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Consider any two internal states, \(s_{1}\) and \(s_{2},\) of an atom. Let \(s_{2}\) be the higher-energy state, so that \(E\left(s_{2}\right)-E\left(s_{1}\right)=\epsilon\) for some positive constant \(\epsilon\) If the atom is currently in state \(s_{2}\), then there is a certain probability per unit time for it to spontaneously decay down to state \(s_{1}\), emitting a photon with energy \(\epsilon\) This probability per unit time is called the Einstein \(A\) coefficient: \(A=\) probability of spontaneous decay per unit time. On the other hand, if the atom is currently in state \(s_{1}\) and we shine light on it with frequency \(f=\epsilon / h,\) then there is a chance that it will absorb a photon, jumping into state \(s_{2}\). The probability for this to occur is proportional not only to the amount of time elapsed but also to the intensity of the light, or more precisely, the energy density of the light per unit frequency, \(u(f)\). (This is the function which, when integrated over any frequency interval, gives the energy per unit volume within that frequency interval. For our atomic transition, all that matters is the value of \(u(f) \text { at } f=\epsilon / h .)\) The probability of absorbing a photon, per unit time per unit intensity, is called the Einstein \(B\) coefficient: \(B=\frac{\text { probability of absorption per unit time }}{u(f)}\) Finally, it is also possible for the atom to make a stimulated transition from \(s_{2}\) down to \(s_{1}\), again with a probability that is proportional to the intensity of light at frequency \(f\). (Stimulated emission is the fundamental mechanism of the laser: Light Amplification by Stimulated Emission of Radiation.) Thus we define a third coefficient, \(B^{\prime},\) that is analogous to \(B\) : \(B^{\prime}=\frac{\text { probability of stimulated emission per unit time }}{u(f)}\) As Einstein showed in 1917 , knowing any one of these three coefficients is as good as knowing them all. (a) Imagine a collection of many of these atoms, such that \(N_{1}\) of them are in state \(s_{1}\) and \(N_{2}\) are in state \(s_{2} .\) Write down a formula for \(d N_{1} / d t\) in terms of \(A, B, B^{\prime}, N_{1}, N_{2},\) and \(u(f)\) (b) Einstein's trick is to imagine that these atoms are bathed in thermal radiation, so that \(u(f)\) is the Planck spectral function. At equilibrium, \(N_{1}\) and \(N_{2}\) should be constant in time, with their ratio given by a simple Boltzmann factor. Show, then, that the coefficients must be related by $$B^{\prime}=B \quad \text { and } \quad \frac{A}{B}=\frac{8 \pi h f^{3}}{c^{3}}$$

Step by step solution

01

Understand the Atoms' Transition Processes

Atoms can transition between states due to spontaneous emission, stimulated emission, and absorption. Spontaneous emission from state \(s_2\) to \(s_1\) occurs at a rate \(A N_2\). Absorption can transition atoms from \(s_1\) to \(s_2\) at a rate \(B N_1 u(f)\). Stimulated emission from \(s_2\) to \(s_1\) occurs at a rate \(B' N_2 u(f)\) and is driven by external radiation.

02

Write the Formula for Change in Population

The rate of change of the number of atoms in state \(s_1\), \(\frac{d N_1}{dt}\), is due to spontaneous and stimulated transitions from \(s_2\) to \(s_1\), and absorption from \(s_1\) to \(s_2\). It is given by the equation:\[\frac{d N_1}{dt} = A N_2 + B' N_2 u(f) - B N_1 u(f)\]

03

Consider Equilibrium Conditions

At equilibrium, \(\frac{d N_1}{dt} = 0\), meaning the transitions between states balance out. In this situation, \(N_1\) and \(N_2\) remain constant. Division by \(N_1\) ensures the ratios involved are included in terms of the Boltzmann factor \(e^{-\epsilon/kT}\).

04

Use the Planck's Law for Energy Density

At thermal equilibrium, the spectral energy density \(u(f)\) is given by Planck's distribution: \[u(f) = \frac{8 \pi h f^3}{c^3 (e^{hf/kT} -1)}\] Using this in the equilibrium condition provides expressions to equate spontaneous and stimulated processes.

