Atoms Flashcards
What are the four complete quantum numbers of an atomic electron
n,l,m,n
n= principle quantum number
l = angular quantum number
m = magnetic quantum number
s = spin quantum number
What is principle quantum number
Quantum number that represents the energy level of an electron in an atom. It determines the size and energy of the electron’s orbital.
What is angular momentum quantum number
Quantum number that specifies the shape of the orbital and the magnitude of the angular momentum. It ranges from 0 to (n-1) for a given principal quantum number.
What is magnetic quantum number
Quantum number that specifies the orientation of the orbital in space. It ranges from -l to +l for a given angular momentum quantum number.
What is spin quantum number
Quantum number that describes the intrinsic angular momentum of a particle. It has a value of either +1/2 (for spin-up) or -1/2 (for spin-down).
What is Pauli Exclusion Principle
Principle stating that no two electrons in an atom can have the same set of quantum numbers, therefore prohibiting electrons from occupying the same quantum state simultaneously.
What is time-independant pertubation theory
Time-independent perturbation theory is a method used to analyze small deviations from a known Hamiltonian.
It helps to understand discrepancies between theoretical predictions and experimental data in quantum systems.
The theory involves expanding the perturbed Hamiltonian in terms of a small parameter λ.
What is non Degenerative pertubation theory
Non-degenerate perturbation theory applies when each energy eigenvalue of the unperturbed
Hamiltonian corresponds to a unique eigenfunction.
First-order perturbation theory yields the first-order energy shift using the formula: E^(1)_n = ⟨Φ_n|H^(0)|Φ_n⟩
The first-order correction to the state |Ψ^(1)_n⟩ is calculated using: |Ψ^(1)n⟩ = ∑(m≠n) ⟨Φ_m|H^(0)|Φ_n⟩/(E^(0)_n - E^(0)_m) |Φ_m⟩
What is second order pertubation theory
Second-order perturbation theory is employed when first-order corrections are insufficient.
The second-order energy shift is given by: E^(2)n = ∑(m≠n) |⟨Φ_m|H^(0)|Φ_n⟩|^2 / (E^(0)_n - E^(0)_m)
Higher-order corrections may be necessary for accurate predictions, especially in strongly perturbed systems.
What is degenerate perturbation theory
Degenerate perturbation theory deals with cases where multiple eigenfunctions share the same eigenvalue.
It involves diagonalizing the perturbation Hamiltonian within the degenerate subspace.
The first-order energy shifts are determined by solving the eigenvalue problem: H^(0) · a_n = E^(1)_n a_n
What is quadratic stark effect
The quadratic Stark effect occurs when an electric field perturbs a ground state hydrogen atom.
First-order energy shifts vanish due to symmetry, but second-order shifts are significant.
The second-order energy shift for the ground state is given by: E^(2)_100 ≈ -1.5 (me a_0^4 e^2 E^2_z / ~^2)
What is linear stark effect
In the linear Stark effect, hydrogen atoms in an electric field are excited to n = 2 states.
Matrix elements of the perturbation Hamiltonian must be computed to find energy shifts.
Only certain states experience energy shifts, while others remain unchanged.
What is Lorentz force
The complete electromagnetic force acting on a particle with charge q and velocity v, given by the expression F = q(E + v × B), where E is the electric field and B is the magnetic field.
Magnetic Dipole Moment equation
µ = IA.
What is the gyrometric ratio
It quantifies the relationship between magnetic moment and angular momentum in particles.
What is the Zeeman Effect
It describes the additional energy levels and spectral splitting observed in the presence of a magnetic field.
caused by the interaction between the magnetic moment of atoms and the external magnetic field.
What is fine structure
It explains discrepancies between predicted and observed spectral lines in atomic spectra.
Small-scale splitting of spectral lines in atomic spectra due to relativistic and quantum mechanical effects, such as spin-orbit coupling and the Lamb shift.
What is spin-orbit coupling
It contributes to the fine structure of atomic spectra, affecting energy levels and spectral line splitting.
