How does the continuity equation mathematically demonstrate conservation of electric charge from Maxwell's equations?

Taking the divergence of the Ampère-Maxwell Law: ∇ · (∇ x B) = μ0 · [∇ · J + ε0 · ∂(∇ · E)/∂t] Since the divergence of any curl is identically zero (∇·(∇xB) = 0), and substituting Gauss's Law (∇·E = ρ/ε0): 0 = ∇ · J + ∂ρ / ∂t ==> ∇ · J + ∂ρ / ∂t = 0 This is the continuity equation, proving local conservation of electric charge.

How do Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein quantum distribution statistics differ?

• Maxwell-Boltzmann: Classical, distinguishable particles; no occupancy restrictions; f(E) = A·e^(-E/k_B T). • Fermi-Dirac: Quantum identical fermions (half-integer spin, e.g., electrons, quarks); obey Pauli Exclusion Principle (max 1 particle per state); f(E) = 1 / [e^((E - μ)/k_B T) + 1]. • Bose-Einstein: Quantum identical bosons (integer spin, e.g., photons, gluons, He-4); no occupancy restrictions (unlimited particles per state); f(E) = 1 / [e^((E - μ)/k_B T) - 1].

What are the two fundamental postulates of Einstein's Special Relativity (1905)?

1. Principle of Relativity: The laws of physics are invariant (identical in form) in all inertial reference frames. 2. Invariance of the Speed of Light: The speed of light in vacuum (c) is a universal constant (299,792,458 m/s) in all inertial reference frames, independent of the motion of the emitting light source or observer.

Why is Iron-56 (and Nickel-62) at the peak of the nuclear binding energy per nucleon curve?

Iron-56 and Nickel-62 exhibit the maximum binding energy per nucleon (~8.8 MeV/nucleon) in the universe. Lighter nuclei release energy through nuclear fusion (moving toward iron), whereas heavier nuclei release energy through nuclear fission (splitting toward iron). Fusing elements heavier than iron or fissioning elements lighter than iron is endothermic (consumes energy).

What does Bell's Theorem state regarding quantum entanglement and local hidden variable theories?

Bell's Theorem proves that no physical theory based on local realism (local hidden variables) can reproduce all statistical predictions of quantum mechanics. Experiments testing Bell inequalities (e.g., Aspect, Clauser, Zeilinger) consistently violate Bell's inequality, confirming quantum entanglement involves non-local correlations that cannot be explained by pre-existing local parameters.

What is Einstein's relativistic velocity addition formula for parallel velocities?

u = (u' + v) / (1 + (u' · v / c²)) • u = Velocity observed in stationary frame S • u' = Velocity observed in moving frame S' • v = Velocity of frame S' relative to S If u' = c (e.g., a photon), u = (c + v) / (1 + v/c) = c, proving the speed of light cannot be exceeded by adding velocities.

What is the spacetime interval (Δs²) in Minkowski spacetime and how are timelike, spacelike, and lightlike intervals defined?

Δs² = (c · Δt)² - (Δx)² - (Δy)² - (Δz)² This interval is invariant under all Lorentz boosts: • Timelike (Δs² > 0): Events can be causally connected; a reference frame exists where events occur at the same spatial position. • Spacelike (Δs² < 0): Events cannot be causally connected without superluminal signals; a frame exists where events are simultaneous. • Lightlike / Null (Δs² = 0): Connected only by signals traveling at speed of light.

What is gauge invariance in classical electrodynamics and how do the Coulomb and Lorenz gauges differ?

Electromagnetic potentials (scalar potential φ, vector potential A) can be transformed by A' = A + ∇λ and φ' = φ - ∂λ/∂t without altering observable physical fields (B = ∇xA, E = -∇φ - ∂A/∂t). • Coulomb gauge: ∇ · A = 0 (simplifies electrostatics; φ(r) determined instantly by instantaneous charge distribution). • Lorenz gauge: ∇ · A + (1/c²)·(∂φ / ∂t) = 0 (explicitly Lorentz-invariant; decouples wave equations for A and φ in special relativity).

How is the electromagnetic wave equation in vacuum derived from Maxwell's equations?

