General Features of Quantum Mechanics

General Features of Quantum Mechanics

Quantum mechanics is a framework to do physics that has replaced classical physics as the correct description of fundamental theory.

Overview

• Quantum mechanics is almost 100 years old (as of the transcript's recording date).
• The centenary of quantum mechanics will likely be in 2025, marking 100 years since Schrodinger and Heisenberg wrote down the equations of quantum mechanics.
• The roots of quantum mechanics began in the late 19th century with Planck and early 20th century with Einstein.

Quantum Physics

• Quantum physics applies the principles of quantum mechanics to various physical phenomena:
  • Quantum electrodynamics: Quantum mechanics applied to electromagnetism.
  • Quantum chromodynamics: Quantum mechanics applied to the strong interaction.
  • Quantum optics: Quantum mechanics applied to photons.
  • Quantum gravity: Quantum mechanics applied to gravitation, leading to string theory.

Topics to be Discussed

1. Linearity of quantum mechanics.
2. Necessity of complex numbers.
3. Laws of determinism.
4. Unusual features of superposition.
5. Entanglement.

1. Linearity

Dynamical Variables and Equations of Motion

• Dynamical variables are those whose values are connected with observation.
• Equations of motion (EOM) are solved for these dynamical variables.

Maxwell's Theory

• Maxwell's theory of electromagnetism is a linear theory.
• If a plane wave propagating in one direction and another propagating in the opposite direction are both solutions, then their combination is also a solution.
• The waves propagate without affecting each other.
• Millions of phone calls can occur simultaneously through cables or via electromagnetic waves without interference due to superposition.

Mathematical Meaning of Linearity

• In Maxwell's theory, electric field E, magnetic field B, charge density \rho, and current density j correspond to a solution if they satisfy Maxwell's equations.
• If (E, B, \rho, j) is a solution, then (\alpha E, \alpha B, \alpha \rho, \alpha j) is also a solution, where \alpha is a real number.
• If (E1, B1, \rho1, j1) and (E2, B2, \rho2, j2) are two solutions, then (E1 + E2, B1 + B2, \rho1 + \rho2, j1 + j2) is also a solution.

Linear Equation

• A linear equation is written as Lu = 0, where u is the unknown and L is a linear operator.
• For multiple equations, we can have several linear operators: L1u = 0, L2u = 0.
• For several unknowns: L(u, v, w) = 0.

Properties of a Linear Operator

• L(\alpha u) = \alpha Lu
• L(u1 + u2) = Lu1 + Lu2
• Consequently, L(\alpha u1 + \beta u2) = \alpha Lu1 + \beta Lu2
• If Lu1 = 0 and Lu2 = 0, then L(\alpha u1 + \beta u2) = 0, meaning that \alpha u1 + \beta u2 is also a solution.

Example: Differential Equation

• Consider the differential equation: \frac{du}{dt} + \frac{1}{\tau}u = 0
• This can be written as Lu = 0 by defining L(u) = \frac{du}{dt} + \frac{1}{\tau}u
• Then L can be written as L = \frac{d}{dt} + \frac{1}{\tau}

Checking Linearity

• Check that L is a linear operator:
  • L(\alpha u) = \frac{d}{dt}(\alpha u) + \frac{1}{\tau}(\alpha u) = \alpha \frac{du}{dt} + \alpha \frac{1}{\tau}u = \alpha Lu
  • Check that L(u1 + u2) = Lu1 + Lu2.