Interaction between the spin and orbital angular momentum of particles, resulting in energy shifts in atomic spectra. It arises from the relativistic correction to the kinetic energy and the magnetic moment of the electron’s spin.
What is lamb shift
The Lamb shift results from the interaction between the electron’s electric field and the quantized electromagnetic field of the vacuum.
What is Darwin term?
Definition: Relativistic correction accounting for finite size of electron’s charge distribution.
What is hyper-fine structure
Description: Arises from interactions between nuclear and electron magnetic moments.
Contributions: Results in additional energy shifts, observable as fine structure level splitting.
Notable Features: Splits fine structure levels into doublets, affecting spectral line patterns.
What is Coulomb repulsion
The electrostatic interaction between electrons leads to Coulomb repulsion, influencing the energy levels of the atom and complicating its theoretical description.
What is non-interacting electron approximation?
In the non-interacting electron approximation, each electron is treated independently, neglecting the electron-electron repulsion. However, this approach leads to an overestimate of the atom’s ionization energy.
Why is coulom repulsion an effective screening mechanism
each electron experiences a reduced nuclear charge due to the presence of the other electron.
Why must the total wave function of a system be antisymmetric under the exchange of particles
Pauli’s exclusion principle
What is product Ansatz
Approximation used to describe quantum states of larger atoms as superpositions of (fully anti-symmetrized) states.
What is Aufbau principle
Principle stating that electrons fill atomic orbitals starting from the lowest energy level, following the order determined by the Madelung rule
What is LS coupling
Coupling scheme where total spin (S) and total orbital angular momentum (L) of electrons are considered, following Hund’s rules for determining ground state configurations.
What is JJ coupling
Coupling scheme dominant in heavier atoms, where spin-orbit coupling dominates over other interactions, leading to more complex energy level structures.
What is Fermi’s golden rule, and how does it preserve energy?
Fermi’s golden rule is a quantum mechanical formula describing transition rates between quantum states in a perturbed system.
It preserves energy by allowing transitions only when the energy difference between initial and final states is approximately equal to the energy of the perturbation.
Why does Fermi’s golden rule become infinite for large times, and how is it useful?
The first-order approximation in Fermi’s golden rule breaks down for long times, leading to an infinite transition rate. However, it’s useful when dealing with dense distributions of states with similar energies, such as in a gas of atoms with slightly different energies due to their local environments.
What is the formula for the total transition rate in Fermi’s golden rule, and how does it relate to absorption and emission?
The total transition rate
𝑊𝑛 is given by an integral of transition rates to nearby states, weighted by the density of states. It describes absorption when the energy of the target state is higher than the original state and emission when the energy of the target state is lower. Additionally, it states that the total transition rate is proportional to the density of states, representing the number of states per energy interval.
What determines the emission or absorption of energy in an atom during a transition?
A resonant transition occurs when an atom moves from a state with energy
𝐸𝑛 to a state with energy 𝐸𝑚 , driven by a perturbation of frequency
𝜔 = ∣𝐸𝑚 − 𝐸𝑛∣/ℏ.
The emission or absorption rate is determined by the matrix element ∣𝐻0𝑚𝑛∣^2, which reflects experimental observations and accounts for factors such as transition probabilities, spectral line strengths, and polarization effects.
What should the pertubation Hamiltonian matrix take account of
Certain transitions, e.g. between a 1S and 2S level, do not normally result in photon emission or absorption.
* Spectral lines can vary in strength by orders of magnitude depending on the levels involved. This implies
large differences in the rates of transitions.
* In the interaction with electromagnetic fields, the polarisation (orientation of those fields) plays a role.
In addition, we need to consider that
* Spectral lines corresp
What condition of the matrix element must be satisfied for transitions to be allowed
Matrix element of H’ must be non zero
What is light linearly polarised along z axis called and why
π-polarised as E-field vector is parallel to quantisation axis therefore ∆m = 0
What is light polarised in the plane orthogonal to quantisation axis called, why and m=?
