Graduate
  1. Physical Chemistry  1.1. Thermodynamics  1.1.1. Laws of Thermodynamics  1.1.2. Enthalpy and heat capacity  1.1.3. Entropy and free energy  1.1.4. Phase Equilibrium  1.1.5. Statistical thermodynamics  1.1.6. Thermodynamic cycle  1.1.7. Fugacity and activity  1.2. Quantum Chemistry  1.2.1. Principles of quantum mechanics  1.2.2. Schrödinger Equation  1.2.3. Particle in a box  1.2.4. Operators and eigenvalues  1.2.5. Atomic orbitals  1.2.6. Molecular orbital theory  1.2.7. Perturbation theory  1.2.8. Valence bond theory  1.2.9. Hybridization and chemical bonding  1.3. Chemical kinetics  1.3.1. Rate laws and reaction mechanisms  1.3.2. Collision theory  1.3.3. Transition state theory  1.3.4. Enzyme Kinetics  1.3.5. Chain Reactions and Polymerization  1.3.6. Photochemical reactions  1.4. Statistical mechanics  1.4.1. Partitioning functions  1.4.2. Molecular distribution functions  1.4.3. Boltzmann Distribution  1.4.4. Bose–Einstein and Fermi–Dirac statistics  1.5. Spectroscopy  1.5.1. Rotational Spectroscopy  1.5.2. Vibrational Spectroscopy  1.5.3. Electronic Spectroscopy  1.5.4. Nuclear magnetic resonance spectroscopy  1.5.5. Mass Spectrometry  1.5.6. Raman Spectroscopy  1.5.7. Electron paramagnetic resonance spectroscopy  1.6. Surface and colloidal chemistry  1.6.1. Absorption isotherm  1.6.2. Surface tension and wetting  1.6.3. Colloidal Stability  1.6.4. Catalysis  1.6.5. Emulsions and micelles  1.7. Electrochemistry  1.7.1. Nernst equation  1.7.2. Electrochemical cells  1.7.3. Conductivity and mobility  1.7.4. War  1.7.5. Fuel cells  1.7.6. Electrolysis
  2. Organic chemistry  2.1. Reaction mechanism  2.1.1. Nucleophilic substitution reactions  2.1.2. Electrophilic addition reactions  2.1.3. Elimination reactions  2.1.4. Rearrangement reactions  2.1.5. Radical reactions  2.1.6. Pericyclic reactions  2.2. Spectroscopy and structural determination  2.2.1. UV-Vis Spectroscopy  2.2.2. IR Spectroscopy  2.2.3. NMR Spectroscopy  2.2.4. Mass Spectrometry  2.2.5. X-ray crystallography  2.3. Stereoscopic  2.3.1. Chirality and optical activity  2.3.2. Structural analysis  2.3.3. Geometrical isomerism  2.3.4. Dynamic Stereochemistry  2.4. Organometallic Chemistry  2.4.1. Organolithium and organomagnesium reagents  2.4.2. Palladium-catalyzed cross-coupling reactions  2.4.3. Transition Metal Complexes  2.4.4. Metal–carbon bond  2.5. Polymer chemistry  2.5.1. Polymerization mechanism  2.5.2. Polymer Characteristics  2.5.3. Biodegradable Polymer  2.5.4. Conducting polymer  2.5.5. Supramolecular polymers  2.6. Medicinal chemistry  2.6.1. Drug design and development  2.6.2. Pharmacokinetics and pharmacodynamics  2.6.3. Structure-activity relationships  2.6.4. Molecular docking and drug screening
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Crystal field theory  3.1.2. Ligand field theory  3.1.3. Spectrochemical Series  3.1.4. Chelation and stability  3.2. Organometallic Chemistry  3.2.1. Metal Carbonyls  3.2.2. Catalysis by organometallic complexes  3.2.3. Metallocenes  3.3. Bio-inorganic chemistry  3.3.1. Metalloproteins and enzymes  3.3.2. The role of metals in biological systems  3.3.3. Metal ion transport and storage  3.4. Solid state chemistry  3.4.1. Crystal Structures  3.4.2. Band theory of solids  3.4.3. Superconductors  3.4.4. Defects in the crystal  3.5. Lanthanides and Actinides  3.5.1. Electronic configuration in lanthanides and actinides  3.5.2. Coordination chemistry of the lanthanides  3.5.3. Magnetic Properties of Lanthanides
  4. Analytical chemistry  4.1. Chromatography  4.1.1. Gas Chromatography  4.1.2. High Performance Liquid Chromatography  4.1.3. Thin Layer Chromatography  4.2. Spectroscopic Techniques  4.2.1. Atomic absorption spectroscopy  4.2.2. X-ray diffraction  4.2.3. Inductively coupled plasma spectroscopy  4.3. Electroanalytical methods  4.3.1. Potentiometry  4.3.2. Voltammetry  4.3.3. Colometry  4.4. Mass Spectrometry  4.4.1. Ionization Technique  4.4.2. Fragmentation Pattern  4.5. Chemometrics  4.5.1. Multidisciplinary analysis  4.5.2. Machine Learning in Chemistry
  5. Theoretical and Computational Chemistry  5.1. Molecular dynamics simulation  5.1.1. Force fields and energy minimization  5.1.2. Monte Carlo simulation in molecular dynamics simulations  5.2. Quantum chemical methods  5.2.1. Hartree–Fock theory  5.2.2. Density functional theory  5.2.3. Semi-empirical methods  5.3. Computational drug design  5.3.1. Molecular Docking  5.3.2. QSAR Modeling  5.3.3. Virtual Screening
  6. Biochemistry  6.1. Enzyme Kinetics  6.1.1. Michaelis-Menten kinetics  6.1.2. Inhibition mechanism  6.2. Metabolism and Bioenergetics  6.2.1. Glycolysis  6.2.2. Citric acid cycle  6.2.3. Electron transport chain  6.3. molecular Biology  6.3.1. DNA replication and repair  6.3.2. Protein synthesis  6.3.3. Gene Regulation  6.4. Structural Biochemistry  6.4.1. Protein Folding  6.4.2. Membrane Biophysics
  7. Environmental Chemistry  7.1. Atmospheric Chemistry  7.1.1. Greenhouse gases  7.1.2. Ozone depletion  7.1.3. Air pollutants and smoke  7.2. Water Chemistry  7.2.1. pH and water hardness  7.2.2. Wastewater treatment  7.2.3. Aquatic Chemistry  7.3. Soil Chemistry  7.3.1. Heavy metal contamination  7.3.2. Soil pH and Buffering  7.3.3. Nutrient cycling  7.4. Toxicology and Chemical Safety  7.4.1. Toxicokinetics  7.4.2. Risk Assessment in Toxicology and Chemical Safety in Environmental Chemistry  7.4.3. Effects of pollutants on the environment

