Quantum mechanics Lecture Summary
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.
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).
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.
Main Topics:
Linearity of quantum mechanics
Necessity of complex numbers
Laws of determinism
Superposition and its unusual features
Entanglement
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.
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.
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).
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.
Main Topics:
Linearity of quantum mechanics
Necessity of complex numbers
Laws of determinism
Superposition and its unusual features
Entanglement
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.