Taking the curl of Faraday's Law in free charge-free vacuum (ρ = 0, J = 0): ∇ x (∇ x E) = -∂(∇ x B) / ∂t Using the vector identity ∇ x (∇ x E) = ∇(∇·E) - ∇²E and Gauss's Law (∇·E = 0): -∇²E = -μ0·ε0 · (∂²E / ∂t²) Yielding the 3D wave equation: ∇²E - (1/c²) · (∂²E / ∂t²) = 0, where c = 1 / sqrt(μ0·ε0).

What was the 'Ultraviolet Catastrophe' in blackbody radiation, and how did Max Planck resolve it?

The classical Rayleigh-Jeans law predicted that blackbody radiation energy density would diverge to infinity at ultraviolet and higher frequencies (u(f) ∝ f²). Max Planck resolved this (1900) by hypothesizing that microscopic harmonic oscillators in cavity walls exchange energy with electromagnetic radiation only in discrete quantized packets: E = n · h · f.

What is Compton scattering and what is the formula for the wavelength shift of scattered photons?

Compton scattering is the inelastic scattering of an X-ray or gamma-ray photon by a free or loosely bound electron, demonstrating that photons carry relativistic momentum (p = h/λ). Wavelength shift: Δλ = λ' - λ = (h / (m_e · c)) · (1 - cos(θ)) where h/(m_e·c) ≈ 2.426 x 10⁻¹² m is the Compton wavelength of the electron, and θ is the photon scattering angle.

What are Maxwell's four differential equations in vector calculus form?

1. ∇ · E = ρ / ε0 (Gauss's Law for Electricity) 2. ∇ · B = 0 (Gauss's Law for Magnetism - no magnetic monopoles) 3. ∇ x E = -∂B / ∂t (Faraday's Law of Induction) 4. ∇ x B = μ0·J + μ0·ε0·(∂E / ∂t) (Ampère-Maxwell Circuital Law)

What is the statistical definition of temperature in microcanonical statistical mechanics?

1 / T = ∂S / ∂E |_V,N where T is absolute temperature, S is entropy, E is internal energy, and volume V and particle number N are held constant. Temperature is the reciprocal of the rate of change of entropy with respect to internal energy; thermal equilibrium between systems occurs when their values of ∂S/∂E equalize.

What is the Bethe-Weizsäcker Semi-Empirical Mass Formula (SEMF) for nuclear binding energy?

B(A, Z) = a_v·A - a_s·A^(2/3) - a_c·(Z²/A^(1/3)) - a_a·((A - 2Z)²/A) ± δ(A, Z) Components: 1. Volume term (a_v·A): Strong nuclear attraction among nearest neighbors. 2. Surface term (-a_s·A^(2/3)): Reduced binding for surface nucleons. 3. Coulomb term (-a_c·Z²/A^(1/3)): Electrostatic repulsion between protons. 4. Asymmetry term (-a_a·(A-2Z)²/A): Pauli exclusion penalty for N ≠ Z. 5. Pairing term (±δ): Enhanced stability for even-even nuclei.

How does the atmospheric muon decay experiment provide empirical verification of special relativity?

Muons produced in Earth's upper atmosphere by cosmic rays have a rest frame mean lifetime of τ0 ≈ 2.2 μs. Classically, even at c, they could only travel ~660 meters before decaying. Yet they reach surface detectors 10 km below because: • Earth frame: Relativistic time dilation stretches muon lifetime to γ·τ0 (~30 μs). • Muon frame: Relativistic length contraction shortens the 10 km atmosphere to L0 / γ (~660 m).

What is the Higgs Mechanism and what fundamental role does the Higgs Boson play in the Standard Model?

The Higgs mechanism describes spontaneous electroweak symmetry breaking via a non-zero vacuum expectation value of the scalar Higgs field (~246 GeV). Elementary particles (quarks, charged leptons, W/Z bosons) acquire inertial mass through Yukawa and gauge couplings with this ubiquitous background field. The Higgs boson (discovered 2012 at LHC, ~125 GeV) is the scalar excitation of this field.

What are the Pauli spin matrices (σx, σy, σz) and what are the spin angular momentum eigenvalues for an electron?

