Statistical Mechanics Concepts Quiz

Science-1: Prep Notes

• Prep notes for Science-1 covering topics like the state of systems, vector calculus, laws of Newton, work, kinetic energy, conservative forces and potential energy, linear momentum, torque and angular momentum, conservation of energy, Galilean relativity, problem solving strategy, single particle constrained motion, simple harmonic oscillator, and limitations of Newtonian mechanics.

General Pointers

• State of system: 3-dimensional Cartesian coordinate system.
  • Scalar quantities (mass, temperature).
  • Vector quantities (position vector, velocity, momentum, angular momentum, torque).
• Vector calculus: Vector equalities, addition.
  • Vector dot product, cross products.
  • Gradient of a scalar function.
  • Line integrals: circulation and curl of vector fields.
  • Surface integrals: flux and divergence of vector fields.

Laws of Newton

• 1st Law: Inertia. Inertial frames. Uniform rectilinear motion in the absence of forces.
• 2nd Law: Definition of momentum. Rate of change of momentum equals the net external force on the object.
• 3rd Law: For central forces, the reaction is equal and opposite to action. Great generality of these laws of motion to all kinds of phenomena.

Work, Kinetic Energy

• Work done on an object by application of external force F(t) is given by: W = ∫F(t) • dr(t)
• Using Newton's 2nd Law, W = mv² -1 mv² = ∆K. Defining K = mv² / 2 as kinetic energy, force acting on system changes the kinetic energy of the system.

Conservative Forces, Potential Energy

• If force is conservative, i.e., F(r) = −∇U(r), then W = −(U2 – U₁).
• Thus conservative forces have an associated potential energy function U.
• For conservative forces, work W is path-independent, depending only on end-point potential energies.
• W = K2 – K₁ = −(U2 – U₁).
• Thus, K₁ + U₁ = K2 + U2 (Law of conservation of mechanical energy).

Law of Conservation of Linear Momentum

• For a vector s, if F • s = 0, then d/dt (p •s) = 0.
• This implies (p •s) is a constant.
• Thus momentum in the direction of s is conserved.

Torque & Angular Momentum & Conservation

• Angular momentum about origin: L = r × p.
• Torque: N = dL/dt = r × F.
• N • s = 0 ⇒ d/dt(L •s) = 0.
• Conservation of Angular momentum along the direction of zero torque.

For Conservative Forces, Show Conservation of Energy

• E = T(v(t)) + U(r(t), t) ⇒ E = T + U
• T = ½ mv² = ½ mv • a = v ⋅ p
• U({r}, t) = Σau(xa) + √ ∫r v • (F + ∇ U) + du/dt
• E = constant when U = U(r).

Galilean Relativity

• All inertial frames have the same form of mechanical law.
• Two inertial frames, K and K' having relative velocity v.
• Then a particle having u in K-frame, u' in K'-frame, u = u' – v.
• Thus clearly, dp/dt = dp'/dt. This gives F = F'.
• Hence, all mechanical laws remain in the same form in both inertial frames.

Problem Solving Strategy

• Identify forces, draw free body diagram; "balance forces" to satisfy constraints.
• Set up dynamical equations (Use 2nd/3rd Laws and/or conservation laws.)
• Solve math of dynamical equations, with appropriate boundary conditions.

Single Particle, Constrained Motion

• Need to introduce "balancing forces". Examples: Block moving on a horizontal plane ("normal force"), block moving down an inclined plane.

Simple Harmonic Oscillator

• Hooke's Law: F(x) = -kx.
• EoM as mx = -kx.
• Rewrite as: x(t) + w₀²x(t) = 0, where w₀ = √k/m.
• Differential Equation: Second order, linear in x.
• One standard practice: Convert to two first-order differential equations.
• Conservation of energy.
• Solution: x(t) = A cos(w₀t) + A sin(w₀t).
• Phase diagram x(t) vs p(t).

Limitations of Newtonian Mechanics

• Handling of constraints (will be revisited later)
• Galilean Invariance (Newtonian Relativity) violated by EM: Special theory of relativity
• Stability of atom: Unexplained by classical physics: Quantum Mechanics.

