause the energy levels of a hydrogen atom to split, leading to different possible transitions between energy states. This results in a more complex time evolution compared to the case without a magnetic field.

## 2. What is the role of the magnetic field strength in the time evolution of a hydrogen atom?

The strength of the magnetic field determines the amount of energy splitting and thus affects the frequency and intensity of the transitions between energy states. A higher magnetic field strength leads to a more pronounced effect on the time evolution of the hydrogen atom.

## 3. How does the time evolution of a hydrogen atom in a magnetic field differ from that without a magnetic field?

Without a magnetic field, the energy levels of a hydrogen atom are degenerate, meaning that multiple energy states have the same energy. In the presence of a magnetic field, this degeneracy is lifted and the energy levels split, resulting in a more complex time evolution with additional possible transitions.

## 4. What is the significance of the time evolution of a hydrogen atom in a magnetic field?

The time evolution of a hydrogen atom in a magnetic field is significant because it helps us understand the behavior of atoms in a more complex physical environment. This has practical applications in fields such as quantum computing and magnetic resonance imaging (MRI).

## 5. How does the time evolution of a hydrogen atom in a magnetic field relate to quantum mechanics?

The time evolution of a hydrogen atom in a magnetic field is a manifestation of quantum mechanics, specifically the concept of energy quantization. The splitting of energy levels and the resulting transitions between states can only be explained by the principles of quantum mechanics.

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