13.1 Selection rules and transition probabilities
4 min read•july 30, 2024
Selection rules and transition probabilities are key to understanding atomic and molecular . They determine which transitions between are allowed and how likely they are to occur, shaping the observed spectra.
These rules stem from conservation laws and quantum mechanics principles. By mastering them, you'll be able to predict and interpret spectral patterns, crucial for analyzing atomic and molecular structures and interactions.
Selection rules for transitions
Conservation of angular momentum and parity
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- Selection rules are based on the conservation of angular momentum and parity during transitions between quantum states
- satisfy the selection rules, while violate them
- The transition probability for allowed transitions is much higher than for forbidden transitions (electric dipole transitions vs magnetic dipole transitions)
- Selection rules arise from the evaluation of transition matrix elements, which involve the overlap of wavefunctions and the transition operator
- The transition operator depends on the type of interaction (electric dipole, magnetic dipole, electric quadrupole) and the symmetry of the system
Transition matrix elements and transition probabilities
- Transition matrix elements determine the probability of a transition occurring between two quantum states
- They involve integrating the product of the initial state wavefunction, the transition operator, and the final state wavefunction over all space
- Non-zero transition matrix elements indicate allowed transitions, while zero matrix elements correspond to forbidden transitions
- The transition probability is proportional to the square of the transition matrix element
- Symmetry considerations (parity, angular momentum) can greatly simplify the evaluation of transition matrix elements
Electric dipole selection rules
Atomic selection rules
- The electric dipole selection rules for atoms are: Δl = ±1, Δm = 0, ±1, and ΔS = 0, where l is the orbital angular momentum quantum number, m is the magnetic quantum number, and S is the spin quantum number
- Transitions between states with the same parity (even to even or odd to odd) are forbidden (Laporte rule)
- Examples: 1s → 2p (allowed), 2s → 2p (forbidden)
Molecular selection rules
- For molecules, the electric dipole selection rules involve changes in the vibrational (Δv = ±1, ±2, ...) and rotational (ΔJ = ±1) quantum numbers, as well as the symmetry of the electronic states (Σ → Σ, Π → Σ)
- The Laporte selection rule states that transitions between states of the same parity (gerade to gerade or ungerade to ungerade) are forbidden for centrosymmetric molecules
- Vibronic coupling and Herzberg-Teller coupling can relax the selection rules, allowing for weakly allowed transitions that would otherwise be forbidden
- Examples: Σ^+ → Σ^+ (allowed), Σ^+ → Σ^- (forbidden), Σ^+ → Π (allowed)
Transition probabilities calculations
Electric dipole approximation
- The transition probability is proportional to the square of the , which is the expectation value of the electric dipole operator between the initial and final states
- In the electric dipole approximation, the transition dipole moment is given by , where ψ_i and ψ_f are the wavefunctions of the initial and final states, respectively, and er is the electric dipole operator
- The electric dipole approximation is valid when the wavelength of the radiation is much larger than the size of the atom or molecule
Einstein coefficients
- The Einstein A coefficient, A_fi, represents the rate and is related to the transition probability by , where ν is the transition frequency, h is Planck's constant, and c is the speed of light
- The Einstein B coefficients, B_fi and B_if, represent and absorption rates, respectively, and are related to A_fi by the Einstein relations: and , where g_i and g_f are the degeneracies of the initial and final states
Transition probabilities vs spectral intensities
Spectral line intensities
- The intensity of a spectral line is proportional to the population of the upper state and the transition probability (Einstein A coefficient) for the corresponding transition
- The relative intensities of spectral lines can be used to determine the population distribution among different energy levels, which is governed by the Boltzmann distribution in thermal equilibrium
- Example: The ratio of intensities for two spectral lines originating from the same upper state is equal to the ratio of their Einstein A coefficients
Oscillator strength and line broadening
- The oscillator strength, f_fi, is a dimensionless quantity that is proportional to the transition probability and is often used to compare the strengths of different transitions
- The line shape and width of spectral lines are influenced by various broadening mechanisms, such as natural broadening (related to the lifetime of the excited state), Doppler broadening (due to the motion of atoms or molecules), and pressure broadening (due to collisions with other particles)
- Example: Natural broadening leads to a Lorentzian line shape, while Doppler broadening results in a Gaussian line shape
Key Terms to Review (16)
Allowed Transitions: Allowed transitions refer to specific quantum mechanical changes between energy states of a system that can occur due to interactions with external fields or forces. These transitions are significant because they are the only ones that can effectively contribute to observable phenomena, such as absorption and emission of radiation, where certain selection rules determine their likelihood. Understanding allowed transitions helps explain how particles behave under various conditions in atomic and nuclear contexts.
Eigenstate: An eigenstate is a specific quantum state of a quantum system that corresponds to a definite value of an observable, represented by an eigenvalue. When a measurement is performed on a quantum system in an eigenstate, the result will always be the associated eigenvalue. This concept is crucial in understanding how systems transition between states and the probabilities associated with these transitions.
