Undergraduate
  1. General chemistry  1.1. Introduction to Chemistry  1.2. Atomic Structure  1.2.1. Subatomic particles  1.2.2. Atomic Model  1.2.3. Quantum numbers  1.2.4. Electron Configuration  1.2.5. Periodic trends  1.2.6. Isotopes and their applications  1.3. Chemical bond  1.3.1. Ionic bond  1.3.2. Covalent bonds  1.3.3. Metal Bond  1.3.4. Hybridization  1.3.5. Molecular geometry and VSEPR theory  1.3.6. Bond polarity and dipole moment  1.4. Stoichiometry  1.4.1. Mole concept  1.4.2. Empirical and molecular formula  1.4.3. Limiting reactant and excess reactant  1.4.4. Percent Yield and Purity  1.4.5. Dilution calculation  1.5. States of matter  1.5.1. Gas Laws  1.5.2. Matter and Intermolecular Forces  1.5.3. Solid and Crystal Structures  1.5.4. Phase diagram  1.5.5. Plasma and supercritical fluid  1.6. Chemical reactions  1.6.1. Types of Chemical Reactions  1.6.2. Balancing Chemical Equations  1.6.3. Thermochemical reactions  1.6.4. Reaction energy profiles  1.7. Solutions and Mixtures  1.7.1. Concentration units  1.7.2. Syndrome properties  1.7.3. Solubility and Solubility Rules  1.7.4. Henry's Law and Raoult's Law  1.7.5. Partition coefficient  1.8. Chemical equilibrium  1.8.1. Law of mass action  1.8.2. Le Chatelier's principle  1.8.3. Reaction quotient  1.8.4. To use this online calculator for Equilibrium Constant, enter Equilibrium Constant (Eq.) and hit the calculate button. Here is how the Equilibrium Constant calculation can be explained with given input values -> 0.05 = 0.05/0.  1.8.5. Acid-base balance  1.9. Acids and bases  1.9.1. Arrhenius, Brønsted–Lowry, and Lewis definitions  1.9.2. pH and pOH  1.9.3. Acid-base titration  1.9.4. Buffer Solution  1.9.5. Acid-base indicator  1.9.6. Polyprotic Acids and Bases  1.10. Electrochemistry  1.10.1. redox reactions  1.10.2. Galvanic and Electrolytic Cells  1.10.3. Nernst equation  1.10.4. Electrolysis  1.10.5. Faraday's laws of electrolysis  1.11. Thermodynamics  1.11.1. Laws of Thermodynamics  1.11.2. Enthalpy, entropy and Gibbs free energy  1.11.3. Spontaneity of reactions  1.11.4. Thermodynamic cycles (Hess's law, Carnot cycle)  1.12. Kinetics  1.12.1. Rate law and reaction order  1.12.2. Activation Energy and Catalyst  1.12.3. Collision theory  1.12.4. Reaction mechanism  1.12.5. Transition state theory
  2. Organic chemistry  2.1. Structure and relationships  2.1.1. Hybridisation in Carbon Compounds  2.1.2. Resonance and aromatization  2.1.3. Molecular orbitals  2.1.4. Inductive and mesomeric effects  2.2. Hydrocarbons  2.2.1. Hydrocarbons  2.2.2. Alkene  2.2.3. Alkynes  2.2.4. Aromatic compounds  2.2.5. Cycloalkanes and Conformation  2.3. Functional Group  2.3.1. Alcohols and phenols  2.3.2. Ethers and epoxides  2.3.3. Aldehyde and ketone  2.3.4. Carboxylic acids and derivatives  2.3.5. Amin  2.3.6. Esters and amides  2.3.7. Thiols and sulfides  2.4. Stereoscopic  2.4.1. Isomerism in Stereochemistry  2.4.2. Chirality and optical activity  2.4.3. Structural analysis  2.4.4. Diastereomers and Enantiomers  2.5.1. Substitution reactions  2.5.2. Elimination reactions  2.5.3. Addition reactions  2.5.4. Radical reactions  2.5.5. Rearrangement reactions  2.5.6. Pericyclic Reactions  2.6. Spectroscopy and structural analysis  2.6.1. Infrared Spectroscopy  2.6.2. Nuclear magnetic resonance spectroscopy  2.6.3. Mass Spectrometry  2.6.4. UV-visible spectroscopy  2.6.5. X-ray crystallography  2.7. Polymer chemistry  2.7.1. Addition and Condensation Polymers  2.7.2. Polymerization mechanism  2.7.3. Biodegradable Polymer  2.7.4. Conducting and smart polymers
  3. Inorganic chemistry  3.1. Coordination chemistry  3.1.1. Ligands and coordination compounds  3.1.2. Crystal field theory  3.1.3. Molecular Orbital Theory in Coordination Compounds  3.1.4. Jahn–Taylor distortion  3.2. Main Group Chemistry  3.2.1. Alkali and alkaline earth metals  3.2.2. Halogens and Noble Gases  3.2.3. Transition metals and their complexes  3.2.4. Lanthanides and Actinides  3.3. Solid state chemistry  3.3.1. Crystal lattices and unit cells  3.3.2. Defects in solids  3.3.3. Electrical and magnetic properties  3.3.4. Superconductivity  3.4. Bio-inorganic chemistry  3.4.1. Metal ions in biology  3.4.2. Enzymatic reactions with metals  3.4.3. Metal poisoning and detoxification  3.4.4. Metalloproteins and Metalloenzymes
  4. Physical chemistry  4.1. Quantum Chemistry  4.1.1. Wave–particle duality  4.1.2. Schrödinger Equation  4.1.3. Quantum Numbers and Orbitals  4.1.4. Atomic and molecular spectroscopy  4.2. Statistical mechanics  4.2.1. Maxwell–Boltzmann distribution  4.2.2. Partitioning functions  4.2.3. Bose–Einstein and Fermi–Dirac statistics  4.3. Chemical thermodynamics  4.3.1. Gibbs Free Energy  4.3.2. Phase transition  4.3.3. Thermodynamic efficiency  4.4. Surface chemistry  4.4.1. Absorption and Catalysis  4.4.2. Colloids and Emulsions
  5. Analytical Chemistry  5.1. Classical methods  5.1.1. Gravimetric Analysis  5.1.2. titrimeter  5.2. Instrumental methods  5.2.1. Chromatography  5.2.2. Spectroscopy  5.2.3. Electroanalytical methods  5.2.4. X-ray diffraction
  6. Industrial Chemistry  6.1. Petrochemistry  6.2. Chemical engineering principles  6.3. Green Chemistry  6.4. Pharmaceuticals and Drug Synthesis
  7. Biochemistry  7.1. Carbohydrates  7.2. Proteins and enzymes  7.3. Lipids and membranes  7.4. Nucleic acids  7.5. Metabolism and Bioenergetics  7.6. Enzyme kinetics and mechanism