Graduate → Physical Chemistry → Quantum Chemistry ↓


Principles of quantum mechanics


Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. Quantum chemistry applies quantum mechanics to problems of chemistry and helps explain the behavior of atoms and molecules. The principles of quantum mechanics form the basis on which this theory is built. They represent a set of principles that govern the behavior of systems at the quantum level.

Postulate 1: The state of a quantum system

The first theorem of quantum mechanics states that the state of a quantum mechanical system is completely specified by a wave function, denoted as ψ. This wave function contains all the information about the system and its position or momentum. The wave function is a complex-valued function of space and time, and it determines the probability amplitude of the system's position and momentum.

Mathematically, the wave function ψ is expressed as:

ψ = ψ(x, t)

The absolute square of the wave function, |ψ(x, t)|², gives the probability density of finding a particle at position x at time t.

Consider a one-dimensional example where the wave function ψ is represented graphically as a sinusoidal wave.

Here, the x-axis represents the position, and the oscillating wave represents the wave function ψ.

Postulate 2: Observables and operators

The second concept states that each observable quantity in a quantum mechanical system, such as position, momentum, and energy, is associated with a mathematical operator. These operators act on the wave function to extract information about the observable.

For example, the position operator ̂x acts on a wave function as follows:

ψx̂x = ψx(x, t)

The momentum operator ̂p is given by:

̂p = −iħ (∂/∂x)

where ħ is the reduced Planck constant, and i is the imaginary unit.

Postulate 3: Measurement and expectation value

The third principle concerns the measurement of observables in quantum systems. According to this principle, the only possible outcome of measuring any observable is one of the eigenvalues of the corresponding operator.

The expected value of an observable A represented by the operator  <A> is given by:

<A> = ∫ψ* Â ψ dx

where ψ* is the complex conjugate of the wave function ψ.

To calculate the expected value, imagine a particle in a potential well. The observable state can be, and its expected value gives an average value of where the particle is likely to be found.

Postulate 4: Time evolution of the system

The fourth principle of quantum mechanics asserts that the time evolution of a quantum system is governed by the Schrödinger equation, which is a fundamental equation in quantum mechanics. This equation describes how the quantum state of a physical system changes over time.

The time-dependent Schrödinger equation is:

∂ψ/∂t = Ĥψ

where Ĥ is the Hamiltonian operator, representing the total energy of the system.

For example, consider the case of a free particle in one dimension. Its time evolution can be calculated using the Schrödinger equation to predict its behavior in the future.

Postulate 5: Quantum Superposition

The fifth concept involves the principle of superposition, which says that if a system can exist in multiple states, then any linear combination of these states is also a possible state of the system.

Mathematically, if ψ₁ and ψ₂ are two solutions of the Schrödinger equation, then their linear combination c₁ψ₁ + c₂ψ₂ is also a solution, where c₁ and c₂ are complex constants.

In the visualization above, the red and green waves represent different states. Their combination, represented in blue, is also a valid state of the system.

Postulate 6: Particle–wave duality

This theory addresses the dual nature of particles. Quantum systems can exhibit both particle-like and wave-like properties, depending on the type of measurement.

A practical example of this principle is the double-slit experiment, which shows that particles such as electrons can create interference patterns, a property of waves.

Textual examples and applications

To further understand the application of quantum principles in quantum chemistry, let us consider some examples:

Example 1: Hydrogen atom
In the hydrogen atom, the electronic state is expressed using wave functions obtained by solving the Schrödinger equation. These wave functions describe possible orbitals, each of which is associated with an energy level. Observing the electron involves measuring the energy using Postulate 2 and predicting its location around the nucleus via Postulate 3.

Example 2: Molecular bond
Quantum mechanics can explain chemical bonding through molecular orbital theory. For example, in a simple molecule such as H₂, the superposition postulate describes the overlap of atomic orbitals, leading to molecular orbitals that dictate bond formation.

Example 3: Heisenberg uncertainty principle
This principle states that it is impossible to know precisely both the position and momentum of a particle at the same time. This connects to measurement principles, as it states how precisely we can measure a quantum system.

As we delve deeper into the principles of quantum mechanics and their applications in chemistry, these fundamental principles enable scientists to unravel a variety of chemical phenomena, from reaction mechanisms to the electronic structure of complex molecules. These principles provide important insights at the most fundamental level of matter behavior and continue to shape the ever-progressing field of quantum chemistry.


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Principles of quantum mechanics

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