Energy Levels: Energy levels refer to the discrete values of energy that an electron in an atom or a molecule can possess. These levels arise from the quantization of the energy states due to the constraints of quantum mechanics, leading to phenomena such as electron transitions, spectral lines, and interactions with external fields.
Fermi's Golden Rule: Fermi's Golden Rule provides a formula for calculating the transition rate between quantum states due to a perturbation, often in the context of time-dependent interactions. This principle is crucial for understanding how systems evolve when subjected to external influences, allowing predictions about probabilities of transitions between initial and final states, particularly in processes like atomic transitions and scattering phenomena.
Forbidden transitions: Forbidden transitions refer to quantum mechanical transitions between energy levels that have very low probabilities of occurring due to selection rules. These transitions are typically associated with changes in quantum numbers that do not adhere to the allowed criteria set by conservation laws or symmetry principles, leading to their rarity in processes such as absorption and emission of light.
Laser operation: Laser operation refers to the process by which a laser emits coherent light through the stimulated emission of radiation. This involves the excitation of atoms or molecules in a gain medium, followed by the release of photons that are amplified and emitted in a focused beam. The principles of laser operation are deeply intertwined with quantum mechanics, particularly through the concepts of selection rules and transition probabilities that govern how atoms transition between energy levels.
Max Planck: Max Planck was a German physicist who is best known for his role in the development of quantum theory, particularly through the introduction of quantized energy levels. His work laid the foundation for modern physics, connecting the discrete nature of energy with various phenomena, which are crucial in understanding measurement, selection rules, and electromagnetic field quantization.
Niels Bohr: Niels Bohr was a Danish physicist known for his foundational contributions to understanding atomic structure and quantum mechanics, particularly through the Bohr model of the atom. His work emphasized the quantization of energy levels in atoms and introduced the idea that electrons orbit the nucleus at fixed distances, which directly ties into the concepts of measurement, operators, transition probabilities, and fine structure in quantum systems.
Parity selection rule: The parity selection rule states that transitions between quantum states must conserve parity, meaning that a system can only transition between states of the same parity or states of opposite parity depending on the nature of the interaction involved. This rule plays a critical role in determining the allowed and forbidden transitions in quantum mechanics, influencing the probability of various processes like emission and absorption of light.
Perturbation theory: Perturbation theory is a mathematical approach used in quantum mechanics to find an approximate solution to a complex problem by starting from the known solutions of a simpler problem and adding small corrections. This technique helps understand how a system behaves under slight disturbances or changes, making it valuable in various contexts, such as energy levels and wave functions.
Quantum electrodynamics: Quantum electrodynamics (QED) is the relativistic quantum field theory that describes how light and matter interact, focusing on the exchange of photons between charged particles. This theory combines the principles of quantum mechanics with special relativity, resulting in a framework that provides insight into phenomena such as electron-photon interactions and the structure of atoms. QED plays a crucial role in understanding selection rules and transition probabilities, as well as in the quantization of the electromagnetic field.
Spectroscopy: Spectroscopy is the study of the interaction between electromagnetic radiation and matter, particularly focusing on how different wavelengths of light can provide information about the energy levels of atoms and molecules. This technique allows scientists to analyze atomic and molecular structures, understand their behaviors, and determine their properties by observing the light emitted or absorbed during transitions between energy states. By applying spectroscopy to various systems, we can derive important insights into atomic, molecular, vibrational, and rotational dynamics.
Spin selection rule: The spin selection rule refers to the constraints on the allowed transitions between quantum states of a system based on the intrinsic angular momentum, or spin, of particles. Specifically, it states that transitions involving changes in the total spin quantum number must adhere to specific rules, which impacts the probabilities of various transitions during processes like electronic excitations. Understanding these rules is essential for predicting the behavior of quantum systems under external perturbations, such as electromagnetic radiation.
Spontaneous emission: Spontaneous emission is the process by which an excited atomic or molecular system transitions to a lower energy state without any external influence, resulting in the release of a photon. This phenomenon is a fundamental aspect of quantum mechanics and plays a crucial role in understanding how atoms interact with electromagnetic fields, leading to various effects such as the emission spectra of materials and the behavior of lasers.
Stimulated Emission: Stimulated emission is the process by which an incoming photon causes an excited atom or molecule to release its energy in the form of a photon, resulting in two photons that are coherent and have the same phase, frequency, and direction. This phenomenon is essential for understanding the operation of lasers and has significant implications in quantum mechanics, particularly in relation to transition probabilities and the interaction between matter and electromagnetic fields.
Transition Dipole Moment: The transition dipole moment is a vector quantity that represents the probability of an electronic transition between two quantum states in a system, typically associated with the absorption or emission of light. This moment depends on the difference in charge distribution between the initial and final states, providing insights into selection rules and transition probabilities during molecular transitions.