Mathematical Physics

Mathematical Physics is the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and the formulation of physical theories.

Key Areas in Mathematical Physics

  • Mechanics: Motion, forces, energy, and Newton’s laws

  • Electromagnetism: Electric and magnetic fields, Maxwell’s equations

  • Waves & Oscillations: Harmonic motion, wave equations

  • Quantum Mechanics: Schrödinger equation, probability amplitudes

  • Relativity: Space-time, Lorentz transformations

Example: Newton's Second Law

F = ma (Force = mass × acceleration)

If a 5 kg object accelerates at 3 m/s², what is the force?

F = 5 × 3 = 15 N

Step-by-Step: Solving a Wave Equation

Problem: Solve the 1D wave equation: ∂²u/∂t² = c²∂²u/∂x²

  1. Assume solution u(x, t) = X(x)T(t)

  2. Separate variables: T''/c²T = X''/X = -λ

  3. Solve both ODEs: X'' + λX = 0, T'' + λc²T = 0

  4. Apply boundary and initial conditions

  5. Combine results for general solution

Practice Problems

1. A particle of mass 2 kg is under a force of 10 N. What is its acceleration?

Solution:

F = ma → a = F/m = 10 / 2 = 5 m/s²

2. The displacement of a wave is given by y(x,t) = A sin(kx - ωt). Identify wave speed.

Solution:

Wave speed v = ω / k

3. Write down the time-independent Schrödinger equation in 1D.

Solution:

-ħ²/2m (d²ψ/dx²) + V(x)ψ = Eψ