Dipole Matrix Elements

Dipole Matrix Elements: Overview and Plan

In the previous sections, we established the relations between the Einstein coefficients by demanding thermal equilibrium and consistency with Planck’s radiation law. These relations are fundamental, but they do not yet tell us the absolute values of the coefficients for a given atomic transition. To make quantitative predictions — e.g. lifetimes of excited states or absorption cross sections — we need explicit formulas for \(A_{21}\), \(B_{12}\), and \(B_{21}\).

This requires us to connect the statistical picture of Einstein with the microscopic dynamics of atoms interacting with the electromagnetic field. The key idea is that the coupling between light and atoms is governed by the electric dipole interaction, and the corresponding matrix elements of the dipole operator determine the transition probabilities.

In this chapter you will learn how to:

1. Write down the Hamiltonian that describes how atoms interact with light.
   
2. Use time-dependent perturbation theory to understand Fermi’s Golden Rule, which tells us the probability for an atom to jump between stationary states.
   
3. Apply the electric dipole approximation and see how transition strengths are determined by the dipole matrix element.
   
4. Connect these results to the Einstein coefficients and work out explicit formulas for \(A_{21}\) and the \(B\)-coefficients in terms of dipole matrix elements.

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In the following sections, we will carry out this program step by step, starting with the derivation of the dipole transition matrix elements and their role in determining the Einstein \(A\) and \(B\) coefficients.

1. Hamiltonian for an Atom in an Electromagnetic Field

To derive the interaction between an atom and electromagnetic radiation, we begin with the Hamiltonian for a charged particle of mass \(m\) and charge \(e\) in the presence of an electromagnetic field. In the minimal-coupling scheme, the canonical momentum \(\hat{\vec{p}}\) is replaced by the generalized momentum

\[ \hat{\vec{p}} \;\;\longrightarrow\;\; \hat{\vec{p}} - e\,\vec{A} , \]

where \(\vec{A}\) is the vector potential. The Hamiltonian then reads

\[ \hat{H} \;=\; \frac{1}{2m}\left( \hat{\vec{p}} - e \vec{A} \right)^2 \;+\; V(\vec{r}), \]

with \(V(\vec{r})\) the Coulomb potential that binds the electron in the atom.

NoteMore on the origin of the generalized momentum

In classical mechanics, the interaction with the field enters naturally through the Lagrangian formalism. There, the canonical momentum is defined as

\[ \vec{p} = \frac{\partial L}{\partial \dot{\vec{r}}}. \]

For a charged particle in an electromagnetic field, the Lagrangian contains not only the kinetic energy but also terms involving the scalar potential \(\Phi\) and the vector potential \(\vec{A}\). As a result, the canonical momentum differs from the mechanical momentum \(m\dot{\vec{r}}\):

\[ \vec{p} = m\dot{\vec{r}} - e\,\vec{A}. \]

This motivates the minimal coupling rule in quantum mechanics, where the canonical momentum operator \(\hat{\vec{p}}\) is replaced by the so-called generalized momentum:

\[ \hat{\vec{p}} \;\;\longrightarrow\;\; \hat{\vec{p}} - e\,\vec{A}. \]

• The minimal coupling rule is not arbitrary: it ensures that the quantum Hamiltonian remains consistent with classical electrodynamics.
  
• If one computes the equations of motion using this Hamiltonian, one recovers the Lorentz force law for a charged particle in an electromagnetic field.
  
• In this sense, minimal coupling provides the simplest and most natural way to introduce the electromagnetic interaction while preserving both gauge invariance and correspondence with classical physics.

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1.1 Simplifications

• Weak-field limit:
  For optical fields in the perturbative regime, the quadratic term in the vector potential can be neglected, i.e. \[ \vec{A}^2 \approx 0. \]

• Coulomb gauge:
  We choose the gauge condition \(\nabla \cdot \vec{A} = 0\).
  In this gauge, the scalar potential vanishes and the interaction Hamiltonian reduces to a simple form.

