⚛️Molecular Electronics Unit 2 – Quantum Mechanics Fundamentals

Key Concepts and Principles

• Quantum mechanics describes the behavior of matter and energy at the atomic and subatomic scales
• Particles exhibit both wave-like and particle-like properties (wave-particle duality)
• The Heisenberg uncertainty principle states that the position and momentum of a particle cannot be simultaneously known with absolute precision
  • The more precisely the position of a particle is determined, the less precisely its momentum can be known, and vice versa
• Quantization is the concept that certain physical quantities can only take on discrete values (energy levels in atoms)
• The Pauli exclusion principle states that no two identical fermions can occupy the same quantum state simultaneously
• The Bohr model introduced the idea of stationary states and energy levels in atoms
• The correspondence principle states that quantum mechanics must agree with classical physics in the limit of large quantum numbers

Quantum States and Wave Functions

• A quantum state is a complete description of a quantum system
• The wave function, denoted as $\Psi(x, t)$, is a mathematical representation of a quantum state
  • The wave function contains all the information about the system
  • The probability of finding a particle at a specific location is proportional to the square of the absolute value of the wave function, $|\Psi(x, t)|^2$
• The normalization condition ensures that the total probability of finding the particle somewhere in space is equal to 1
• The expectation value of an observable is the average value of the observable over many measurements on identically prepared systems
• The collapse of the wave function occurs when a measurement is made on a quantum system, causing the system to "jump" into one of the possible eigenstates of the observable being measured
• The time evolution of a quantum state is determined by the time-dependent Schrödinger equation

Schrödinger Equation and Its Applications

• The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system
• The time-independent Schrödinger equation is used to find the stationary states and energy levels of a quantum system
  • It is an eigenvalue equation, where the eigenvalues correspond to the allowed energy levels and the eigenfunctions represent the corresponding wave functions
• The time-dependent Schrödinger equation describes the time evolution of a quantum state
• The Schrödinger equation can be applied to various systems, such as the particle in a box, the harmonic oscillator, and the hydrogen atom
• The particle in a box model is used to understand the behavior of electrons in confined structures (quantum dots)
• The harmonic oscillator model is used to describe the vibrations of atoms in molecules and the behavior of photons in a cavity
• The hydrogen atom is a fundamental system in quantum mechanics, and its solution using the Schrödinger equation led to the understanding of atomic structure and spectra

Quantum Operators and Observables

• Quantum operators are mathematical entities that represent physical observables in quantum mechanics
• Observables are measurable quantities, such as position, momentum, energy, and angular momentum
• The eigenvalues of an operator correspond to the possible outcomes of a measurement of the associated observable
• The eigenfunctions of an operator represent the quantum states in which the system has a definite value of the observable
• The commutator of two operators, $[A, B] = AB - BA$, determines whether the observables represented by the operators can be simultaneously measured with arbitrary precision
  • If the commutator is zero, the observables are said to commute and can be simultaneously measured
  • If the commutator is non-zero, the observables are said to be incompatible and cannot be simultaneously measured with arbitrary precision (Heisenberg uncertainty principle)
• The expectation value of an observable is calculated using the operator and the wave function, $\langle A \rangle = \int \Psi^* \hat{A} \Psi dx$

Quantum Tunneling and Barriers

• Quantum tunneling is a phenomenon in which a particle can pass through a potential barrier that it classically could not surmount
• The probability of a particle tunneling through a barrier depends on the barrier height, width, and the particle's energy
• The transmission coefficient, $T$, represents the probability of a particle tunneling through a barrier
  • It is calculated using the wave functions on either side of the barrier and the barrier potential
• Quantum tunneling has important applications in scanning tunneling microscopy (STM), where electrons tunnel between a sharp tip and a surface to create images with atomic resolution
• Tunneling is also the basis for many electronic devices, such as tunnel diodes and resonant tunneling diodes (RTDs)
• The Josephson effect is a manifestation of quantum tunneling in superconductors, where Cooper pairs can tunnel through a thin insulating barrier between two superconductors
• The Fowler-Nordheim tunneling describes the tunneling of electrons from a metal surface into vacuum under the influence of a strong electric field

Quantum Superposition and Entanglement

• Quantum superposition is the principle that a quantum system can exist in multiple states simultaneously until a measurement is made
• The Schrödinger's cat thought experiment illustrates the concept of quantum superposition, where a cat is both alive and dead until observed
• Quantum entanglement is a phenomenon in which two or more particles are correlated in such a way that the quantum state of each particle cannot be described independently
  • Entangled particles exhibit correlations that cannot be explained by classical physics
  • The Einstein-Podolsky-Rosen (EPR) paradox highlighted the strange nature of quantum entanglement
• Bell's theorem provided a way to test for the presence of entanglement and ruled out local hidden variable theories as an explanation for the correlations
• Quantum entanglement has applications in quantum cryptography (secure communication) and quantum teleportation (transferring quantum states)
• The no-cloning theorem states that it is impossible to create an identical copy of an arbitrary unknown quantum state, which is a consequence of the linearity of quantum mechanics

Molecular Orbital Theory

• Molecular orbital theory describes the behavior of electrons in molecules using quantum mechanics
• Molecular orbitals are formed by the linear combination of atomic orbitals (LCAO) from the constituent atoms
• The wave function of a molecular orbital is expressed as a linear combination of the atomic orbital wave functions, $\Psi_{MO} = c_1 \phi_1 + c_2 \phi_2 + ...$
  • The coefficients $c_i$ represent the contribution of each atomic orbital to the molecular orbital
• Bonding orbitals are lower in energy than the constituent atomic orbitals and have a higher probability of electron occupancy between the nuclei, resulting in a stable bond
• Antibonding orbitals are higher in energy than the constituent atomic orbitals and have a node (zero probability) between the nuclei, resulting in an unstable bond
• The highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) are important in determining the chemical and optical properties of molecules
• The HOMO-LUMO gap represents the minimum energy required to excite an electron from the HOMO to the LUMO, which is related to the absorption and emission spectra of the molecule

Quantum Effects in Electronic Devices

• Quantum effects become significant when the dimensions of electronic devices approach the nanoscale
• The quantum confinement effect occurs when the size of a material is comparable to the de Broglie wavelength of the electrons, leading to discrete energy levels and modified optical and electronic properties
  • Quantum dots, nanowires, and nanosheets exhibit quantum confinement effects
• Single-electron transistors (SETs) operate based on the controlled transfer of individual electrons, utilizing the Coulomb blockade effect
• Quantum well lasers confine electrons and holes in a thin layer (quantum well), resulting in improved efficiency and reduced threshold current compared to bulk semiconductor lasers
• Quantum cascade lasers (QCLs) are based on the transitions of electrons between quantum well states in a superlattice structure, enabling emission at specific wavelengths in the mid-infrared and terahertz regions
• Spintronics exploits the spin degree of freedom of electrons in addition to their charge, leading to the development of spin-based devices (spin valves, magnetic tunnel junctions)
• Topological insulators are materials that exhibit conducting states on their surface while being insulating in their bulk, due to the topology of their electronic band structure
• Quantum computing utilizes quantum bits (qubits) and quantum gates to perform computations, potentially offering a significant speedup over classical computers for certain problems (factoring, optimization)