Quantum Mechanics: Wave-Particle Duality

• Quantum mechanics studies the physics of atoms, and subatomic particles.
• It is a mathematical framework used to develop physical theories.
• Quantum mechanics incorporates both particle and wave properties to describe matter.
• Quantum mechanics is required to understand the behavior of systems at very small scales (e.g. atoms, molecules).
• Classical mechanics provides accurate results when applied to macroscopic objects.

Wave–particle duality

• One of the basic postulates of quantum mechanics is wave–particle duality.
• Wave–particle duality states that all particles exhibit wave-like properties and vice versa.
• A single quantum object can be described as either a particle or a wave.
• Particles can display wave-like behavior such as diffraction and interference.
• Waves can display particle-like behavior under certain conditions.
• The wave-like behavior is noticeable when the size of the object is comparable to its wavelength.

Quantum Superposition

• Quantum superposition states that if a physical system can be in multiple configurations, it can also be in a combination of these states.
• The state of the system is a combination of all possible states until it is measured.
• Mathematically, superposition means that a quantum state can be represented as a sum of two or more other distinct states.

Quantum Entanglement

• Quantum entanglement occurs when two or more particles become correlated.
• The quantum state of each particle cannot be described independently of the others.
• The measurement of the quantum state of one particle determines the possible results of the measurement of the other particles.
• The result is true even when the particles are separated by large distances.

Quantum Measurement

• Quantum measurement refers to the process of determining a property of a quantum system.
• The act of measuring a quantum system invariably disturbs it.
• Quantum measurement affects the superposition of states, and forces the system to "choose" one state.
• The wave function collapses during a quantum measurement.
• Quantum measurements are probabilistic.

Heisenberg Uncertainty Principle

• The Heisenberg uncertainty principle states that there is a fundamental limit to the precision with which certain pairs of physical properties of a particle can be known simultaneously.
• For position and momentum, the more accurately one property is known, the less accurately the other can be known.
• Mathematically, the uncertainty principle is expressed as Δx * Δp ≥ ħ/2
• Δx represents the uncertainty in position, and Δp represents the uncertainty in momentum.
• ħ is the reduced Planck constant (h/2π).

Schrödinger Equation

• The Schrödinger equation is a mathematical equation that describes the time evolution of quantum mechanical systems.
• The time-dependent Schrödinger equation is iħ(∂ψ/∂t) = Hψ
• ψ is the wave function of the quantum system.
• H is the Hamiltonian operator, corresponding to the total energy of the system.
• The Schrödinger equation is a fundamental equation in quantum mechanics and can be used to predict the behavior of quantum systems.
• Solving the Schrödinger equation is essential for understanding the dynamics of quantum systems.

Wave Function

• The wave function describes the quantum state of a system.
• It contains information about the probability amplitude of finding a particle at a specific position and time.
• The wave function is a mathematical function denoted by ψ(r, t).
• The square of the absolute value of the wave function gives the probability density of finding the particle at a given point in space and time.
• The wave function must satisfy certain mathematical conditions, such as being single-valued, continuous, and finite.

Quantum Tunneling

• Quantum tunneling is the phenomenon where a particle passes through a potential energy barrier that it classically cannot surmount.
• The probability of tunneling depends on the width and height of the barrier, as well as the particle's energy.
• Tunneling is more probable for particles with smaller mass, lower barrier heights, and narrower barriers.
• Quantum tunneling plays a crucial role in various physical phenomena, such as nuclear fusion in stars.

Quantum Harmonic Oscillator

• The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator.
• It describes systems such as vibrating molecules or atoms in a lattice.
• The energy levels are quantized and evenly spaced, given by En = (n + 1/2)ħω, where n is a non-negative integer and ω is the angular frequency of the oscillator.
• The quantum harmonic oscillator is a fundamental concept in quantum mechanics.

Operators in Quantum Mechanics

• In quantum mechanics, physical quantities are represented by mathematical operators.
• An operator acts on a wave function to produce another wave function.
• Examples of operators include the position operator, momentum operator, and energy operator (Hamiltonian).
• The eigenvalues of an operator correspond to the possible values of the physical quantity that can be measured.

Quantum Numbers

• Quantum numbers are a set of numbers that describe the properties of an atomic orbital.
• There are four main quantum numbers: principal quantum number (n), azimuthal quantum number (l), magnetic quantum number (ml), and spin quantum number (ms).
• The principal quantum number (n) determines the energy level of the electron.
• The azimuthal quantum number (l) determines the shape of the electron's orbital.
• The magnetic quantum number (ml) determines the orientation of the electron's orbital in space.
• The spin quantum number (ms) describes the intrinsic angular momentum of the electron.

Time-Independent Schrödinger Equation

• The time-independent Schrödinger equation is a specific form of the Schrödinger equation used for systems where the potential energy does not change with time.
• It is given by Hψ = Eψ, where H is the Hamiltonian operator, ψ is the wave function, and E is the energy of the system.
• Solving the time-independent Schrödinger equation yields the stationary states of the system and their corresponding energies.
• These stationary states represent time-independent probability distributions.

Identical Particles

• In quantum mechanics, identical particles are indistinguishable from one another.
• Identical particles must be either bosons or fermions.
• Bosons have integer spin and obey Bose-Einstein statistics.
• Fermions have half-integer spin and obey Fermi-Dirac statistics.
• The wave function of a system of identical bosons is symmetric under particle exchange.
• The wave function of a system of identical fermions is antisymmetric under particle exchange.

Perturbation Theory

• Perturbation theory is a method used to approximate solutions to quantum mechanical problems that cannot be solved exactly.
• It involves treating a small disturbance (perturbation) to a system that can be solved exactly.
• The perturbation is typically an external field or a small change in the potential energy.
• Perturbation theory allows to calculate approximate energy levels and wave functions for the perturbed system.

Born Approximation

• The Born approximation is an approximation used in scattering theory to calculate the scattering amplitude.
• It assumes that the incident wave is only weakly scattered by the potential.
• The Born approximation simplifies the scattering problem allowing an easier calculation of the scattering cross-section.