With these assumptions, the interaction part of the Hamiltonian becomes

\[ \hat{V}(t) \;=\; -\frac{e}{m}\, \vec{A}(t)\cdot \hat{\vec{p}} . \]

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1.2 Time-dependent perturbation

We now consider a monochromatic radiation field with angular frequency \(\omega\) and polarization vector \(\vec{\varepsilon}\). In a quantum mechanical treatment, it is common to model this field within a quantization volume \(V\) (e.g., a box with periodic boundary conditions). This leads to the following form for the vector potential:

\[ \vec{A}(t) \;=\; \sqrt{\frac{\hbar}{2\varepsilon_0 \omega V}}\,\vec{\varepsilon}\, e^{-i(\omega t - \vec{k}\cdot\vec{r})} + \text{c.c.} \]

The first term, with the negative exponent in the time-dependent phase, corresponds to the absorption of a photon, as its frequency term is \(−i\omega t\). The complex conjugate (c.c.) term, with a positive exponent, is responsible for stimulated emission. The coefficient is derived from quantizing the electromagnetic field and relates the classical amplitude to the energy of a single photon, \(\hbar \omega\).

This vector potential is then used to construct the interaction Hamiltonian, which describes how the radiation field perturbs an atomic system:

\[ \hat{V}(t) \;=\; - \frac{e}{m}\, \sqrt{\frac{\hbar}{2\varepsilon_0 \omega V}}\, e^{-i(\omega t - \vec{k}\cdot\vec{r})}\, \vec{\varepsilon}\cdot \hat{\vec{p}}. \tag{1}\]

This result highlights two key features:

1. The perturbation couples the electromagnetic field polarization to the electron momentum operator.
   
2. The quantization of the field ensures that such an interaction exists even in the vacuum, reflecting the presence of vacuum fluctuations of the electromagnetic field.

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In the next step, we will see how to handle the Schrödinger equation when the Hamiltonian contains a time-dependent perturbation. This will give us the tools to calculate transition amplitudes and probabilities.

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2. Time-Dependent Perturbation Theory

We now want to connect the interaction Hamiltonian with transition probabilities between atomic states. The appropriate framework is time-dependent perturbation theory, where the Hamiltonian is written as

\[ \hat{H}(t) = \hat{H}_0 + \hat{V}(t), \]

with \(\hat{H}_0\) the unperturbed Hamiltonian and \(\hat{V}(t)\) a weak, time-dependent perturbation.

The stationary eigenstates of \(\hat{H}_0\) are defined by

\[ \hat{H}_0 \, \ket{\phi_n} = E_n \, \ket{\phi_n}. \]

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2.1 Ansatz for the time-dependent wavefunction

We expand the general solution of the time-dependent Schrödinger equation (TDSE) in terms of the eigenstates of \(\hat{H}_0\):

\[ |\Psi(t)\rangle = \sum_k c_k(t)\, e^{-i\omega_k t}\, \ket{\phi_k}, \]

where \(\omega_k = E_k/\hbar\). The question is: how do the expansion coefficients \(c_k(t)\) evolve in time under the perturbation \(\hat{V}(t)\)?

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2.2 Equation of motion for the coefficients

Inserting this ansatz into the TDSE,

\[ i\hbar \frac{\partial}{\partial t} |\Psi(t)\rangle = \hat{H}(t)|\Psi(t)\rangle, \]

we get

\[ i\hbar \sum_k \dot{c}_k(t)\,|\phi_k\rangle e^{-i\omega_k t} = \sum_k c_k(t)\, \hat{V}(t)\,|\phi_k\rangle e^{-i\omega_k t}. \]

NoteIntermediate steps

Substituting the ansatz

\[ |\Psi(t)\rangle = \sum_k c_k(t)\,|\phi_k\rangle e^{-i\omega_k t} \]

gives

\[ i\hbar \frac{\partial}{\partial t} \left( \sum_k c_k(t)\,|\phi_k\rangle e^{-i\omega_k t} \right) = \left( \hat{H}_0 + \hat{V}(t) \right) \sum_k c_k(t)\,|\phi_k\rangle e^{-i\omega_k t}. \]

Since \(\hat{H}_0|\phi_k\rangle = E_k|\phi_k\rangle = \hbar\omega_k|\phi_k\rangle\), the \(\hat{H}_0\) terms cancel with the derivative of \(e^{-i\omega_k t}\).

