Graduate
  1. Classical mechanics  1.1. Advanced Kinematics  1.1.1. Generalized coordinate systems  1.1.2. Parametric equations of motion  1.1.3. Non-inertial frames and fictitious forces  1.1.4. Covariant formulation of momentum  1.1.5. Action-angle variable  1.2. Lagrangian and Hamiltonian mechanics  1.2.1. Principle of minimum action  1.2.2. Euler–Lagrange equations  1.2.3. Noether's theorem and conservation laws  1.2.4. Normalized Speed  1.2.5. Canonical conversion  1.2.6. Poisson bracket and Hamilton–Jacobi theory  1.2.7. Hamiltonian chaos and integrability  1.3. Rigid body mobility  1.3.1. Euler's equations of motion  1.3.2. Principal moments of inertia  1.3.3. Gyroscopic motion and precession  1.3.4. Inertia tensor  1.3.5. Stability of rotational speed  1.4. Nonlinear dynamics and chaos  1.4.1. Phase space and stability analysis  1.4.2. Poincaré sections and bifurcation theory  1.4.3. Strange Attractors and Fractals  1.4.4. Lyapunov exponent  1.4.5. KAM theorem and quasi-periodic motion
  2. Electromagnetism  2.1. Advanced Electrodynamics  2.1.1. Green's function and potential theory  2.1.2. Multipole expansion  2.1.3. Image charging method  2.1.4. Boundary conditions and uniqueness theorem  2.2. Electromagnetic wave propagation  2.2.1. Plane waves in dielectrics and conductors  2.2.2. Waveguides and cavity resonators  2.2.3. Radiation pressure and optical tweezers  2.2.4. Polarization and Jones calculus  2.3. Relativistic electrodynamics  2.3.1. Covariant formulation of electrodynamics  2.3.2. Lienard–Wiechert potentials  2.3.3. Synchrotron radiation and bremsstrahlung  2.3.4. Electromagnetic field tensor and Lorentz transformations  2.4. Plasma physics  2.4.1. Magnetohydrodynamics  2.4.2. Debye shielding and plasma oscillations  2.4.3. Alfvén waves and tokamak confinement  2.4.4. Plasma instabilities and turbulence
  3. Statistical mechanics and thermodynamics  3.1. Advanced Thermodynamics  3.1.1. Legendre transforms and thermodynamic potentials  3.1.2. Fluctuation–dissipation theorem  3.1.3. Onsager interpersonal relations  3.1.4. Critical events and phase transitions  3.2. Quantum statistical mechanics  3.2.1. Density Matrix and Ensemble Theory  3.2.2. Bose–Einstein condensates  3.2.3. Quantum phase transition  3.2.4. Fermi–Dirac and Bose–Einstein statistics  3.2.5. Partition functions and grand canonical ensembles
  4. Quantum mechanics  4.1. Advanced wave mechanics  4.1.1. WKB approximation  4.1.2. Path integral formulation  4.1.3. Quantum harmonic oscillator and coherent states  4.2. Angular momentum and spin  4.2.1. Spherical Harmonics  4.2.2. Clebsch–Gordan coefficient  4.2.3. Wigner–Eckart theorem  4.2.4. Spin–orbit coupling  4.3. Quantum scattering theory  4.3.1. Born approximation  4.3.2. Partial wave analysis  4.3.3. Optical theorem  4.3.4. S-matrix theory  4.4. Quantum field theory  4.4.1. Second Quantization  4.4.2. Feynman diagrams and propagators  4.4.3. Renormalization theory  4.4.4. Gauge theory and Yang–Mills fields  4.4.5. Spontaneous symmetry breaking and the Higgs mechanism
  5. General relativity and cosmology  5.1. Tensor calculus and differential geometry  5.1.1. Einstein field equations  5.1.2. Schwarzschild and Kerr metrics  5.1.3. Geodesics and Christoffel symbols  5.2. Cosmology and the Universe  5.2.1. The FLRW metric and cosmic inflation  5.2.2. Dark energy and structure formation  5.2.3. Cosmic microwave background radiation
  6. Condensed matter physics  6.1. Band structure and transport theory  6.1.1. Block theorem and the Kronig–Penney model  6.1.2. Fermi surface topology  6.1.3. Quantum Hall Effect  6.2. Superconductivity  6.2.1. BCS theory and Cooper pairs  6.2.2. Meissner effect and flux quantization  6.2.3. The Josephson Effect and SQUIDs  6.3. Topological phases of matter  6.3.1. Topological Insulators  6.3.2. Majorana fermions in topological phases of matter
  7. Nuclear and Particle Physics  7.1. Quantum Chromodynamics (QCD)  7.1.1. Quark–gluon plasma  7.1.2. Asymptotic freedom  7.1.3. Confinement and hadronization  7.2. Beyond the Standard Model  7.2.1. Grand Unified Theories  7.2.2. Supersymmetry and extra dimensions  7.2.3. Neutrino oscillations

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Quantum mechanics


Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature on the scale of atoms and subatomic particles. It is the basis of all quantum physics, including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

The birth of quantum mechanics

Quantum mechanics emerged in the early 20th century when scientists discovered that classical physics could not explain certain phenomena. Classical physics, such as Newtonian mechanics, is excellent for large-scale observations, but fails at the atomic and subatomic levels.

