Overview of Quantum Mechanics

• Introduction to Quantum Mechanics

  • Key features and foundations of quantum mechanics.

  • Historical context leading to the development of quantum mechanics.

  • Introduction of principles relevant for the current semester and future courses.

Historical Background

• Age of Quantum Mechanics

  • Nearly 100 years old; established in the 1920s.

  • 2016 marks the centenary of General Relativity; the quantum mechanics centenary expected in 2025.

  • Major contributors include Planck (late 19th century), Einstein, Schrodinger, and Heisenberg (1925).

Definition of Quantum Mechanics

• Framework of Quantum Mechanics

  • Replaces classical physics for fundamental theory.

  • Classical physics is an approximation; quantum mechanics offers a more accurate and conceptually different approach.

  • Applications of quantum mechanics include:

    • Quantum Electrodynamics (QED): related to electromagnetism.

    • Quantum Chromodynamics (QCD): concerns strong interactions.

    • Quantum Optics: addresses the behavior of photons.

    • Quantum Gravity: applies quantum mechanics to gravitational systems.

    • String Theory: a potential theory encompassing all interactions including gravity.

Key Topics to Cover

• Main Topics:

  1. Linearity of quantum mechanics

  2. Necessity of complex numbers

  3. Laws of determinism

  4. Superposition and its unusual features

  5. Entanglement

Linearity in Quantum Mechanics

• Definition of Linearity

  • Involves dynamical variables connected to observations and equations of motion (EO

  • Example: Maxwell's theory of electromagnetism is a linear theory.

• Properties of Linear Theories

  • Superposition Principle: If solutions exist, their linear combinations are also solutions.

  • Practically useful: Electromagnetic waves can overlap without interfering with each other, allowing multiple signals (e.g., phone calls) to coexist simultaneously.

• Mathematical Representation

  • Set of variables connected by Maxwell's equations (electric field, magnetic field, charge density, current density).

  • Linearity implies:

    • If a solution exists, scaling it by a constant results in another solution.

    • The sum of two solutions yields another valid solution (superposition).

• Understanding Linear Operators

  • Schematically represented as L*u = 0,

    • L: Linear operator,

    • u: Unknown variable.

  • Linearity properties:

    • L(au) = aL(u)

    • L(u1 + u2) = L(u1) + L(u2)

    • Extends to linear combinations: L(αu1 + βu2) = αL(u1) + βL(u2).

• Example of a Linear Equation

  • Differential equation form:

    • Example: du/dt + (1/τ)*u = 0

  • Defined as L on u to be du/dt + (1/τ)u.

  • Check linearity properties for validation.