We are left with

\[ i\hbar \sum_k \dot{c}_k(t)\,|\phi_k\rangle e^{-i\omega_k t} = \sum_k c_k(t)\, \hat{V}(t)\,|\phi_k\rangle e^{-i\omega_k t}. \]

Projecting this onto \(\langle \phi_n|\), we obtain the coupled equations

\[ i\hbar \frac{\partial c_n(t)}{\partial t} = \sum_k \langle \phi_n | \hat{V}(t) | \phi_k \rangle \, c_k(t)\, e^{i(\omega_n - \omega_k)t}. \]

This equation shows explicitly how amplitudes \(c_n(t)\) get “fed” from other states \(k\) through the perturbation matrix elements \(\langle \phi_n|\hat{V}(t)|\phi_k\rangle\).

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2.3 First-order approximation

To make progress, we now look at the lowest-order approximation of the coupled equations.
We assume the system starts in a specific state \(|i\rangle\), which means

\[ c_i(0) = 1, \qquad c_{k\neq i}(0) = 0. \]

This expresses the physical situation: the system is initially in state \(|i\rangle\) with full probability, and all other states are unoccupied.

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Simplification of the coupled equations

In the general expression

\[ i\hbar \frac{\partial c_n(t)}{\partial t} = \sum_k \langle \phi_n|\hat{V}(t)|\phi_k\rangle \, c_k(t)\, e^{i(\omega_n - \omega_k)t}, \]

the sum over \(k\) contains contributions from all states.
However, to first order in perturbation theory, we only need to keep the term where \(k=i\), because all other coefficients \(c_k(t)\) remain very small and can be neglected.

This leads to

\[ i\hbar \frac{\partial c_n^{(1)}(t)}{\partial t} \;\approx\; \langle \phi_n|\hat{V}(t)|\phi_i\rangle \, c_i(t)\, e^{i(\omega_n - \omega_i)t}. \]

Since \(c_i(t) \approx 1\) to zeroth order, we arrive at the simple first-order equation

\[ \frac{\partial c_n^{(1)}(t)}{\partial t} = \frac{1}{i\hbar}\, \langle \phi_n|\hat{V}(t)|\phi_i\rangle \, e^{i\omega_{ni}t}, \tag{2}\]

where

\[ \omega_{ni} = \frac{E_n - E_i}{\hbar}. \]

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This equation tells us how the amplitude of state \(|n\rangle\) grows in time due to the perturbation, starting from an initial state \(|i\rangle\).

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2.4 Transition probability for an oscillating perturbation: Fermis Golden Rule

If the perturbation is sinusoidal, for example of the form

\[ \hat{V}(t) \propto e^{-i\omega t} + e^{+i\omega t}, \]

one finds that the transition rate between states \(|i\rangle\) and \(|n\rangle\) is given by

TipFermis Golden Rule

\[ \Gamma_{i\to n} = \frac{2\pi}{\hbar} \, \big|\langle \phi_n | \hat{V} | \phi_i \rangle\big|^2 \, \delta(E_n - E_i - \hbar\omega). \tag{3}\]

This result is known as Fermi’s golden rule.
It states that the transition rate is proportional to the square of the matrix element of the perturbation and enforces energy conservation through the delta function.

NoteStep-by-step derivation of Fermi’s Golden Rule

For a sinusoidal perturbation of the form

\[ \hat{V}(t) \propto e^{-i\omega t} + e^{+i\omega t}, \]

the matrix elements in the equation for \(c_n(t)\) pick up oscillatory time-dependence. Physically, these terms represent the absorption (with \(e^{-i\omega t}\)) and emission (with \(e^{+i\omega t}\)) of photons.

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Step 1: Transition amplitude

For a system that starts in \(|i\rangle\) at \(t=0\), the first-order expression for the amplitude of finding the system in another state \(|n\rangle\) at time \(t\) is (from Equation 2)

\[ c_n^{(1)}(t) = \frac{1}{i\hbar} \int_0^t dt'\, \langle \phi_n | \hat{V}(t') | \phi_i \rangle \, e^{i\omega_{ni}t'}, \]

where \(\omega_{ni} = (E_n - E_i)/\hbar\).