Wave–particle duality

One of the fundamental concepts in quantum mechanics is wave-particle duality. This theory holds that every particle or quantum entity can be described as either a particle or a wave. Consider the behavior of light:

When light is shone through a narrow slit, it creates an interference pattern typical for waves rather than particles. However, when individual photons are counted, they appear as discrete packets of energy, which behave like particles. This duality is illustrated in the famous double-slit experiment.

Kiln 1 Kiln 2

In the double-slit experiment, when particles such as electrons are directed at a screen with two slits, they create an interference pattern on the detector screen, which suggests wave behavior. However, if one looks at which slit the electrons pass through, they revert to particle behavior, and the interference pattern disappears.

Uncertainty principle

Another core idea of quantum mechanics is the Heisenberg uncertainty principle, which states that certain pairs of properties, such as position and momentum, cannot both be known to arbitrary precision. That is, the more precisely a particle's position is known, the less precisely its momentum can be known, and vice versa. This is not due to imperfections in measurement, but is a fundamental property of quantum systems.

Δx * Δp ≥ ħ/2

In this equation, Δx is the uncertainty in position, Δp is the uncertainty in momentum, and ħ (h-bar) is the reduced Planck constant, which is approximately equal to 1.054571 x 10^-34 Js.

Quantum superposition

Quantum superposition is the principle according to which a quantum system can exist in multiple states simultaneously until it is measured. The act of measuring forces the system into one of the possible states. A classic thought experiment illustrating this is Schrödinger's cat.

Schrödinger imagined a scenario in which a cat was placed inside a sealed box containing a radioactive atom that had a 50% chance of decaying and releasing a poison that would kill the cat. Until the box is opened, the cat is simultaneously alive and dead according to quantum mechanics.

Box Cat Alive + dead

This thought experiment highlights the paradox and strange nature of quantum superposition.

Quantum entanglement

Quantum entanglement is a phenomenon in which particles become intertwined in such a way that the state of one particle instantaneously affects the state of another, no matter how far apart they are. This interesting behavior was questioned by Einstein, who called it "spooky action at a distance."

Entangled particles share information in a way that defies classical intuition and appear to operate as a single system, with changes in one having an immediate effect on the others.

Mathematics of quantum mechanics

The mathematics of quantum mechanics is complex, relying heavily on linear algebra, calculus, and complex numbers. The state of a quantum system is described by a wave function, represented by the Greek letter psi (ψ).

Ψ(x, t) = A e^(i(kx - ωt))

In this wave function, A is the amplitude, e is the exponential function, i is the imaginary unit, k is the wave number and is related to momentum, ω is the angular frequency related to energy, and x and t represent space and time.

The Schrödinger equation is an important equation in quantum mechanics, which provides a way to calculate the wave function of a quantum system. It is expressed as:

iħ ∂Ψ/∂t = -ħ²/2m ∇²Ψ + VΨ

Here, the left part includes the time derivative of the wave function while the right part includes terms representing kinetic and potential energy. Solutions to the Schrödinger equation help predict how quantum systems evolve over time.

Operators and observables

In quantum mechanics, physical quantities are represented by operators, mathematical objects that act on wave functions. When an operator acting on a wave function returns the same wave function multiplied by a constant, that constant represents a possible value (eigenvalue) of the observable associated with the operator.

Ĥψ = Eψ

In this equation, Ĥ is the Hamiltonian operator, which represents the total energy of the system, ψ is the wave function, and E is the energy eigenvalue.

Quantum tunneling

Quantum tunneling is a quantum mechanics phenomenon in which a particle tunnels through a barrier it cannot cross. In classical physics, if a ball does not have enough energy to reach the top of a hill, it will roll back. In contrast, quantum mechanics allows particles to tunnel through hills even without the required energy, exposing one of the paradoxical aspects of the theory.

Start Ending

Quantum tunneling is important in processes such as nuclear fusion in stars and in the operation of devices such as tunnel diodes.

Applications of quantum mechanics

Quantum mechanics is fundamental to modern technology and scientific understanding. It is the basis for many developments in fields such as electronics, computing and materials science. Technologies such as semiconductors, transistors, lasers and MRI machines are based on the principles of quantum mechanics.

Conclusion

Quantum mechanics is one of the most fascinating and challenging areas of physics, reshaping our understanding of the universe at the smallest scales. Its principles challenge classical ideas about reality and demonstrate the complex and sometimes unpredictable behaviour of matter and energy.


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Quantum mechanics

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