Spontaneous emission is the quantum mechanical process by which an excited atomic system drops to a lower energy state by emitting a photon without external stimulation. The emitted photons are characterized by random phases, polarization, and directions of propagation.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Basic definition, distinction between spontaneous and stimulated emission, role in ordinary light sources, and connection to energy levels.
PGT extension
Einstein A and B coefficient derivations, transition probabilities, lifetime of excited states, and spectral line broadening mechanisms.
Prerequisites
Bohr's atomic model and energy quantization
Boltzmann distribution law
Basic quantum mechanics and probability transition rates
Mechanism of Spontaneous Emission
An electron in an excited state of energy E&sub2; naturally decays to a lower energy state E&sub1; after a characteristic mean lifetime (τ).
The energy difference Δ$E = E_{2} - E_{1}$ is released in the form of a single photon of frequency $\nu = \DeltaE / h$.
The process occurs entirely independently of any external radiation field, driven by zero-point fluctuations of the electromagnetic vacuum.
Einstein's A Coefficient
The transition rate per unit time for an atom to spontaneously drop from state 2 to state 1 is denoted by the Einstein A coefficient, A&sub2;&sub1;.
The number of atoms decaying per second per unit volume is given by dN&sub2;/d$t = -A_{21}N_{2}$.
The lifetime of the excited state is inversely proportional to the spontaneous emission rate, such that $\tau = 1 / A_{21}$.
Coherence and Directionality Characteristics
Photons emitted via spontaneous emission are mutually incoherent because each atom emits independently with a random phase.
The radiation propagates in random directions, leading to high spatial divergence characteristic of ordinary thermal and discharge sources.
The spectral output possesses a natural linewidth due to the finite lifetime of the excited state governed by the Heisenberg uncertainty principle.
Comparison with Stimulated Emission
Unlike spontaneous emission, stimulated emission requires an incident photon of matching energy to trigger the downward transition.
Stimulated emission produces coherent photons identical in phase, frequency, polarization, and direction to the triggering photon, forming the basis of laser action.
At optical frequencies, spontaneous emission dominates over stimulated emission in thermal equilibrium due to the low population of high-energy states.
Derivations
Relation Between Einstein A and B Coefficients
Conditions: System in thermal equilibrium with a radiation field at absolute temperature T.
Write the upward absorption rate as R&sub1;$_{2} = N_{1}\rho(\nu)B_{12}$ and downward emission rate as R&sub2;$_{1} = N_{2}[A_{21} + \rho(\nu)B_{21}]$, where ρ(ν) is the energy density of radiation.
Equate upward and downward rates in thermal equilibrium: N&sub1;ρ(ν)B&sub1;$_{2} = N_{2}A_{21} + N_{2}\rho(\nu)B_{21}$.
Rearrange to solve for energy density: ρ(ν) = A&sub2;&sub1; / [B&sub2;&sub1; ((N&sub1;/N&sub2;) - (B&sub1;&sub2;/B&sub2;&sub1;))].
Substitute the Boltzmann population ratio N&sub1;/N$_{2} = (g_{2}/g_{1})exp(h\nu/kT)$ and match with Planck's radiation law ρ(ν) = (8πhν³/c³) / [exp(hν/kT) - 1].
Equating terms yields the relation between coefficients: A&sub2;&sub1; / B&sub2;$_{1} = 8\pih\nu^{3}/c^{3}$ and g&sub1;B&sub1;$_{2} = g_{2}B_{21}$.
Result: A_21 / $B_21 = \frac{8pi h \nu^3}{c^3}$
Exponential Decay of Excited State Population
Conditions: Absence of pumping and stimulated processes; pure spontaneous decay.
Set up the rate equation for the population N&sub2; of the excited state: dN_2 / d$t = -A_21 N_2$.
Separate variables to integrate: int (1 / N_2) d$N_2 = -A_21 \int dt$.
Perform integration to obtain ln($N_2) = -A_21 t + C$.
Exponentiate both sides to get the final decay law $N_2(t) = N_0 e^{-A_21 t}$.