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Step 2: Oscillatory integral

If \(\hat{V}(t)\) contains a factor \(e^{-i\omega t'}\), the integral becomes

\[ \int_0^t e^{i(\omega_{ni}-\omega)t'} \, dt' = \frac{e^{i(\omega_{ni}-\omega)t} - 1}{i(\omega_{ni}-\omega)}. \]

This looks like a ratio of a sine function to its argument when written in real form:

\[ \frac{e^{i(\omega_{ni}-\omega)t} - 1}{i(\omega_{ni}-\omega)} = e^{i(\omega_{ni}-\omega)t/2} \frac{\sin\!\big[(\omega_{ni}-\omega)t/2\big]}{(\omega_{ni}-\omega)/2}. \]

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Step 3: Probability and the long-time limit

The transition probability is proportional to \(|c_n^{(1)}(t)|^2\).
This gives a function sharply peaked around resonance (\(\omega_{ni} = \omega\)):

\[ P_{i\to n}(t) \propto \frac{\sin^2[(\omega_{ni}-\omega)t/2]}{[(\omega_{ni}-\omega)/2]^2}. \]

• For finite \(t\), this is a peak of finite width.
  
• For large \(t\), the peak becomes extremely sharp and tall.

In the limit \(t \to \infty\), this expression behaves like a delta function:

\[ \lim_{t \to \infty} \frac{\sin^2[(\omega_{ni}-\omega)t/2]}{[(\omega_{ni}-\omega)/2]^2} = 2\pi t \, \delta(\omega_{ni}-\omega). \]

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Step 4: Transition rate

Since the probability grows linearly in time, we define the rate as

\[ \Gamma_{i\to n} = \lim_{t \to \infty} \frac{P_{i\to n}(t)}{t}. \]

This gives the celebrated result:

\[ \Gamma_{i\to n} = \frac{2\pi}{\hbar} \big|\langle \phi_n | \hat{V} | \phi_i \rangle\big|^2 \delta(E_n - E_i - \hbar\omega). \]

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Step 5: Physical interpretation

• The matrix element tells us how strongly the perturbation couples the two states.
  
• The delta function enforces energy conservation: a transition only occurs if the photon energy \(\hbar\omega\) exactly matches the energy gap \(E_n - E_i\).
  
• The factor \(\tfrac{2\pi}{\hbar}\) is a universal constant, giving the overall scale.

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Technical background: the delta function identity

The crucial step is recognizing that

\[ \lim_{t \to \infty} \frac{\sin^2(x t/2)}{(x/2)^2} = 2\pi t \, \delta(x). \]

This is not an ordinary function, but a distributional limit.
It encodes the fact that oscillatory integrals average to zero, except when the frequency mismatch \(x\) vanishes.

In practice, this means:
- The system only makes transitions if the driving frequency matches the Bohr frequency between states.
- The “infinite time” assumption justifies why real experiments see sharp resonance peaks.

Historical aside: Why “golden”?

This result was first written down in detail by Dirac (1927) in his theory of radiation, and later reformulated by Enrico Fermi, who emphasized its practical power.
Fermi himself nicknamed it the “golden rule” because it was universally useful: with one compact formula, physicists could calculate transition rates for atoms, nuclei, and even particle decays.

It quickly became a workhorse of quantum mechanics, showing how microscopic dynamics connect to experimentally measurable rates — from atomic spectroscopy to modern quantum optics.

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In the next section, we will apply this framework to the dipole interaction Hamiltonian, thereby obtaining explicit expressions for the transition matrix elements that determine the Einstein coefficients.

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3. Electric Dipole Matrix Element

In the previous section, we derived Fermi’s Golden Rule (3), which tells us that the transition rate between two atomic states is proportional to the squared modulus of a matrix element of the perturbation Hamiltonian.