Pauli matrices: σx = [[0, 1], [1, 0]], σy = [[0, -i], [i, 0]], σz = [[1, 0], [0, -1]] Spin angular momentum operators: S = (ħ / 2) · σ. For spin-1/2 fermions like electrons: • Total spin magnitude: S² = s·(s + 1)·ħ² = (3/4)·ħ² • Z-axis spin projection Sz eigenvalues: ms · ħ = ± (1/2) · ħ ('spin up' and 'spin down').

What are the energy eigenvalues and wavefunctions for a particle of mass m in a 1D infinite square well of width L?

• Energy eigenvalues: En = (n² · h²) / (8 · m · L²) = (n² · π² · ħ²) / (2 · m · L²) (for n = 1, 2, 3...) • Stationary wavefunctions: Ψn(x) = sqrt(2 / L) · sin(n·π·x / L) Energy is strictly quantized; the ground state (n = 1) possesses non-zero zero-point energy E1 > 0, consistent with Heisenberg's Uncertainty Principle.

What are the nuclear equations and emission products of Alpha decay, Beta-minus decay, and Beta-plus decay?

• Alpha decay: (A, Z)X -> (A-4, Z-2)Y + ⁴₂He²⁺ (emits helium nucleus) • Beta-minus (β⁻) decay: ¹₀n -> ¹₁p + ⁰₋₁e⁻ + anti-ν_e (neutron converts to proton, emitting electron and electron antineutrino) • Beta-plus (β⁺) decay: ¹₁p -> ¹₀n + ⁰₊₁e⁺ + ν_e (proton converts to neutron, emitting positron and electron neutrino) • Gamma (γ) decay: De-excitation of metastable isomer emitting high-energy photon with no change in A or Z.

What is nuclear mass defect (Δm) and how does it relate to nuclear binding energy?

Mass defect is the difference between the sum of the individual free nucleon masses and the total mass of the assembled nucleus: Δm = [Z · m_proton + (A - Z) · m_neutron] - M_nucleus According to E = Δm · c², this mass defect represents the nuclear binding energy released when nucleons coalesce into a bound nucleus, and the energy required to disassemble it into free nucleons.

Why is molar heat capacity at constant pressure (Cp) always greater than at constant volume (Cv) for an ideal gas, and what is Mayer's relation?

When heat is added at constant volume, all energy increases internal thermal energy (ΔU = Cv·ΔT). At constant pressure, the gas expands against surroundings, performing mechanical work (P·ΔV); additional heat is required to do this work while raising temperature by the same ΔT. Mayer's relation for an ideal gas: Cp - Cv = R (where R is the ideal gas constant, 8.314 J/(mol·K)).

What are the four fundamental interactions and their corresponding gauge mediator bosons in the Standard Model?

1. Electromagnetic Force: Mediated by the massless Photon (γ); infinite range; couples to electric charge. 2. Weak Force: Mediated by massive W⁺, W⁻, and Z⁰ bosons; subatomic range (~10⁻¹⁸ m); responsible for flavor changes and beta decay. 3. Strong Force: Mediated by 8 massless Gluons (g); binds quarks into hadrons via color charge; effective range ~10⁻¹⁵ m. 4. Gravitational Force: Hypothetically mediated by spin-2 Graviton; infinite range; couples to mass-energy (not yet unified in Standard Model).

What is the radioactive decay law and what is the mathematical relation between decay constant (λ) and half-life (t_1/2)?

Decay law: N(t) = N0 · e^(-λ · t) • N(t) = Number of undecayed radioactive parent nuclei at time t • N0 = Initial number of parent nuclei • λ = Decay constant (probability of decay per unit time) Half-life (time required for half of parent nuclei to decay): t_1/2 = ln(2) / λ ≈ 0.693 / λ

What is the complete relativistic energy-momentum relationship?

E² = (p · c)² + (m0 · c²)² • E = Total relativistic energy (E = γ·m0·c²) • p = Relativistic momentum (p = γ·m0·v) • m0 = Invariant rest mass • c = Speed of light For a particle at rest (p = 0): E0 = m0·c². For a massless particle like a photon (m0 = 0): E = p·c.

What is the Larmor Formula in electrodynamics and what does it calculate?