05

Relate Coefficients Using Boltzmann Distribution

At equilibrium, \(\frac{N_2}{N_1} = e^{-\epsilon/kT}\) where \(\epsilon = hf\). Substituting this into the balance of rates and using \(u(f)\) from Step 4:\[A = B' \frac{8 \pi h f^3}{c^3} \quad \text{and} \quad \frac{B'}{B} = 1\]Thus, these simplify to \(B' = B\) and \(\frac{A}{B} = \frac{8 \pi h f^3}{c^3}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Spontaneous Emission

Spontaneous emission is a fundamental process in which an atom in a higher-energy state, like state \(s_2\), randomly drops to a lower energy state, such as \(s_1\), without external influence. During this transition, it emits a photon, the energy of which equals the energy difference \(\epsilon\) between the two states. This emission process is intrinsic to the atom itself and occurs naturally, without any external prompting.
To quantify spontaneous emission, we use the Einstein \(A\) coefficient. This coefficient represents the probability per unit time that the atom will make this transition. The relaxation of this atom from the excited to the ground state releases energy in the form of electromagnetic radiation.

• It leads to photon emission.
• Its probability is intrinsic and constant for a given atom.
• Occurs independently of external light or radiation fields.

Understanding spontaneous emission is crucial for comprehending how atoms lose energy naturally, reverting to more stable states.

Stimulated Emission

Stimulated emission occurs when an atom in an excited state, \(s_2\), makes a transition to a lower energy state, \(s_1\), due to the presence of external electromagnetic radiation of the correct frequency. This process not only drops the energy state of the atom but also emits an additional photon, which is coherent with the incident light.
This is where the Einstein \(B'\) coefficient becomes vital. It expresses the probability of this stimulated transition occurring per unit time and per unit energy density \(u(f)\) at the frequency that matches the energy difference \(\epsilon\). Stimulated emission is crucial in devices like lasers, where light is amplified by the stimulated emission of radiation.

• Requires external radiation to occur.
• Leads to coherent photon emission.
• Crucial for laser operation.

In essence, stimulated emission is about using existing light to enhance its own intensity and produce identical photons.

Atomic Transitions

Atomic transitions refer to the process of atoms changing from one energy state to another, such as from \(s_1\) to \(s_2\) or vice versa. These transitions involve the absorption or emission of photons and are the basis for all spectroscopic techniques.
In addition to spontaneous and stimulated emission, atoms can also undergo absorption. When light of a specific frequency shines on an atom in a lower energy state \(s_1\), it can absorb a photon and transition to an excited state \(s_2\). This phenomenon is quantified by the Einstein \(B\) coefficient, representing the probability per unit time and per unit energy density that the atom will absorb a photon.

• Involves photon absorption and emission.
• Distinct probabilities characterized by A, B, and B' coefficients.
• Defines the fundamental mechanisms of spectroscopy.

By studying atomic transitions, we gain insights into the energy levels within an atom, contributing to our understanding of atomic structure and the behavior of electrons.

Planck Spectral Function

The Planck spectral function describes the distribution of electromagnetic radiation as a function of frequency, particularly in thermal equilibrium. It's foundational in quantum physics, providing insight into blackbody radiation's spectral density and is instrumental in defining energy density \(u(f)\) in many processes, including atomic transitions.
At thermal equilibrium, the spectral energy density is given by:\[ u(f) = \frac{8 \pi h f^3}{c^3 (e^{hf/kT} -1)} \]This describes how energy is distributed across different frequencies at a given temperature \(T\). It plays a crucial role in determining the energy available for stimulated processes in atoms.

• Critical for understanding thermal radiation.
• Links to energy density crucial for Einstein coefficients.
• Explains radiation as a function of frequency and temperature.

Understanding the Planck spectral function allows us to appreciate how light interacts with matter at different temperatures, bridging classical and quantum physics.