Reasons & Postulates of Special Theory of Relativity

• Electromagnetic phenomena seemingly in violation.
• EM wave equation is not Galilean invariant
• Same phenomena in different inertial frames: Moving coil, magnet at rest; moving magnet, coil at rest.
• Postulates of Special Theory of Relativity (Einstein 1905):
  • Same physical laws in all inertial frames
  • Speed of light is same value in all inertial frames.

Events Simultaneous in all Frames?

• Concept: Relativity of simultaneity.
• Observer at the middle of a train, receiving flashes from front and back.

How do Space-Time Coordinates Transform?

• Frame K' with relative velocity v w.r.t. frame K.

• x' = γ(x - vt); y' = y; z' = z; t' = γ(t - vx/c²),

• Space-time invariant: (Δs)² = (Δx)² + (Δy)² + (Δz)² – c²(Δt)²

Notes on Lorentz Transform

• K'-frame moving with velocity v w.r.t. K-frame
• Given an event e = (x, y, z, t), use forward transform to get e' =(x', y', z', t').
• Strategy is to use difference in events as experimental measurable; Note that interchanging primes and changing v to -v gives inverse.

What is the time difference between ticks of a clock as measured from a moving frame?

• Time dilation. e.g.; Take event e₀ = (0, 0) and e₁ = (0, T₀).
• In K-frame, using Lorentz transform, we get e'n = (-γvT₀, γT₀).
• Hence, time difference between two consecutive ticks of the stationary clock will be measured in K'-frame.

What is the length of a rod as measured from a moving frame?

• Length contraction. e.g., From back end e₁ = (0, t) of rod to front end e₂ = (L₀, t) for any t.
• We need to find e₁,2 with t₁=t₂ using inverse Lorentz transform.

What is the frequency measured by a moving observer?

• Relativistic Doppler Effect.

What about moving rod and a moving clock?

• Symmetry: both observers will agree. Stationary clock measured to have longer time; moving clock to have longer.
• Stationary rod is measured (by moving observer) to be shorter than rest length.
• Moving rod is measured (by stationary observer) to be shorter than rest length.

Derive relative velocity formulae

• Particle with x(t) = ut, what is its velocity in K'-frame?
• dx'/dt' = γ (dx/dt - v)

What is relativistic momentum?

• p = (γm)u
• Momentum change of A is -2m u, but not for B

From momentum, find energy of particle

• K = (γ-1)mc²
• E² = (pc)² + (mc²)²

Space-time invariant

• Define (As)² = (Δx)² + (Δy)² + (Δz)² – c²(Δt)²
• Lorentz transform.
• Define (As')².
• Space-time 4-vector s = (x, y, z, ict)

Show that electric and magnetic effects are equivalent by STR.

• Consider an infinite line of charges; force on charge q due to this line; now let the wire of charge move; electric force; magnetic force; relation.
• Length contraction in K'-frame, resulting in larger electric force.

Review: One particle Newtonian Mechanics

• Identify forces; Free body diagram; "balance forces" for constraints.
• Dynamical equation set up and/or conservation laws use.
• Solving for appropriate boundary conditions.
• Examples of equation of motions: Freely falling body, charged particle in magnetic field, block sliding down an incline, simple pendulum.

From EoM to path properties

• Newton's laws: Forces change state of system (x, v) = (v, F/m for each time point t)
• Ex: Harmonic Oscillator (1-dim): F(x) = -kx = mx
• Trajectory: x(t) = A cos(wt), p(t) = -Amω sin(wt).
• Alternate description: (x/A)² + (p/B)² = 1.
• Conservation Laws E = T + U along path.
• Straight line path for free particle.
• Also for free particle min average KE path.
• Fermat minimum time principle for path of light: Snell's law of refraction.