The Larmor formula calculates the total power P radiated as electromagnetic waves by a non-relativistic point charge q undergoing linear acceleration a in vacuum: P = (q² · a²) / (6π · ε0 · c³) Radiated power is proportional to the square of acceleration. This explained why classical physics predicted Rutherford's planetary atom should collapse in ~10⁻¹¹ seconds, necessitating quantum mechanics.

What is the canonical partition function (Z) and why is it central to statistical thermodynamics?

Z = Σ_i e^(-E_i / (k_B · T)) = Σ_i e^(-β · E_i) where E_i represents energy levels, k_B is Boltzmann's constant, and β = 1/(k_B·T). The partition function encodes all thermodynamic properties of a system in contact with a heat bath: Helmholtz free energy F = -k_B·T·ln(Z), average energy = -∂ln(Z)/∂β, and entropy S = -∂F/∂T.

What is the Clausius-Clapeyron equation and what physical phase equilibrium does it describe?

dP / dT = ΔH_phase / (T · ΔV_phase) • dP/dT = Slope of the phase coexistence boundary on a P-T diagram • ΔH_phase = Latent heat of the phase transition (e.g., vaporization) • T = Absolute temperature • ΔV_phase = Change in molar or specific volume across the phase transition Governs how equilibrium vapor pressure varies with temperature.

What is the Lorentz factor (γ) and what are the Lorentz transformation equations for coordinates moving along the x-axis at speed v?

Lorentz factor: γ = 1 / sqrt(1 - v² / c²) Transformations from frame S to S' moving at velocity v along x: • x' = γ · (x - v·t) • y' = y • z' = z • t' = γ · (t - (v·x / c²)) As v 1 and the equations reduce to classical Galilean transformations.

What is the energy spectrum of a 1D quantum harmonic oscillator of angular frequency ω?

En = (n + 1/2) · ħ · ω (for n = 0, 1, 2, 3...) Energy levels are equispaced by ΔE = ħω. The ground state (n = 0) has a non-vanishing zero-point energy: E0 = (1/2)·ħ·ω. This zero-point motion prevents quantum particles from ever coming to complete rest, even at absolute zero (0 K).

What are the elementary fermions (quarks and leptons) in the Standard Model of particle physics?

• Quarks (spin-1/2, feel strong force, color charge): - Generation 1: Up (u, +2/3e), Down (d, -1/3e) - Generation 2: Charm (c, +2/3e), Strange (s, -1/3e) - Generation 3: Top (t, +2/3e), Bottom (b, -1/3e) • Leptons (spin-1/2, do not feel strong force): - Generation 1: Electron (e⁻, -1e), Electron neutrino (ν_e, 0) - Generation 2: Muon (μ⁻, -1e), Muon neutrino (ν_μ, 0) - Generation 3: Tau (τ⁻, -1e), Tau neutrino (ν_τ, 0)

What is the relationship between pressure and volume during a quasi-static reversible adiabatic expansion of an ideal gas?

P · V^γ = constant where γ = Cp / Cv is the heat capacity ratio (adiabatic index; γ = 5/3 for monatomic gases, γ = 7/5 for diatomic gases). Also expressed as T · V^(γ - 1) = constant and T^γ · P^(1 - γ) = constant. The curve is steeper than an isothermal P-V curve because gas cools during adiabatic expansion.

What are relativistic time dilation and length contraction equations, and what is proper time / proper length?

• Time dilation: Δt = γ · Δt0 where Δt0 is 'proper time' measured by a clock at rest relative to the events (Δt >= Δt0; moving clocks tick slower). • Length contraction: L = L0 / γ where L0 is 'proper length' measured in the rest frame of the object (L <= L0; objects appear contracted along the axis of relative motion).

Ingeniería Mecánica — Avanzado (C1/C2) 1: Thermodynamics, Maxwell's Equations, Relativity & Quantum Physics

[Español] Ingeniería Mecánica: Advanced physics covering statistical thermodynamics, Maxwell's differential equations and field radiation, Einstein's special relativity, quantum mechanics, and nuclear/subatomic particle physics.

