Mathematical physics Study Guide

📖 Core Concepts Mathematical physics – development of mathematical tools for physics and the use of physics‑inspired ideas in mathematics. Lagrangian mechanics – dynamics from the scalar Lagrangian \(L(q,\dot q,t)=T-V\) via the principle of stationary action \(\delta S=0\) with \(S=\int L\,dt\). Hamiltonian mechanics – phase‑space reformulation; Hamiltonian \(H(q,p,t)=\sumi pi\dot qi-L\) generates Hamilton’s equations \(\dot qi=\partial H/\partial pi,\;\dot pi=-\partial H/\partial qi\). Noether’s theorem – every continuous symmetry \(\rightarrow\) a conserved quantity (e.g., time‑translation \(\rightarrow\) energy). Hilbert space – complete inner‑product space; quantum states are unit vectors \(|\psi\rangle\). Operators & spectral theory – observables are self‑adjoint operators; their spectra give possible measurement outcomes. Fourier analysis – any suitable function can be written as a sum/integral of sines & cosines; solves linear PDEs with constant coefficients. Partition function \(Z=\displaystyle\sum{i}e^{-\beta Ei}\) (or \(\int e^{-\beta H}\,d\Gamma\)) encodes all equilibrium thermodynamic information. Lorentz transformation – relates coordinates between inertial frames moving at relative velocity \(v\): \[ \begin{aligned} t' &= \gamma\!\left(t-\frac{vx}{c^{2}}\right),\\ x' &= \gamma\,(x-vt),\qquad \gamma=\frac{1}{\sqrt{1-v^{2}/c^{2}}}. \end{aligned} \] --- 📌 Must Remember Euler‑Lagrange equation: \(\displaystyle\frac{d}{dt}\!\left(\frac{\partial L}{\partial\dot qi}\right)-\frac{\partial L}{\partial qi}=0\). Legendre transform links \(L\) and \(H\): \(pi=\partial L/\partial\dot qi\), \(H=\sumi pi\dot qi-L\). Noether: continuous symmetry \(\Rightarrow\) conserved current \(J^\mu\); for time invariance, \(E\) is conserved. Schrödinger operator: \(\hat H=-\frac{\hbar^{2}}{2m}\nabla^{2}+V(\mathbf r)\). Spectral theorem: a self‑adjoint operator \(\hat A\) can be written \(\hat A=\int \lambda\,dE{\lambda}\) with real eigenvalues \(\lambda\). Partition function relations: Free energy \(F=-k{B}T\ln Z\). Average energy \(\langle E\rangle = -\partial\ln Z/\partial\beta\). Entropy \(S = k{B}(\ln Z + \beta\langle E\rangle)\). Lorentz factor \(\gamma\) grows without bound as \(v\to c\); length contracts \(L'=L/\gamma\), time dilates \(\Delta t'=\gamma\Delta t\). --- 🔄 Key Processes Deriving equations of motion (Lagrangian) Write \(L(q,\dot q,t)\). Compute \(\partial L/\partial qi\) and \(\partial L/\partial\dot qi\). Apply Euler‑Lagrange → differential equations. From Lagrangian to Hamiltonian Define momenta \(pi=\partial L/\partial\dot qi\). Solve for \(\dot qi(p,q)\). Perform Legendre transform to get \(H(q,p)\). Write Hamilton’s equations. Applying Noether’s theorem Identify a continuous symmetry (e.g., \(q\to q+\epsilon\)). Compute the corresponding conserved quantity \(Q = \sumi \frac{\partial L}{\partial\dot qi}\delta qi\). Solving a linear PDE with Fourier series Assume separable solution \(u(x,t)=X(x)T(t)\). Expand \(X(x)\) in sines/cosines satisfying boundary conditions. Solve resulting ODEs for each mode; superpose. Constructing the partition function List all microstates \(\{i\}\) and their energies \(Ei\). Compute \(Z=\sumi e^{-\beta Ei}\) (discrete) or \(Z=\int e^{-\beta H(p,q)}\,d\Gamma\) (continuous). --- 🔍 Key Comparisons Lagrangian vs. Hamiltonian Lagrangian: uses \((q,\dot q)\); natural for constraints and relativistic formulations. Hamiltonian: uses \((q,p)\); suited for phase‑space analysis, canonical quantization. Classical vs. Quantum observables Classical: functions on phase space, commute. Quantum: self‑adjoint operators, generally non‑commuting \([\hat A,\hat B]\neq0\). Lorentz vs. Galilean transformation Lorentz: mixes space & time, preserves speed of light. Galilean: adds velocities, assumes absolute time. Microcanonical vs. Canonical ensemble Microcanonical: fixed energy, volume, particle number. Canonical: fixed temperature, volume, particle number; uses \(Z\). --- ⚠️ Common Misunderstandings “Lagrangian = kinetic energy.” It is kinetic minus potential; may include