Result: $N_2(t) = N_0 e^{-A_21 t}$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Spontaneous Emission Rate Equation
$$\frac{dN_2}{dt} = -A_{21}N_2$$
$N_2 = Population of excited state$, $A_21 = Einstein A coefficient$ s^{-1}
Pure spontaneous decay
Excited State Population Decay
$$N_2(t) = N_0 e^{-t/\tau}$$
$N_0 = Initial population$, \tau = Lifetime of excited state Dimensionless population
t >= 0
Spontaneous Lifetime
$$\tau = \frac{1}{A_{21}}$$
\tau = Mean lifetime, $A_21 = Spontaneous transition probability$ seconds (s)
Single isolated transition
Einstein Coefficient Ratio
$$\frac{A_{21}}{B_{21}} = \frac{8pi h \nu^3}{c^3}$$
A_21, $B_21 = Einstein coefficients$, h = Planck constant, \nu = Frequency, c = Speed of light J s m^{-3}
Thermal equilibrium
Energy of Emitted Photon
$$E = h\nu = E_2 - E_1$$
h = Planck constant, \nu = Frequency, E_2, $E_1 = Energy levels$ Joules (J) or eV
Bohr frequency condition
Frequency-Wavelength Relation
$$\nu = \frac{c}{\lambda}$$
c = Speed of light, lambda = Wavelength Hertz (Hz)
Vacuum propagation
Degeneracy Relation for B Coefficients
$$g_1 B_{12} = g_2 B_{21}$$
g_1, $g_2 = Statistical weights (degeneracies) of states 1$ and 2 Dimensionless ratio
Equilibrium conditions
Natural Linewidth (Energy-Time Uncertainty)
$$\delta E \cdot \delta t \approx hbar$$
Delta E = Uncertainty in energy, Delta t = Lifetime \tau, hbar = Reduced Planck constant Joule seconds (J s)
Heisenberg uncertainty principle
Graphs and experiments
Exponential Decay of Excited State Population
Draw: A plot of population N_2(t) on the y-axis versus time t on the x-axis, showing a downward concave exponential curve intersecting the y-axis at N_0.
Interpret: Shows that the number of excited atoms decreases exponentially over time, with the time required for population to drop to 1/e of its initial value defined as the lifetime τ.
Exam use: Tested in MCQs to identify decay curves and calculate half-life or mean lifetime from graphical data.
Energy Level Transition Diagram for Spontaneous Emission
Draw: Two horizontal lines representing energy levels E_1 and E_2 (with E_2 above E_1), connected by a wavy downward vertical arrow labeled with photon emission energy hν.
Interpret: Depicts the downward transition of an electron from a higher unstable state to a lower stable state accompanied by the emission of a single photon.
Exam use: Frequently used in conceptual questions distinguishing spontaneous emission, absorption, and stimulated emission.
Spectral Line Profile (Natural Broadening)
Draw: A symmetrical bell-shaped intensity (I) versus frequency (ν) curve, with full width at half maximum (FWHM) labeled as Δν.
Interpret: Illustrates that spontaneous emission is not perfectly monochromatic; due to the finite lifetime of the state, frequencies span a narrow Lorentzian distribution.
Exam use: Tested in PGT level problems concerning line broadening mechanisms and uncertainty relations.
Solved numericals
TGT core
An excited state of an atom has a mean lifetime of 10 ns. Calculate the Einstein A coefficient for this transition.
Identify the given lifetime: $\tau = 10 \text{ ns} = 10 \times 10^{-9} \text{ s}$.
Use the relation between lifetime and Einstein A coefficient: $A_{21} = \frac{1}{\tau}$.
For a atomic transition at wavelength lambda = 6000 \text{ AA}, the Einstein A coefficient is 2 \times 10^7 \text{ s}^{-1}. Find the value of the Einstein B coefficient B_{21}.
If the initial population of an excited energy level is 5 \times 10^{18} \text{ atoms/cm}^3 and the spontaneous decay rate constant is 5 \times 10^7 \text{ s}^{-1}, find the population remaining after 20 \text{ ns}.