Level: advanced
Languages: es → es
Cards: 32
Markdown: Ingeniería Mecánica — Avanzado (C1/C2) 1: Thermodynamics, Maxwell's Equations, Relativity & Quantum Physics (.md)

Flashcards (32)

#	Front (Term / Question)	Back (Definition / Answer)
1	How does the continuity equation mathematically demonstrate conservation of electric charge from Maxwell's equations?	Taking the divergence of the Ampère-Maxwell Law: ∇ · (∇ x B) = μ0 · [∇ · J + ε0 · ∂(∇ · E)/∂t] Since the divergence of any curl is identically zero (∇·(∇xB) = 0), and substituting Gauss's Law (∇·E = ρ/ε0): 0 = ∇ · J + ∂ρ / ∂t ==> ∇ · J + ∂ρ / ∂t = 0 This is the continuity equation, proving local conservation of electric charge.
2	How do Maxwell-Boltzmann, Fermi-Dirac, and Bose-Einstein quantum distribution statistics differ?	• Maxwell-Boltzmann: Classical, distinguishable particles; no occupancy restrictions; f(E) = A·e^(-E/k_B T). • Fermi-Dirac: Quantum identical fermions (half-integer spin, e.g., electrons, quarks); obey Pauli Exclusion Principle (max 1 particle per state); f(E) = 1 / [e^((E - μ)/k_B T) + 1]. • Bose-Einstein: Quantum identical bosons (integer spin, e.g., photons, gluons, He-4); no occupancy restrictions (unlimited particles per state); f(E) = 1 / [e^((E - μ)/k_B T) - 1].
3	What are the two fundamental postulates of Einstein's Special Relativity (1905)?	1. Principle of Relativity: The laws of physics are invariant (identical in form) in all inertial reference frames. 2. Invariance of the Speed of Light: The speed of light in vacuum (c) is a universal constant (299,792,458 m/s) in all inertial reference frames, independent of the motion of the emitting light source or observer.
4	Why is Iron-56 (and Nickel-62) at the peak of the nuclear binding energy per nucleon curve?	Iron-56 and Nickel-62 exhibit the maximum binding energy per nucleon (~8.8 MeV/nucleon) in the universe. Lighter nuclei release energy through nuclear fusion (moving toward iron), whereas heavier nuclei release energy through nuclear fission (splitting toward iron). Fusing elements heavier than iron or fissioning elements lighter than iron is endothermic (consumes energy).
5	What does Bell's Theorem state regarding quantum entanglement and local hidden variable theories?	Bell's Theorem proves that no physical theory based on local realism (local hidden variables) can reproduce all statistical predictions of quantum mechanics. Experiments testing Bell inequalities (e.g., Aspect, Clauser, Zeilinger) consistently violate Bell's inequality, confirming quantum entanglement involves non-local correlations that cannot be explained by pre-existing local parameters.
6	What is Einstein's relativistic velocity addition formula for parallel velocities?	u = (u' + v) / (1 + (u' · v / c²)) • u = Velocity observed in stationary frame S • u' = Velocity observed in moving frame S' • v = Velocity of frame S' relative to S If u' = c (e.g., a photon), u = (c + v) / (1 + v/c) = c, proving the speed of light cannot be exceeded by adding velocities.
7	What is the spacetime interval (Δs²) in Minkowski spacetime and how are timelike, spacelike, and lightlike intervals defined?	Δs² = (c · Δt)² - (Δx)² - (Δy)² - (Δz)² This interval is invariant under all Lorentz boosts: • Timelike (Δs² > 0): Events can be causally connected; a reference frame exists where events occur at the same spatial position. • Spacelike (Δs² < 0): Events cannot be causally connected without superluminal signals; a frame exists where events are simultaneous. • Lightlike / Null (Δs² = 0): Connected only by signals traveling at speed of light.