For the atom–light interaction in the minimal-coupling scheme, this gives (for the potential given in Equation 1):

\[ \Gamma_{i \to n} = \frac{\pi e^2}{m^2 \varepsilon_0 \omega V} \Big| \langle \phi_n | e^{-i\vec{k}\cdot\vec{r}} \, \hat{\vec{\varepsilon}}\cdot \hat{\vec{p}} | \phi_i \rangle \Big|^2 \delta(E_n - E_i \pm \hbar\omega). \]

The total rate is obtained by summing over all photon momenta and polarizations:

\[ \Gamma_{\text{tot}} = \sum_{\vec{k}, \text{pol}} \Gamma_{i \to n}. \]

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3.1 Simplifying the matrix element

The central challenge is the matrix element

\[ \langle \phi_n | e^{-i\vec{k}\cdot\vec{r}} \, \hat{\vec{\varepsilon}} \cdot \hat{\vec{p}} | \phi_i \rangle. \tag{4}\]

Two simple but powerful ideas allow us to turn this into something physically transparent:

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1. Commutator trick

We recall the commutator between the unperturbed Hamiltonian \(\hat{H}_0\) and the position operator \(\hat{\vec{r}}\):

\[ [\hat{H}_0, \hat{\vec{r}}] = \frac{\hbar}{i m}\hat{\vec{p}}. \tag{5}\]

This relation can be derived directly from \(\hat{H}_0 = \hat{\vec{p}}^2 / 2m + V(\vec{r})\) and the canonical commutation rules.
It effectively tells us that momentum matrix elements can be rewritten in terms of energy differences and position operators.

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2. Electric dipole approximation

For visible light, the wavelength (\(\lambda \sim 500\) nm) is much larger than the size of an atom (\(a_0 \sim 0.05\) nm).
Thus, within the atom, the spatial dependence of the electromagnetic field is negligible:

\[ e^{-i\vec{k}\cdot\vec{r}} \approx 1 - i\vec{k}\cdot \vec{r} + \dots \;\;\approx 1. \tag{6}\]

This is called the electric dipole approximation.

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3. Putting it together

Using (5) and (6), the matrix element (4) reduces to

\[ \langle \phi_n | e^{-i\vec{k}\cdot\vec{r}} \, \hat{\vec{\varepsilon}}\cdot \hat{\vec{p}} | \phi_i \rangle \;\approx\; \hat{\vec{\varepsilon}} \cdot \langle \phi_n | \hat{\vec{p}} | \phi_i \rangle. \]

Then we apply the commutator relation:

\[ \hat{\vec{\varepsilon}} \cdot \langle \phi_n | \hat{\vec{p}} | \phi_i \rangle = \hat{\vec{\varepsilon}} \cdot \frac{im}{\hbar} \, \langle \phi_n | [\hat{H}_0, \hat{\vec{r}}] | \phi_i \rangle. \]

Finally, since \(\hat{H}_0|\phi_i\rangle = E_i|\phi_i\rangle\), we find

\[ = \frac{im}{\hbar} (E_n - E_i) \langle \phi_n | \hat{\vec{\varepsilon}} \cdot \hat{\vec{r}} | \phi_i \rangle. \]

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3.2 The dipole matrix element

We have arrived at the compact and physically transparent form:

\[ \langle \phi_n | e^{-i\vec{k}\cdot\vec{r}} \hat{\vec{\varepsilon}} \cdot \hat{\vec{p}} | \phi_i \rangle \;\;\propto\;\; \langle \phi_n | \hat{\vec{\varepsilon}} \cdot \hat{\vec{r}} | \phi_i \rangle. \]

The quantity

\[ \langle \phi_n | \hat{\vec{\varepsilon}} \cdot \hat{\vec{r}} | \phi_i \rangle \]

is called the electric dipole matrix element.
It measures how “aligned” the electron’s motion in the atom is with the oscillating electric field.

• If the dipole matrix element is zero (due to symmetry), the transition is forbidden in the dipole approximation.
  
• If it is nonzero, it determines the strength of the transition rate, and thus the absolute values of the Einstein \(A\) and \(B\) coefficients.

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In summary:
Fermi’s Golden Rule tells us that transition rates depend on matrix elements of the interaction Hamiltonian. By applying the electric dipole approximation and the commutator relation, these matrix elements reduce to simple position-operator terms. This makes the electric dipole matrix element the central quantity for describing atomic transitions induced by light.

4. Einstein Coefficients from Dipole Matrix Elements

We are now in a position to connect the microscopic dipole matrix element to the macroscopic Einstein coefficients that describe spontaneous emission, absorption, and stimulated emission.