Hand-wavy derivation of a path principle

• F = p → [F(t) - p(t)]η(t) = 0
• J = ∫F - p ηdt
• Use (v ⋅ η) = v ⋅ η + v ⋅ η to get J = ∫(-∇U⋅η + mv⋅η) dt
• Set η(t₁) = 0, η(t2) = 0, Set η(t) = x(t) – x*(t).
• L(x, v) = -U(x) + mv²/2, J = ∫8L(x, v)dt
• Action S = ∫L(x, v)dt thus J = 8S = 0 for true path.

Issues with derivation

• Issue-1: Fc is not included in above derivation when constraints are present, F = −∇U + Fc.
• Issue-2: Non-simple constraints (like Pendulum).
• Mathematically rigorous derivation of Euler-Lagrange equations.

Calculus of variations: Find stationary soln.

• J = ∫x⁴ dy dx f(y(x), y'(x); x)
• Path space parametrization.
• With y = yo + an, we have dy/da = η, dy'/da = η'

Calculus of variations (cntd.)

• d/dx (∂f/∂y') = ∂f/∂y,

Euler equation of 'second' kind

• f = f(y(x), y'(x)); hence ∂f/∂x = 0 , then - df/dy' = constant.

Brachistochrone problem

• What is shortest time curve for particle traveling from point A to B, starting from rest?
• mv²/2 = mgx → v = √2gx, ds = vdt

Path Principle: General Pointers

• Generalize to N-particle system.
• L = KE - PE.
• S = ∫ L dt δS = 0.

Generalised coordinates

• Simple pendulum, Wall of death, block sliding down raising incline.

Simple Pendulum

• Single generalized coordinates.
• L = ½ m l² θ² – mgl cos θ.
• E-L: ml² θ̈ = -mgl sin θ.

Wall of death

• Two generalized coordinates (r, θ).
• Z = r cot a, (x, y, z) = (r cos θ, r sin θ, r cot a).
• L = m(r²csc²a + r²θ² – mgr cot a).
• E-L equations.

Block sliding down raising incline

• Block is located on incline whose angle is increasing linearly with time θ = at.

ML: training as variational problem

• Given dataset (xₖ, yₖ) to find function f such that y = f(x).
• Standard methodology is to minimize J = ∫[y – f(x)]² dx

E-L with undetermined multipliers

• Constraints f(x, t) = 0
• Example: Disk rolling without slipping.

Example: Disk rolling down incline

• Generalized coordinates (y, θ ).
• Constraint g(y, θ) = y - R θ = 0.
• L = (1/2) My² + (1/2)I θ² - Mg(-y sin α) = L(y, θ, ẏ, θ̇).
• E-L equations.

Comparison: Newtonian and Lagrangian formulations

• Newtonian vs Lagrangian.
• Equation of motion order, Coordinates, Interactions, Constraints

Time symmetry leads to conservation of Energy

• Time symmetry: path is 'translated'.
• dL/dt =0; H = constant.

Translation symmetry leads to conservation of Momentum

• Translation symmetry; path 'translated' in space, δL = 0.

Rotational symmetry

• Rotational symmetry; path 'rotated' in space; δL=0.

Hamiltonian Dynamics

• H = Σ pₖqₖ – L(qₖ, q̇ₖ, t).
• Hamilton's equations of motion: q̇k = ∂H/∂pₖ and ṗₖ = -∂H/∂qₖ.

Two-body problem

• L = ½m₁v₁² + ½m₂v₂² – U(|r₁ – r₂|).
• Center of Mass; diff vector → r.

Two-body problem: Continued

• µr² θ̇ = l constant, where and µ.
• Ueff = l²/(2µr²) – U(r) = ½µr² θ̇² – U(r) .

Two-body problem: Implications on Solar System

• Kepler's laws.
• Newton's proof of relationship with law of universal gravitation in solar systems.
• Problems of several fields, Lagrange and Poincaré.
• Perturbations to two-body trajectories of planets.

Solar system

• Ancients; Uranus; Pluto and the Kuiper belt.

Three-body problem

• No general solution with closed orbits.

Multi-particle systems: Center of Mass

• Total mass.
• Center of Mass (CoM) definition.
• Momentum of particle k; total linear momentum of system.
• Total angular momentum of system.
• Net torque about origin of system.