8	What is gauge invariance in classical electrodynamics and how do the Coulomb and Lorenz gauges differ?	Electromagnetic potentials (scalar potential φ, vector potential A) can be transformed by A' = A + ∇λ and φ' = φ - ∂λ/∂t without altering observable physical fields (B = ∇xA, E = -∇φ - ∂A/∂t). • Coulomb gauge: ∇ · A = 0 (simplifies electrostatics; φ(r) determined instantly by instantaneous charge distribution). • Lorenz gauge: ∇ · A + (1/c²)·(∂φ / ∂t) = 0 (explicitly Lorentz-invariant; decouples wave equations for A and φ in special relativity).
9	How is the electromagnetic wave equation in vacuum derived from Maxwell's equations?	Taking the curl of Faraday's Law in free charge-free vacuum (ρ = 0, J = 0): ∇ x (∇ x E) = -∂(∇ x B) / ∂t Using the vector identity ∇ x (∇ x E) = ∇(∇·E) - ∇²E and Gauss's Law (∇·E = 0): -∇²E = -μ0·ε0 · (∂²E / ∂t²) Yielding the 3D wave equation: ∇²E - (1/c²) · (∂²E / ∂t²) = 0, where c = 1 / sqrt(μ0·ε0).
10	What was the 'Ultraviolet Catastrophe' in blackbody radiation, and how did Max Planck resolve it?	The classical Rayleigh-Jeans law predicted that blackbody radiation energy density would diverge to infinity at ultraviolet and higher frequencies (u(f) ∝ f²). Max Planck resolved this (1900) by hypothesizing that microscopic harmonic oscillators in cavity walls exchange energy with electromagnetic radiation only in discrete quantized packets: E = n · h · f.
11	What is Compton scattering and what is the formula for the wavelength shift of scattered photons?	Compton scattering is the inelastic scattering of an X-ray or gamma-ray photon by a free or loosely bound electron, demonstrating that photons carry relativistic momentum (p = h/λ). Wavelength shift: Δλ = λ' - λ = (h / (m_e · c)) · (1 - cos(θ)) where h/(m_e·c) ≈ 2.426 x 10⁻¹² m is the Compton wavelength of the electron, and θ is the photon scattering angle.
12	What are Maxwell's four differential equations in vector calculus form?	1. ∇ · E = ρ / ε0 (Gauss's Law for Electricity) 2. ∇ · B = 0 (Gauss's Law for Magnetism - no magnetic monopoles) 3. ∇ x E = -∂B / ∂t (Faraday's Law of Induction) 4. ∇ x B = μ0·J + μ0·ε0·(∂E / ∂t) (Ampère-Maxwell Circuital Law)
13	What is the statistical definition of temperature in microcanonical statistical mechanics?	1 / T = ∂S / ∂E \|_V,N where T is absolute temperature, S is entropy, E is internal energy, and volume V and particle number N are held constant. Temperature is the reciprocal of the rate of change of entropy with respect to internal energy; thermal equilibrium between systems occurs when their values of ∂S/∂E equalize.
14	What is the Bethe-Weizsäcker Semi-Empirical Mass Formula (SEMF) for nuclear binding energy?	B(A, Z) = a_v·A - a_s·A^(2/3) - a_c·(Z²/A^(1/3)) - a_a·((A - 2Z)²/A) ± δ(A, Z) Components: 1. Volume term (a_v·A): Strong nuclear attraction among nearest neighbors. 2. Surface term (-a_s·A^(2/3)): Reduced binding for surface nucleons. 3. Coulomb term (-a_c·Z²/A^(1/3)): Electrostatic repulsion between protons. 4. Asymmetry term (-a_a·(A-2Z)²/A): Pauli exclusion penalty for N ≠ Z. 5. Pairing term (±δ): Enhanced stability for even-even nuclei.
15	How does the atmospheric muon decay experiment provide empirical verification of special relativity?	Muons produced in Earth's upper atmosphere by cosmic rays have a rest frame mean lifetime of τ0 ≈ 2.2 μs. Classically, even at c, they could only travel ~660 meters before decaying. Yet they reach surface detectors 10 km below because: • Earth frame: Relativistic time dilation stretches muon lifetime to γ·τ0 (~30 μs). • Muon frame: Relativistic length contraction shortens the 10 km atmosphere to L0 / γ (~660 m).
16	What is the Higgs Mechanism and what fundamental role does the Higgs Boson play in the Standard Model?	The Higgs mechanism describes spontaneous electroweak symmetry breaking via a non-zero vacuum expectation value of the scalar Higgs field (~246 GeV). Elementary particles (quarks, charged leptons, W/Z bosons) acquire inertial mass through Yukawa and gauge couplings with this ubiquitous background field. The Higgs boson (discovered 2012 at LHC, ~125 GeV) is the scalar excitation of this field.