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4.1 Transition Rates in the Dipole Approximation

From Fermi’s Golden Rule, the transition rate into a final state \(|n\rangle\) from an initial state \(|i\rangle\) depends on the dipole matrix element projected onto the photon polarization \(\hat{\vec{\varepsilon}}\):

\[ \Gamma_{i \to n}^{(\hat{\varepsilon})} \;\propto\; \big| \langle n | \, \hat{\vec{\varepsilon}} \cdot \vec{r} \, | i \rangle \big|^2 . \]

This expression gives the partial rate for emission into a fixed polarization and solid angle.
The total transition rate is obtained by summing over both photon polarizations and all emission directions:

\[ \Gamma_{i \to n} \;=\; \frac{\alpha \, \omega^3}{2 \pi c^2} \sum_{\hat{\vec{\varepsilon}}} \int d\Omega_{\gamma}\, \big| \langle n | \, \hat{\vec{\varepsilon}} \cdot \vec{r} \, | i \rangle \big|^2 . \]

Here:
- \(d\Omega_{\gamma}\) is the solid angle element,
- \(\alpha=e^2/(4\pi \varepsilon_0\hbar c)\) is the fine-structure constant.

Carrying out the angular integration and polarization sum yields the compact result:

\[ \Gamma_{i \to n} \;=\; \frac{\omega^3 e^2}{3\pi \varepsilon_0 \hbar c^3}\, \big| \langle n | \vec{r} | i \rangle \big|^2 . \]

This formula makes explicit that the transition rate grows with \(\omega^3\) and is proportional to the square of the dipole matrix element.

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4.2 Spontaneous emission coefficient

The Einstein coefficient for spontaneous emission \(A\) is equal to the transition rate we just calculated and it is

Tip\(A\) coefficient

\[ A_{21} = \frac{\omega^3 e^2}{3\pi \varepsilon_0 \hbar c^3} \, \big| \langle \phi_1 | \hat{\vec{r}} | \phi_2 \rangle \big|^2, \]

where \(\omega = (E_2 - E_1)/\hbar\) is the transition frequency.

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4.3 Absorption and stimulated emission coefficients

Using the relation between \(B_{21}\) and \(A_{21}\) derived from Planck’s law,

\[ A_{21} = \frac{8\pi h\nu^3}{c^3} B_{21}, \]

we can substitute the microscopic expression for \(A_{21}\) and obtain the absorption coefficient \(B_{12}\) and the stimulated emission coefficient \(B_{21}\) (for equal degeneracy)

Tip\(B\) coefficients

\[ B_{12} = B_{21} = \frac{e^2}{6 \varepsilon_0 \hbar^2} \, \big|\langle \phi_2 | \hat{\vec{r}} | \phi_1 \rangle\big|^2. \]

• The equality \(B_{12} = B_{21}\) (up to degeneracy factors) is the symmetry relation derived earlier.
  
• The result shows that all Einstein coefficients are directly proportional to the squared dipole matrix element.

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4.3 Spontaneous emission coefficient

Using the relation between \(A_{21}\) and \(B_{21}\) derived from Planck’s law,

\[ A_{21} = \frac{8\pi h\nu^3}{c^3} B_{21}, \]

we can substitute the microscopic expression for \(B_{21}\) and obtain

Tip\(A\) coefficient

\[ A_{21} = \frac{\omega^3 e^2}{3\pi \varepsilon_0 \hbar c^3} \, \big| \langle \phi_1 | \hat{\vec{r}} | \phi_2 \rangle \big|^2, \]

where \(\omega = (E_2 - E_1)/\hbar\) is the transition frequency.

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4.4 Physical meaning

• \(B\) coefficients (absorption and stimulated emission):
  measure how strongly the atom couples to an external light field.

• \(A\) coefficient (spontaneous emission):
  gives the natural decay rate (inverse lifetime) of an excited state, entirely determined by the dipole matrix element and the transition frequency.

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5. Summary

• The Einstein coefficients are no longer just abstract statistical parameters.
  
• They can be calculated from quantum mechanics, using the electric dipole matrix element.
  
• This provides the link between the microscopic wavefunctions of atoms and the macroscopic behavior of light and matter, such as absorption spectra, emission lifetimes, and the operation of lasers.