Collection of particles

• Forces: Fk = Fe + Fj
• Effects of these forces: net force; effective mass; angular momentum; internal forces.

Show P conserved, when no external forces

• d/dt(P) = ∑F(ext forces)

Show L conserved when no external forces

• L = Σ rk × pk = (rCOM × pCOM) + Σ (rk × pk),
• dL/dt = Σ rk × Fk.

Show energy is conserved

• W = ∫ ∑ F⋅dr = Change in KE.

Summary: Multi-particle Systems

• Center of Mass of collection of particles.
• Total momentum of system: P.
• Net force on system F.
• If no external forces, linear momentum of system is conserved. Total angular momentum of system about origin.
• Net torque on system N = dL/dt.
• If no external torque, angular momentum is conserved.

Light

• Huygens proposed wave theory.
• Newton proposed corpuscular theory.
• Refraction, diffraction, internal reflection.
• Young's double-slit experiment.
• Fresnel's mathematical Huygens-Fresnel theory.
• Maxwell's equations.

Light in vacuum

• Plane wave solutions for E and B: E = Eₐei(kz - wt)
• Transversal waves, c² = 1/μ₀ε₀.
• Superposition of plane waves ⇒ Polarisation

Light: Color depends on wavelength

• Wavelength from diffraction gratings (like Young's double slit)
• Visible range (Human).
• Sunlight after refraction through prism Contains non-visible infrared and UV radiation)
• When metals are heated, they emit light.
• Kirchhoff's black body problem.

Black body radiation

• Stefan-Boltzmann radioactive power law.
• Wein displacement law.
• Wein distribution law.
• Rayleigh-Jeans law.

Planck's black-body radiation formula

• Planck's attempted expression / model.
• Planck's model required statistical mechanics, with discrete energy.

Photoelectric effect

• In vacuum, charged metal objects discharge when UV light falls on it.
• Counter-intuitive results from contemporary physics.
• Einstein's model that uses Planck energy quantization.

X-ray production

• High KE electrons impinge on metal, light is emitted
• X-rays are light of very small wavelength (0.1 nm)
• X-Ray diffraction probes chemical structure.

Compton experiment

• Showed photon as a particle,

Summary: Light

• Light as wave; light as particle; light wave-particle duality
• Wavelength; frequency; speed; c = λν
• Mass m = 0, E = pc, Planck's E = hv, p = h/λ.

de Broglie hypothesis: Matter waves

• Inspired by wave-particle duality of light, de Broglie proposed wave-particle duality of particles.
• Specifically that a wave of wavelength λ = h/p is associated with matter having momentum p.
• Einstein's strong support
• Experiments conclusively showed diffraction.

Davisson-Germer experiment

• Particle Diffraction
• Diffraction peaks of electrons from the nickel crystal surface.
• Using n = 2dsinθ, λ = h/p.

Matter waves

• For 1 µg mass moving at 1µm/s, λ ~ 10⁻²⁰ m.
• Regime of small: in energy/momentum/mass
• Localization of wave; particle behavior: group wave
• 1D wave: A cos(2πx/λ + 2πνt) = A cos(kx + w t).

Matter waves: Group wave

• Superposition: A cos(wt - kx) + A cos((w + ∆w)t – (k + ∆k)x).
• Group wave vg = dw/dk.

Matter waves: Group wave (2)

• Many individual waves; more localization in group wave
• ω = 2πν = 2πhγmc², And k = 2π/λ = 2πγmν/ħ
• Phase velocity Vp = ω/k = c².
• Group velocity vg = dw/dk = dv/dp =Vγ/m.

Uncertainty Principle

• Waves have fundamental property: ΔxΔk ≥ 1
• ΔxΔp ≥ ħ/2

Old Quantum Theory

• Planck (1900AD) hypothesized: energy of oscillator is quantized E = nhv.
• Bohr (1913AD) hypothesized: angular momentum of electron quantized mvr = nh/2π; η = 1, 2, 3, ...
• Somerfield.