17	What are the Pauli spin matrices (σx, σy, σz) and what are the spin angular momentum eigenvalues for an electron?	Pauli matrices: σx = [[0, 1], [1, 0]], σy = [[0, -i], [i, 0]], σz = [[1, 0], [0, -1]] Spin angular momentum operators: S = (ħ / 2) · σ. For spin-1/2 fermions like electrons: • Total spin magnitude: S² = s·(s + 1)·ħ² = (3/4)·ħ² • Z-axis spin projection Sz eigenvalues: ms · ħ = ± (1/2) · ħ ('spin up' and 'spin down').
18	What are the energy eigenvalues and wavefunctions for a particle of mass m in a 1D infinite square well of width L?	• Energy eigenvalues: En = (n² · h²) / (8 · m · L²) = (n² · π² · ħ²) / (2 · m · L²) (for n = 1, 2, 3...) • Stationary wavefunctions: Ψn(x) = sqrt(2 / L) · sin(n·π·x / L) Energy is strictly quantized; the ground state (n = 1) possesses non-zero zero-point energy E1 > 0, consistent with Heisenberg's Uncertainty Principle.
19	What are the nuclear equations and emission products of Alpha decay, Beta-minus decay, and Beta-plus decay?	• Alpha decay: (A, Z)X -> (A-4, Z-2)Y + ⁴₂He²⁺ (emits helium nucleus) • Beta-minus (β⁻) decay: ¹₀n -> ¹₁p + ⁰₋₁e⁻ + anti-ν_e (neutron converts to proton, emitting electron and electron antineutrino) • Beta-plus (β⁺) decay: ¹₁p -> ¹₀n + ⁰₊₁e⁺ + ν_e (proton converts to neutron, emitting positron and electron neutrino) • Gamma (γ) decay: De-excitation of metastable isomer emitting high-energy photon with no change in A or Z.
20	What is nuclear mass defect (Δm) and how does it relate to nuclear binding energy?	Mass defect is the difference between the sum of the individual free nucleon masses and the total mass of the assembled nucleus: Δm = [Z · m_proton + (A - Z) · m_neutron] - M_nucleus According to E = Δm · c², this mass defect represents the nuclear binding energy released when nucleons coalesce into a bound nucleus, and the energy required to disassemble it into free nucleons.
21	Why is molar heat capacity at constant pressure (Cp) always greater than at constant volume (Cv) for an ideal gas, and what is Mayer's relation?	When heat is added at constant volume, all energy increases internal thermal energy (ΔU = Cv·ΔT). At constant pressure, the gas expands against surroundings, performing mechanical work (P·ΔV); additional heat is required to do this work while raising temperature by the same ΔT. Mayer's relation for an ideal gas: Cp - Cv = R (where R is the ideal gas constant, 8.314 J/(mol·K)).
22	What are the four fundamental interactions and their corresponding gauge mediator bosons in the Standard Model?	1. Electromagnetic Force: Mediated by the massless Photon (γ); infinite range; couples to electric charge. 2. Weak Force: Mediated by massive W⁺, W⁻, and Z⁰ bosons; subatomic range (~10⁻¹⁸ m); responsible for flavor changes and beta decay. 3. Strong Force: Mediated by 8 massless Gluons (g); binds quarks into hadrons via color charge; effective range ~10⁻¹⁵ m. 4. Gravitational Force: Hypothetically mediated by spin-2 Graviton; infinite range; couples to mass-energy (not yet unified in Standard Model).
23	What is the radioactive decay law and what is the mathematical relation between decay constant (λ) and half-life (t_1/2)?	Decay law: N(t) = N0 · e^(-λ · t) • N(t) = Number of undecayed radioactive parent nuclei at time t • N0 = Initial number of parent nuclei • λ = Decay constant (probability of decay per unit time) Half-life (time required for half of parent nuclei to decay): t_1/2 = ln(2) / λ ≈ 0.693 / λ
