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# 物理代写|原子物理学代考Atomic Physics代考|PHYS421 Problem definition and some useful approximations

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## 物理代写|原子物理学代考Atomic Physics代考|Problem definition and some useful approximations

The interaction of a hydrogenic atom with static electric or magnetic fields has been extensively discussed in chapter 3: we have learned that their effects basically consist in shifting and/or splitting the energy levels of the unperturbed system. Now we want to extend our knowledge to the effects induced by electromagnetic (e.m.) radiation, namely to the case in which the electric and magnetic fields vary in time.

The most rigorous theoretical framework to address this problem is quantum electrodynamics, where both the atomic system and the radiation field are treated quantum mechanically [1]. This treatment, however, falls far beyond the scope of this primer and, therefore, we will rely on a semi-classical picture according to which the atomic system is described quantum mechanically, while the e.m. radiation is discussed as a classical field fulfilling Maxwell equations [2]. We will also assume that the intensity of the radiation is not too high. This implies that its effects can be considered as perturbations on the spectrum of the isolated atom. More specifically, we will assume that the energy spectrum of the atom remains unaffected by the radiation field, whose net effect will only be to promote electronic transitions between stationary states. These transitions will take place between discrete energy levels and, for simplicity, we will neglect relativistic effects. Despite these approximations, the resulting model will be accurate enough to catch the main features underlying the mechanisms of absorption and emission of light by an atomic system.

## 物理代写|原子物理学代考Atomic Physics代考|Emission and absorption

Let us consider two atomic levels with energy $E_1$ and $E_2$, respectively, described by the hydrogenic wavefunctions $\psi_1$ and $\psi_2$. If we further assume that $E_1<E_2$ we will refer to $\psi_2$ as the ‘excited state’, while $\psi_1$ will be named the ‘ground state’ ${ }^{\prime}$.

We begin our discussion by considering the case of an isolated atom, i.e. the case in which no radiation field is present. If the atom initially occupies the excited state with energy $E_2$, then a spontaneous decay $\psi_2 \rightarrow \psi_1$ is observed, accompanied by the emission of a photon with frequency $\nu_{21}=\left|E_1-E_2\right| / h$. This process is usually referred to as spontaneous emission. The situation is sketched in figure $4.1$ We must duly remark that, strictly speaking, within our semi-classical model such a spontaneous emission should not occur, since $\psi_2$ is a stationary state of the timeindependent Hamiltonian operator describing the atom (see discussion in section 2.3.2). On the other hand, within a quantum electrodynamics treatment it is proved that $\psi_2$ is not a stationary state of the full Hamiltonian operator describing both the atom and the radiation field. This fundamental issue is translated into our semiclassical model by admitting that the excited state-although stationary-may undergo a transition through a photon emission.

On the other hand, whenever the atom is subjected to the action of an e.m. field two different processes may occur: (i) if it initially occupies the ground state, the transition $\psi_1 \rightarrow \psi_2$ is in fact observed, due to the absorption by the atom of a photon with frequency $\nu_{12}=\left|E_2-E_1\right| / h$ subtracted from the radiation bath; (ii) in the opposite situation where the state initially occupied is the excited one, the transition $\psi_2 \rightarrow \psi_1$ is observed accompanied by the emission of a photon with frequency $\nu_{21}=\left|E_1-E_2\right| / h$. The two processes are called stimulated absorption and stimulated emission, respectively. The word ‘stimulated’ means that such processed are activated by the radiation field.

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