Bohr Model of Hydrogen atom

• Electron orbiting fixed nucleus; postulate mvr = nħ.
• Orbit of electron, E = ½mv² - Kₑe²/r; balancing electrostatic.
• E = -Rₙ/n²; Rₙ = 2π²mₑe⁴/ (4πε₀ħ²).

Frank-Hertz experiment

• Electrons in a vacuum tube sent through vapor of mercury.
• Light of 253.6 nm is emitted; this corresponds to KE of 4.9 eV electron.

Quantum Theory: Math Formulation

• Insists that there is a "wave function" ψ(x, t) that is the state of the system.
• It satisfies the Schrödinger equation (evolution of ψ).

Motivate proof of Schrodinger equation

• Waves satisfy wave equation: ∇²ψ(x) + k²ψ(x) = 0
• de Broglie: λ = h/p → k = 2π/λ = 2πp/ħ.
• Suggestion (x) = exp(ipx/ħ), E = p²/2m.
• For free particle, E = p²/2m; Using (2) gives 1ħ²/2m∇²ψ(x) = Eψ(x).

Another motivating derivation

• Wave can be represented as Ψ(x, t) = exp i(kx - wt).
• Planck-Einstein: E = hv ⇒ w = E/ħ.
• Ψ = exp(i(px-Et)/ħ), momentum p, energy E
• E = p²/2m + V(x).
• d/dt Ψ = (-ħ²/2m∇² + V(x)) Ψ

Features of Schrodinger equation

• Linearity and superposition
• Normalization
• Stationary solutions; Hamiltonian operators.
• Expectation values
• Correspondence principle: CM limit.

Momentum operator

• (pₓ) = ih∂/∂x.

Particle in 1-D box

• Particle confined to x ∈ [0, L], V(x) = 0 ∀x ∈ [0, L], V(x) = ∞ ∀x ∉ [0, L].
• SE is -ħ²/2m d²ψ/dx² + V(x)ψ = Eψ .

Particle in 3-D box

• Box defined x ∈[0,Lx], y∈[0,Ly], z ∈[0, Lz].
• Potential V(x, y, z) = 0 inside box and V=∞ outside.
• Separation of variables: ψ(x,y, z) = ψ₁(x) ψ₂(y) ψ₃(z).
• Result: ψ(x, y, z), E = h² (n₁² + n₂² + n₃²)/(8mL²)

Harmonic Oscillator

• SE: -ħ²/2m d²ψ/dx² + ½ kx²ψ = Eψ
• Find the wavefunction to solve the SHO

Tunneling

• Non-zero probability beyond classical turning points in harmonic oscillators (with KE ≠0).
• Starting in Region-1 with energy E < U; conceptually particle cannot penetrate Region-2.
• Conditions at the boundaries (x=0, x=L).

Tunnel effect

• Region-1: Dxxψ₁ + C₁Eψ₁ = 0
• Region-2: Dxxψ₂ + C₂(E - U)ψ₂ = 0,
• Region-3: Dxxψ₃ + C₃Eψ₃ = 0
• Boundary conditions: ψ₁(x=0) = ψ₂(x=0); ψ₁(x=L) = ψ₃(x=L), etc.
• Transmission coefficient.

Hydrogen atom

• SE: -ħ²/2m(∇²ψ) + Vψ = Eψ; V = e²/4πε₀r.
• Separation of variables ψ (r, θ, φ) = R(r) Θ(θ) Φ($).

Hydrogen atom (2&3)

• Equating terms gives radial equation: 1/r² d (r² dR)/ dr + ... = 0, and angular equation.
• Solutions for m₁, l and n

Hydrogen atom (4)

• Normalized wave functions,

Hydrogen atom (5)

• Shapes of orbitals for Hydrogen atom.

Hydrogen atom (6): Transitions

• Energy levels and allowed transitions.

Zeeman effect

• Magnetic dipole of atom interacts with external magnetic field.

Multi-electron atoms

• SE for multi-electron atoms (numerical methods).
• Wave function (even/odd symmetry).
• Electron spin.
• Exclusion principle.

Periodic Table

• Atomic number; atomic mass.

