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In quantum mechanics, electrons in a hydrogen atom occupy specific energy levels or states, each identified by quantum numbers \( n \), \( l \), and \( m \). When subjected to an external influence, such as an electric field, electrons can jump from one energy level to another in what is known as a transition. For our hydrogen atom initially in its ground state \((n, l, m) = (1,0,0)\), we are particularly interested in transitions to the \(2p\) and \(2s\) states, represented by quantum numbers \((2,1,m)\) with \(m = -1, 0, 1\), and \((2,0,0)\) respectively.
This process involves absorption or emission of energy, corresponding to changes in the electron's energy state. The transitions of interest in this exercise are influenced by a time-dependent electric field, impacting the probability of electrons moving between these states.

When an external electric field is applied to a hydrogen atom, it interacts with the atom, affecting the electron's dynamics. In this scenario, a time-dependent electric field \( \mathbf{E}(t) = \mathbf{E}_0 e^{-t/\tau} \) is applied in the \( z \)-direction.
The interaction with the atomic electron is captured by the perturbation Hamiltonian, \( H'(t) = -e \mathbf{r} \cdot \mathbf{E}(t) \), where \( e \) is the electron charge, and \( \mathbf{r} \) is the electron's position vector. More specifically, the perturbation term simplifies to \( H'(t) = -eE_0 z e^{-t/\tau} \) due to the field's direction.
This perturbation changes the atom's state by providing or absorbing energy, facilitating transitions to higher energy levels. The perturbation depends on both the electron's position and the characteristics of the electric field, such as its strength (\( E_0 \)) and decay rate (\( \tau \)).

In perturbation theory, the likelihood of an electron transitioning from an initial state \(|i\rangle\) to a final state \(|f\rangle\) due to an external influence is described by a probability. This transition probability \(P_{i \to f}\) is given mathematically by:

• The square of the matrix element \( \left| (1/i\hbar) \int_{0}^{\infty} \langle f|H'(t)|i\rangle e^{i\omega_{fi}t} dt \right|^2 \), which includes the overlap between the initial and final states with respect to the perturbation, \( H'(t) \).
• Here, \( \omega_{fi} = (E_f - E_i) / \hbar \) is the angular frequency associated with the energy difference between final and initial states.
• The matrix element \( \langle f|z|i\rangle \) reflects the transition only occurs if \( \Delta l = \pm 1 \), showing non-zero probability for \((2,1,-1)\), \((2,1,0)\), and \((2,1,1)\) states, but zero for \(2s\).

The time integral evaluates to a simple form \( \frac{1}{i\omega_{fi} - 1/\tau} \), especially after time \( t \gg \tau \), helping to precisely estimate transition probabilities.

Perturbation theory provides a structured approach for understanding how quantum systems behave under small disturbances. In this context, it helps calculate the transition probabilities induced by the electric field perturbing the hydrogen atom.
First-order perturbation theory simplifies the challenge of solving complex quantum systems by focusing on the primary order of change in the system caused by the perturbation. It's used here to compute the likelihood of finding the electron in different excited states after the electric field's influence.
By expressing the transition probability \( P_{1 \rightarrow 2, l, m} \) as

• \[ P_{1 \rightarrow 2, l, m} = \left| \frac{-eE_0}{\hbar} \langle 2, l, m|z|1,0,0 \rangle \frac{e^{i\tau\omega_{fi}}}{i\tau\omega_{fi} - 1} \right|^2 \],

we incorporate the electric field's effect and the quantum states' properties for precise calculation. Notably, these calculations highlight the significance of the matrix elements and time integrations over the electron's transition probability during electric field exposure.