Assumes two-level atomic systems interacting with isotropic blackbody radiation fields in thermodynamic equilibrium for Einstein coefficient derivations.
Neglects multi-photon emission processes and non-radiative decay mechanisms such as collisional quenching or Auger transitions.
Common misconceptions
Common error: Spontaneous emission produces coherent light just like stimulated emission in lasers.
Correct: Spontaneous emission produces incoherent light with random phases and directions.
Each atom decays independently without external phase referencing, unlike stimulated emission where the incoming photon forces phase coherence.
Common error: Spontaneous emission requires an external triggering photon.
Correct: Spontaneous emission occurs entirely on its own without any external photon stimulus.
It is triggered by vacuum electromagnetic field fluctuations rather than incident radiation.
Common error: Einstein A coefficient depends on temperature.
Correct: Einstein A coefficient is an intrinsic atomic property independent of temperature.
A_21 depends only on atomic wavefunctions and transition dipole moments, whereas radiation density depends on temperature.
Quick revision
Spontaneous emission is the unprovoked downward transition of an electron emitting an incoherent photon: $A_21 = 1 / \tau$.
Excited State Population Decay: $N_2(t) = N_0 e^{-t/\tau}$
Must remember
Spontaneous emission is the natural decay of an electron from an excited state $E_2$ to a lower state $E_1$, releasing a photon of energy $\Delta E = E_2 - E_1$.
The process occurs independently of any external radiation field, driven by zero-point fluctuations of the electromagnetic vacuum.
The rate of population decay of the excited state is given by $\frac{dN_2}{dt} = -A_{21}N_2$, where $A_{21}$ is the Einstein A coefficient.
The mean lifetime of the excited state is inversely proportional to the Einstein A coefficient: $\tau = \frac{1}{A_{21}}$.
Photons emitted via spontaneous emission are mutually incoherent, possessing random phases and propagating in random directions.
The Einstein A coefficient is an intrinsic atomic property and is independent of temperature.
The natural linewidth of the emitted spectral line is governed by the Heisenberg uncertainty principle: $\Delta E \cdot \Delta t \approx \hbar$.
At optical frequencies, spontaneous emission dominates over stimulated emission in thermal equilibrium due to low high-energy state populations.
Glossary
Spontaneous EmissionThe random, unforced decay of an atomic system from a higher energy state to a lower energy state with the emission of a photon.
Einstein A Coefficient ($A_{21}$)The transition probability per unit time for spontaneous emission from state 2 to state 1.
Einstein B CoefficientsCoefficients representing the probability rates for stimulated absorption ($B_{12}$) and stimulated emission ($B_{21}$).
Excited State Lifetime ($\tau$)The average time an atom spends in an excited energy state before decaying, equal to $\frac{1}{A_{21}}$.
Natural LinewidthThe inherent spread in frequency or energy of a spectral line caused by the finite lifetime of the excited energy state.
Vacuum FluctuationsZero-point fluctuations of the electromagnetic field that trigger spontaneous emission even in the absence of external radiation.
Spontaneous vs. Stimulated Emission
Feature
Spontaneous Emission
Stimulated Emission
Trigger Mechanism
Vacuum electromagnetic fluctuations
Incident photon matching transition energy
Phase & Direction
Random phase, random direction (incoherent)
Identical phase, frequency, and direction (coherent)
Governing Coefficient
Einstein A coefficient ($A_{21}$)
Einstein B coefficient ($B_{21}$)
Application
Ordinary lamps, discharge tubes, LEDs
Lasers, masers
Practice Quiz
Choose an option to reveal the explanation.
1. What is the primary driving mechanism behind spontaneous emission in atoms?
Explanation: Spontaneous emission occurs entirely independently of external radiation, driven fundamentally by the zero-point fluctuations of the electromagnetic vacuum.
2. If the mean lifetime of an excited atomic state is $\tau = 10\text{ ns}$, what is the value of the Einstein A coefficient ($A_{21}$)?