Spin-orbit coupling

• Electron spin magnetic moment interaction with magnetic field due to circular motion around the nucleus.

Energy level ordering

• Quantum states in an atom.

Molecular covalent bond

• Potential energy diagram.

Statistical Mechanics

• Atoms / molecules approximated as interacting particles. (Lennard-Jones, electrostatic, bonds (as springs) etc).
• Classical Mechanics; Issue: N ~ 10²³ atoms. Too large for numerical tracking.
• Statistics; Boltzmann.
• Statistical Mechanics gives probabilistic description of system; Moving away from deterministic Classical Mechanics.

Simple Harmonic oscillator

• mx = −kx
• x = A cos(wt), w = √k/m.
• Phase plot (x vs p): point on circle.
• Equal probability for system to be in any of the possible phase points.
• Mathematical rigorous proof for Hamiltonian systems: Liouville theorem.

Postulates of Statistical mechanics

• Phase-space point x = ({qₖ}ₖ=₁, {pₖ}ₖ=₁) in R⁶ᴺ space.
• Postulate: Equal a priori for states with equal energy.
• Equivalently: Probability of a state ∝ e⁻ᴱ(x)/kT.
• Ensemble: micro-canonical, canonical, grand canonical.
• Temperature(Ergodicity): Time average equals ensemble average.

Canonical ensemble: general setup

• Phase point x ∈ R³ᴺ × R³ᴺ space.
• Probability density p(x) ∝e⁻ᴱ(x)/kT
• Partition function Z = ∫ dx e⁻ᴱ(x)/kT= ∫ ... e⁻ᴱ/kT. Thermodynamic connection

Degeneracy of energy level, Entropy

• Degeneracy of an energy level.
• Entropy S(T)= k ln W(E)∆E

System of non-interacting particles

• Total energy E = ΣEₖ
• Probability density formula
• Barometric law / Maxwell-Boltzmann distribution for single particle energy Eₖ..

System of non-interacting particles (cntd.)

• Equipartition theorem: Avg. energy of a quadratic term in E, proportional to kT .
• Calculation of partition function Z for a system of non-interacting particles

Grand-canonical ensemble statistics

• Constant chemical potential µ, volume V and Temperature T; Varying number of particles N.
• Each state can have Fermi-Dirac statistics (ns = 0, 1) or Bose-Einstein statistics (ns = 0, 1, 2,… N).
• Occupation probability: prob(ns) = 1/exp((Eₛ₋µ)/kT) ± 1.

+ sign in occupation statistics

• Maxwell-Boltzmann works well if particles are distinguishable.
• Problems arise with identical particles that are indistinguishable by the wavefunctions.
• Fermi-Dirac and Bose-Einstein statistics; origin.

Black-body radiation: Boson photon gas

• Rayleigh-Jeans: u(v) dν = (8πν²/c²) kT dν
• Standing wave density: g(j) dj = 2 ⋅ (4πj²) dj; density of stand wave g(v) dν.
• With energy per wave ~ kT, u(v) dν = (8πν²/c³) kT dν.

Planck Radiation Formula

• Planck's assumption about the quantization of energy.
• Average energy per oscillator E = Σ (n hν) e⁻nhv/kT.
• Energy density: u(v) dv = (8πh/c³) v³/(exp(hv/kT) - 1) dv.

LASER

• Stimulated emission, same probability as absorption.
• Ratio of spontaneous and stimulated emission is proportional to v³.

Specific heat of solids

• CM model of solid: Each atom is a 3D oscillator ⇒ energy E = 3Nk T , Cᵥ= 3Nk
• Einstein's model: E = hv/(e^(hv/kT) - 1)

Electrons in metal

• Electrons are fermions ⇒ NFD(e) = 1/(e^(e-εF)/kT) + 1).
• Relationship between g(e) de and n(e).

Test your knowledge on key concepts in statistical mechanics, including the Maxwell-Boltzmann distribution, equipartition theorem, and various ensembles. This quiz challenges you to match definitions, understand relationships, and explore fundamental principles in thermodynamics. Perfect for students studying advanced physics topics.