velocity‑dependent potentials. “Hamiltonian always equals total energy.” Only when \(L\) has no explicit time dependence and the coordinates are standard. “Any symmetry gives a conservation law.” Noether applies only to continuous symmetries; discrete symmetries give selection rules, not conserved currents. “Partition function is a probability.” \(Z\) is a normalization factor; probabilities are \(pi=e^{-\beta Ei}/Z\). “Lorentz factor \(\gamma\) is always > 1.” It equals 1 only at \(v=0\); it diverges as \(v\to c\). --- 🧠 Mental Models / Intuition Action as “least effort” – the system chooses a path that makes the total “effort” (action) stationary, analogous to a marble rolling to the lowest point. Symmetry → Conservation – think of symmetry as a hidden “balance” that the system cannot change, so something (energy, momentum) stays fixed. Hilbert space as an infinite‑dimensional vector space – just like \(\mathbb{R}^n\) but with infinitely many components (wavefunction coefficients). Operators as machines – they take a state vector, “process” it, and output another state (or a number if the state is an eigenstate). Fourier series as “musical decomposition” – any periodic signal can be heard as a sum of pure tones (sine/cosine). --- 🚩 Exceptions & Edge Cases Time‑dependent Hamiltonian – energy is not conserved; use extended phase space or Noether’s theorem with explicit time symmetry breaking. Discrete symmetries (e.g., parity) → no continuous Noether current, but can enforce selection rules. Non‑periodic boundary conditions – Fourier integrals (continuous spectrum) replace series. Systems with long‑range interactions – partition function may diverge; require regularization or alternative ensembles. Relativistic speeds – Newtonian kinetic energy \(T=\frac12mv^2\) fails; use \(T= (\gamma-1)mc^2\). --- 📍 When to Use Which Lagrangian → when constraints are easier expressed in generalized coordinates or when dealing with relativistic actions. Hamiltonian → for phase‑space methods, canonical quantization, or when energy conservation is central. Fourier analysis → linear PDEs with constant coefficients and periodic or infinite domains. Spectral theorem → any problem requiring eigenvalues/eigenvectors of self‑adjoint operators (quantum measurements, stability analysis). Partition function → equilibrium statistical mechanics; choose canonical ensemble if temperature is fixed, microcanonical if energy is fixed. Group theory → to identify conserved quantities or classify particle states (e.g., angular momentum from rotational symmetry). --- 👀 Patterns to Recognize Symmetry → conserved quantity – spot a continuous invariance → write down the associated conserved momentum/energy/angular momentum. Separable PDEs – look for products of functions of single variables; leads to eigenvalue problems solved by Fourier or Sturm–Liouville theory. Hermitian operators → real eigenvalues → physically observable. Non‑analytic points in thermodynamic potentials → signals of a phase transition (first‑order: discontinuous first derivative; second‑order: discontinuous second derivative). Commutator \([\hat A,\hat B]=0\) → simultaneous eigenstates → compatible observables. --- 🗂️ Exam Traps Choosing the wrong energy expression – mixing kinetic energy \(T\) with the Lagrangian \(L=T-V\); remember \(H\) may differ from total energy if \(L\) depends on time. Assuming every symmetry yields a conserved scalar quantity – continuous vector symmetries give vector conservation laws (e.g., rotational symmetry → angular momentum vector). Mis‑applying Lorentz factor – using \(\gamma\) for speeds much less than \(c\) when Galilean formulas suffice; can over‑complicate simple problems. Treating partition function as a probability – leads to incorrect normalization of thermodynamic averages. Confusing eigenvalues with eigenvectors – exam may give a matrix; answer must be the observable value (eigenvalue), not the state vector. Forgetting boundary conditions in Fourier solutions – ignoring them yields incorrect mode spectra (e.g., using sine series when both ends are free). ---