Explanation: The Einstein A coefficient is the inverse of the lifetime: $A_{21} = \frac{1}{\tau} = \frac{1}{10 \times 10^{-9}\text{ s}} = 10^8\text{ s}^{-1}$.
3. Which of the following statements correctly describes the coherence of photons produced by spontaneous emission?
Explanation: Photons emitted via spontaneous emission are mutually incoherent because each atom decays independently with random phase and random spatial direction.
4. How does the Einstein A coefficient ($A_{21}$) vary with absolute temperature $T$ in thermal equilibrium?
Explanation: The Einstein A coefficient is an intrinsic atomic property determined by atomic wavefunctions and transition dipole moments; it is completely independent of temperature.
5. The differential equation governing the decay of the excited state population $N_2$ via spontaneous emission is given by:
Explanation: The rate of decrease of the excited state population due to spontaneous decay alone is expressed as $\frac{dN_2}{dt} = -A_{21} N_2$.
6. The natural linewidth of a spectral line arising from spontaneous emission is fundamentally connected to which physical principle?
Explanation: The natural linewidth arises due to the finite lifetime $\tau$ of the excited state, satisfying the energy-time uncertainty relation $\Delta E \cdot \Delta t \approx \hbar$.
7. What is the relationship between the Einstein coefficients $A_{21}$ and $B_{21}$ for isotropic radiation in thermal equilibrium?
Explanation: Equating the rates of absorption and emission in thermal equilibrium using Planck's radiation law yields the ratio $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$.
8. Why does spontaneous emission dominate over stimulated emission at optical frequencies in ordinary thermal sources?
Explanation: At optical frequencies, the thermal equilibrium population of high-energy states is exponentially small (Boltzmann factor), making spontaneous emission the dominant relaxation pathway.
9. If the initial population of an excited state is $N_0$ at $t = 0$, what is the population at $t = \tau$ (where $\tau$ is the lifetime)?
Explanation: The population decay follows $N_2(t) = N_0 e^{-t/\tau}$. At $t = \tau$, $N_2(\tau) = N_0 e^{-1} = N_0 / e$.
10. Which of the following light sources primarily relies on spontaneous emission?
Explanation: An incandescent lamp and standard discharge tubes emit light predominantly through incoherent spontaneous emission, unlike lasers which rely on stimulated emission.
11. What are the SI units of the Einstein A coefficient ($A_{21}$)?
Explanation: Since $A_{21}$ represents a transition probability per unit time (decay rate per atom), its unit is inverse seconds ($\text{s}^{-1}$).
12. What is the energy of the photon released when an atom makes a spontaneous transition between two levels of energy $E_2$ and $E_1$?
Explanation: By conservation of energy, the energy of the emitted photon is exactly equal to the energy difference between the two states: $\Delta E = E_2 - E_1 = h\nu$.
13. In the presence of level degeneracy, how are the Einstein B coefficients $B_{12}$ and $B_{21}$ related for states with statistical weights $g_1$ and $g_2$?
Explanation: Detailed balance in thermal equilibrium requires that $g_1 B_{12} = g_2 B_{21}$, accounting for the statistical weights (degeneracies) of the energy levels.
14. Which misconception about spontaneous emission is incorrect?
Explanation: The statement that spontaneous emission produces coherent light is false (it is a misconception); spontaneous emission produces incoherent light.
15. If an excited state has a very short lifetime $\tau$, what happens to its natural spectral linewidth ($\Delta \nu$)?
Explanation: From the uncertainty principle $\Delta E \cdot \Delta t \approx \hbar$, a shorter lifetime $\Delta t$ (or $\tau$) results in a larger uncertainty in energy and frequency, hence a broader natural linewidth.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. The mean lifetime of an excited atomic state is $2\times 10^{-8}\text{ s}$. The Einstein A coefficient for this transition is:
Explanation: Using the relation $A_{21} = \frac{1}{\tau}$, we get $A_{21} = \frac{1}{2 \times 10^{-8}\text{ s}} = 0.5 \times 10^8\text{ s}^{-1} = 5 \times 10^7\text{ s}^{-1}$بيه.