σ-polarisation, direction of propagation can be along z -direction.
so For σ±-polarised light: ∆m = ±1
What are single photon transitions
Transitions under the electric dipole approximation
What is the parity selection rule, and how does it affect wave function coupling?
The parity selection rule states that wave functions can only be coupled if they have opposite parity: one even and one odd. This means that the product of wave functions must be even for coupling to occur.
How does the angular momentum of an electron relate to photon spin, and what is the selection rule for electric dipole transitions?
Angular momentum conservation requires that the angular momentum quantum number
𝑙 changes by one unit upon absorption or emission of a photon. The selection rule for electric dipole transitions is Δ𝑙 = ±1, indicating that the angular momentum changes by one unit.
What happens to the total orbital angular momentum during electric dipole transitions, and what is the selection rule for total angular momentum?
During electric dipole transitions, the total orbital angular momentum
𝐿 may remain unchanged or change by Δ𝐿=0 or ±1. However, transitions where L=0 remains
L=0 are not allowed. This indicates that the relative orientation of orbital angular momenta may change during transitions.
Does electric dipole interactiom change spin and why?
No because Hamiltonian that contains electric fields does not depend on spin
What is Kirchhoff’s law of thermal radiation
Emissivity and absorbitivity of any surface will be equal at a given temperature and frequency
What is a black body
onject aborbing all incoming radiation
What is the spectral density of modes
Number of ways an EM field can oscillate
d^2n/dV dν = (8πv^2)/c^3
Energy spectral density according to Rayleigh- Jeans law
ρν(ν) = ( d2n/dVdν )kBT = 8πkBTv^2/ c^3
What are the three processes described by Einstein coefficients, and how are they related to transitions between energy levels?
The three processes are absorption (1 → 2), stimulated emission (2 → 1), and spontaneous emission (2 → 1). Absorption and stimulated emission depend on the energy spectral density
𝜌𝜈(𝜈), while spontaneous emission is independent. The absorption rate per atom is 𝐵12𝜌𝜈(𝜈), stimulated emission rate is 𝐵21𝜌𝜈(𝜈), and the spontaneous emission rate is
𝐴21.
How are absorption, stimulated emission, and spontaneous emission depicted?
Absorption and stimulated emission processes are coherent and depend on the energy spectral density of the light field.
Spontaneous emission is independent and incoherent.
where the first two processes exhibit fixed phases between the atomic dipole oscillation and the local electric field, while spontaneous emission is random and directionally independent.
interaction energy of an atom’s dipole moment in an electromagnetic field
𝐻0=−𝑑⋅𝐸
What do the Einstein coefficients A21and B12 represent?
A21 represents the rate of spontaneous emission from state 2 to state 1,
while B12
represents the rate of absorption from state 1 to state 2 per unit spectral energy density.
How do absorption and stimulated emission differ from spontaneous emission in terms of coherence?
Absorption and stimulated emission are coherent processes, meaning they exhibit fixed phases between the atomic dipole oscillation and the local electric field, while spontaneous emission is incoherent.
In a hydrogen atom, how is the interaction energy of the dipole moment described in an electromagnetic field?
The interaction energy is described by the perturbation Hamiltonian
𝐻0=−𝑒𝑟, where
𝑒
e is the elementary charge and
𝑟
r is the position vector of the electron relative to the nucleus.
How is the matrix element
𝐻𝑚𝑛0 related to the electromagnetic energy density
𝑈
U?
The total transition rate
𝑊𝑚𝑛 is proportional to ∣𝐻𝑚𝑛0∣^2, where
𝑈
U represents the electromagnetic energy density.
What is Kirchhoff’s Law of thermal radiation
Emissivity and absorbitivity of any surface will be equal at a given temperature and frequency
what is a blackbody
an object that absorbs all incoming radiation
What is the equipartition thereom
Each vibrating mode contributes to an average energy of <E> = 2×kBT /2</E>
