Atoms

Question 1

What are the four complete quantum numbers of an atomic electron

Answer 1

n,l,m,n
n= principle quantum number
l = angular quantum number
m = magnetic quantum number
s = spin quantum number

Question 2

What is principle quantum number

Answer 2

Quantum number that represents the energy level of an electron in an atom. It determines the size and energy of the electron’s orbital.

Question 3

What is angular momentum quantum number

Answer 3

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.

Question 4

What is magnetic quantum number

Answer 4

Quantum number that specifies the orientation of the orbital in space. It ranges from -l to +l for a given angular momentum quantum number.

Question 5

What is spin quantum number

Answer 5

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).

Question 6

What is Pauli Exclusion Principle

Answer 6

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.

Question 7

What is time-independant pertubation theory

Answer 7

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.

Question 8

What is non Degenerative pertubation theory

Answer 8

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⟩

Question 9

What is second order pertubation theory

Answer 9

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.

Question 10

What is degenerate perturbation theory

Answer 10

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

Question 11

What is quadratic stark effect

Answer 11

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)

Question 12

What is linear stark effect

Answer 12

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.

Question 13

What is Lorentz force

Answer 13

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.

Question 14

Magnetic Dipole Moment equation

Answer 14

µ = IA.

Question 15

What is the gyrometric ratio

Answer 15

It quantifies the relationship between magnetic moment and angular momentum in particles.

Question 16

What is the Zeeman Effect

Answer 16

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.

Question 17

What is fine structure

Answer 17

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.

Question 18

What is spin-orbit coupling

Answer 18

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.

Question 19

What is lamb shift

Answer 19

The Lamb shift results from the interaction between the electron’s electric field and the quantized electromagnetic field of the vacuum.

Question 20

What is Darwin term?

Answer 20

Definition: Relativistic correction accounting for finite size of electron’s charge distribution.

Question 21

What is hyper-fine structure

Answer 21

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.

Question 22

What is Coulomb repulsion

Answer 22

The electrostatic interaction between electrons leads to Coulomb repulsion, influencing the energy levels of the atom and complicating its theoretical description.

Question 23

What is non-interacting electron approximation?

Answer 23

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.

Question 24

Why is coulom repulsion an effective screening mechanism

Answer 24

each electron experiences a reduced nuclear charge due to the presence of the other electron.

Question 25

Why must the total wave function of a system be antisymmetric under the exchange of particles

Answer 25

Pauli’s exclusion principle

Question 26

What is product Ansatz

Answer 26

Approximation used to describe quantum states of larger atoms as superpositions of (fully anti-symmetrized) states.

Question 27

What is Aufbau principle

Answer 27

Principle stating that electrons fill atomic orbitals starting from the lowest energy level, following the order determined by the Madelung rule

Question 28

What is LS coupling

Answer 28

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.

Question 29

What is JJ coupling

Answer 29

Coupling scheme dominant in heavier atoms, where spin-orbit coupling dominates over other interactions, leading to more complex energy level structures.

Question 30

What is Fermi’s golden rule, and how does it preserve energy?

Answer 30

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.

Question 31

Why does Fermi’s golden rule become infinite for large times, and how is it useful?

Answer 31

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.

Question 32

What is the formula for the total transition rate in Fermi’s golden rule, and how does it relate to absorption and emission?

Answer 32

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.

Question 33

What determines the emission or absorption of energy in an atom during a transition?

Answer 33

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.

Question 34

What should the pertubation Hamiltonian matrix take account of

Answer 34

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

Question 35

What condition of the matrix element must be satisfied for transitions to be allowed

Answer 35

Matrix element of H’ must be non zero

Question 36

What is light linearly polarised along z axis called and why

Answer 36

π-polarised as E-field vector is parallel to quantisation axis therefore ∆m = 0

Question 37

What is light polarised in the plane orthogonal to quantisation axis called, why and m=?

Answer 37

σ-polarisation, direction of propagation can be along z -direction.
so For σ±-polarised light: ∆m = ±1

Question 38

What are single photon transitions

Answer 38

Transitions under the electric dipole approximation

Question 39

What is the parity selection rule, and how does it affect wave function coupling?

Answer 39

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.

Question 40

How does the angular momentum of an electron relate to photon spin, and what is the selection rule for electric dipole transitions?

Answer 40

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.

Question 41

What happens to the total orbital angular momentum during electric dipole transitions, and what is the selection rule for total angular momentum?

Answer 41

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.

Question 42

Does electric dipole interactiom change spin and why?

Answer 42

No because Hamiltonian that contains electric fields does not depend on spin

Question 43

What is Kirchhoff’s law of thermal radiation

Answer 43

Emissivity and absorbitivity of any surface will be equal at a given temperature and frequency

Question 44

What is a black body

Answer 44

onject aborbing all incoming radiation

Question 45

What is the spectral density of modes

Answer 45

Number of ways an EM field can oscillate

d^2n/dV dν = (8πv^2)/c^3

Question 46

Energy spectral density according to Rayleigh- Jeans law

Answer 46

ρν(ν) = ( d2n/dVdν )kBT = 8πkBTv^2/ c^3

Question 47

What are the three processes described by Einstein coefficients, and how are they related to transitions between energy levels?

Answer 47

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.

Question 48

How are absorption, stimulated emission, and spontaneous emission depicted?

Answer 48

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.

Question 49

interaction energy of an atom’s dipole moment in an electromagnetic field

Answer 49

𝐻0=−𝑑⋅𝐸

Question 50

What do the Einstein coefficients A21and B12 represent?

Answer 50

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.

Question 51

How do absorption and stimulated emission differ from spontaneous emission in terms of coherence?

Answer 51

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.

Question 52

In a hydrogen atom, how is the interaction energy of the dipole moment described in an electromagnetic field?

Answer 52

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.

Question 53

How is the matrix element
𝐻𝑚𝑛0 related to the electromagnetic energy density
𝑈
U?

Answer 53

The total transition rate
𝑊𝑚𝑛 is proportional to ∣𝐻𝑚𝑛0∣^2, where
𝑈
U represents the electromagnetic energy density.

Question 54

What is Kirchhoff’s Law of thermal radiation

Answer 54

Emissivity and absorbitivity of any surface will be equal at a given temperature and frequency

Question 55

what is a blackbody

Answer 55

an object that absorbs all incoming radiation

Question 56

What is the equipartition thereom

Answer 56

Each vibrating mode contributes to an average energy of <E> = 2×kBT /2</E>