2. In the relation between Einstein coefficients $\frac{A_{21}}{B_{21}} = \frac{8\pi h \nu^3}{c^3}$, the dimension of the ratio $A_{21}/B_{21}$ is equivalent to:
Explanation: The ratio $A_{21} / B_{21}$ has the dimensions of spectral energy density $\rho(\nu)$, which is expressed in $\text{J}\cdot\text{m}^{-3}\cdot\text{Hz}^{-1}$ or $\text{J}\cdot\text{s}\cdot\text{m}^{-3}$.
3. If $N_0$ is the initial number of atoms in an excited state at $t = 0$, the number of atoms remaining in that state after a time equal to two mean lifetimes ($t = 2\tau$) is:
4. Which of the following characteristics distinguishes spontaneous emission from stimulated emission?
Explanation: Spontaneous emission is inherently random and independent for each atom, triggered by vacuum fluctuations, whereas stimulated emission is prompted by an incident photon.
5. The natural broadening of a spectral line is directly tied to:
Explanation: Natural line broadening is an intrinsic property resulting from the finite lifetime of the atomic energy levels, governed by $\Delta E \cdot \Delta t \approx \hbar$.
6. What is the physical significance of the Einstein A coefficient ($A_{21}$)?
Explanation: The Einstein A coefficient $A_{21}$ defines the spontaneous transition probability per unit time for an atom in state 2 to decay to state 1.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. In a collection of excited atoms undergoing spontaneous emission, the decay rate of the excited state population is proportional to:
Explanation: The rate equation is $\frac{dN_2}{dt} = -A_{21}N_2$, showing that the decay rate is directly proportional to the instantaneous population $N_2$ (first-order kinetics).
2. If the wavelength of a spectral line is $\lambda = 500\text{ nm}$, what is the frequency of the emitted photon during spontaneous emission?
3. Which of the following factors does NOT affect the Einstein A coefficient of a specific atomic transition?
Explanation: Ambient temperature affects thermal radiation density and population distributions, but the Einstein A coefficient is an isolated atomic property independent of temperature.
4. The time required for the population of an excited state to decay to $1/e^2$ of its initial value is:
Explanation: Since $N_2(t) = N_0 e^{-t/\tau}$, for $N_2(t) = N_0 / e^2$, we must have $t/\tau = 2$, which gives $t = 2\tau$.
5. Why are light bulbs and fluorescent tubes classified as sources of spontaneous emission rather than stimulated emission?
Explanation: Bulbs and discharge tubes produce light through independent, random de-excitations of atoms, resulting in incoherent, multi-directional radiation characteristic of spontaneous emission.
6. What is the relation between the half-life ($T_{1/2}$) of an excited state and its mean lifetime ($\tau$)?
7. According to Einstein's derivation, how does the ratio of spontaneous to stimulated emission rate depend on the energy density of radiation $\rho(\nu)$?
Explanation: Spontaneous emission rate is $A_{21}$, while stimulated emission rate is $B_{21}\rho(\nu)$. Their ratio is $\frac{A_{21}}{B_{21}\rho(\nu)}$, which is inversely proportional to $\rho(\nu)$.
8. If two energy levels have degeneracies $g_1 = 2$ and $g_2 = 4$, what is the relation between their Einstein B coefficients $B_{12}$ and $B_{21}$?
Explanation: Using $g_1 B_{12} = g_2 B_{21}$, substituting $g_1 = 2$ and $g_2 = 4$ yields $2 B_{12} = 4 B_{21}$, which simplifies to $B_{12} = 2 B_{21}$.
9. Which physical phenomenon provides the microscopic 'kick' that initiates spontaneous decay of an excited atomic electron?
Explanation: Quantum electrodynamics shows that spontaneous emission is actually stimulated by the zero-point fluctuations of the vacuum electromagnetic field.
10. In terms of atomic transition probability, a larger Einstein A coefficient ($A_{21}$) implies:
Explanation: Since $\tau = \frac{1}{A_{21}}$, a larger $A_{21}$ means a shorter lifetime $\tau$ and a much faster rate of spontaneous decay.