24	What is the complete relativistic energy-momentum relationship?	E² = (p · c)² + (m0 · c²)² • E = Total relativistic energy (E = γ·m0·c²) • p = Relativistic momentum (p = γ·m0·v) • m0 = Invariant rest mass • c = Speed of light For a particle at rest (p = 0): E0 = m0·c². For a massless particle like a photon (m0 = 0): E = p·c.
25	What is the Larmor Formula in electrodynamics and what does it calculate?	The Larmor formula calculates the total power P radiated as electromagnetic waves by a non-relativistic point charge q undergoing linear acceleration a in vacuum: P = (q² · a²) / (6π · ε0 · c³) Radiated power is proportional to the square of acceleration. This explained why classical physics predicted Rutherford's planetary atom should collapse in ~10⁻¹¹ seconds, necessitating quantum mechanics.
26	What is the canonical partition function (Z) and why is it central to statistical thermodynamics?	Z = Σ_i e^(-E_i / (k_B · T)) = Σ_i e^(-β · E_i) where E_i represents energy levels, k_B is Boltzmann's constant, and β = 1/(k_B·T). The partition function encodes all thermodynamic properties of a system in contact with a heat bath: Helmholtz free energy F = -k_B·T·ln(Z), average energy <E> = -∂ln(Z)/∂β, and entropy S = -∂F/∂T.
27	What is the Clausius-Clapeyron equation and what physical phase equilibrium does it describe?	dP / dT = ΔH_phase / (T · ΔV_phase) • dP/dT = Slope of the phase coexistence boundary on a P-T diagram • ΔH_phase = Latent heat of the phase transition (e.g., vaporization) • T = Absolute temperature • ΔV_phase = Change in molar or specific volume across the phase transition Governs how equilibrium vapor pressure varies with temperature.
28	What is the Lorentz factor (γ) and what are the Lorentz transformation equations for coordinates moving along the x-axis at speed v?	Lorentz factor: γ = 1 / sqrt(1 - v² / c²) Transformations from frame S to S' moving at velocity v along x: • x' = γ · (x - v·t) • y' = y • z' = z • t' = γ · (t - (v·x / c²)) As v << c, γ -> 1 and the equations reduce to classical Galilean transformations.
29	What is the energy spectrum of a 1D quantum harmonic oscillator of angular frequency ω?	En = (n + 1/2) · ħ · ω (for n = 0, 1, 2, 3...) Energy levels are equispaced by ΔE = ħω. The ground state (n = 0) has a non-vanishing zero-point energy: E0 = (1/2)·ħ·ω. This zero-point motion prevents quantum particles from ever coming to complete rest, even at absolute zero (0 K).
30	What are the elementary fermions (quarks and leptons) in the Standard Model of particle physics?	• Quarks (spin-1/2, feel strong force, color charge): - Generation 1: Up (u, +2/3e), Down (d, -1/3e) - Generation 2: Charm (c, +2/3e), Strange (s, -1/3e) - Generation 3: Top (t, +2/3e), Bottom (b, -1/3e) • Leptons (spin-1/2, do not feel strong force): - Generation 1: Electron (e⁻, -1e), Electron neutrino (ν_e, 0) - Generation 2: Muon (μ⁻, -1e), Muon neutrino (ν_μ, 0) - Generation 3: Tau (τ⁻, -1e), Tau neutrino (ν_τ, 0)
31	What is the relationship between pressure and volume during a quasi-static reversible adiabatic expansion of an ideal gas?	P · V^γ = constant where γ = Cp / Cv is the heat capacity ratio (adiabatic index; γ = 5/3 for monatomic gases, γ = 7/5 for diatomic gases). Also expressed as T · V^(γ - 1) = constant and T^γ · P^(1 - γ) = constant. The curve is steeper than an isothermal P-V curve because gas cools during adiabatic expansion.
32	What are relativistic time dilation and length contraction equations, and what is proper time / proper length?	• Time dilation: Δt = γ · Δt0 where Δt0 is 'proper time' measured by a clock at rest relative to the events (Δt >= Δt0; moving clocks tick slower). • Length contraction: L = L0 / γ where L0 is 'proper length' measured in the rest frame of the object (L <= L0; objects appear contracted along the axis of relative motion).