Undergraduate → Physical chemistry → Quantum Chemistry ↓


Schrödinger Equation


The Schrödinger equation is a fundamental concept in quantum chemistry that describes how the quantum state of a physical system changes over time. It is a key element in understanding the behavior of atoms, molecules, and subatomic particles in the field of quantum mechanics. In simple terms, the equation is used to predict the probability of finding a particle in a particular state as well as its energy levels.

Introduction to quantum mechanics

Before delving into the intricacies of the Schrödinger equation, it is important to understand some basic concepts of quantum mechanics. Unlike classical mechanics, which deals with the motion of macroscopic objects, quantum mechanics explains the behaviour of particles on a microscopic scale, such as electrons in an atom.

Quantum mechanics introduces the idea of wave-particle duality, which means that particles such as electrons exhibit both wave-like and particle-like properties. This dual nature is important for the formulation and application of the Schrödinger equation.

Schrödinger equation

The Schrödinger equation is often considered the cornerstone of quantum mechanics. There are two forms of the equation: the time-dependent Schrödinger equation and the time-independent Schrödinger equation.

Time-dependent Schrödinger equation

The time-dependent Schrödinger equation is used to describe the evolution of a quantum state over time. It is given as:

    ∂ψ/∂t = Ĥψ

In this equation:

  • i is the imaginary unit.
  • ħ (h-bar) is the reduced Planck constant.
  • ψ is the wave function of the system.
  • Ĥ is the Hamiltonian operator, which represents the total energy of the system.
  • ∂ψ/∂t denotes the partial derivative of the wave function with respect to time.
  • The equation states that the change in the wave function over time is related to the energy of the system.

Time-independent Schrödinger equation

For systems where the potential energy does not change with time, we can use the time-independent Schrödinger equation. It is derived from the time-dependent version and looks like this:

    Ĥψ = eψ

Here:

  • E represents the energy eigenvalue of the system.
  • This equation is mainly used to find the allowed energy levels (also called eigenvalues) of a quantum system.

The time-independent Schrödinger equation is commonly used to solve various cases in quantum chemistry, such as the hydrogen atom, potential wells, and other molecular systems.

Visualization example: wave function in a 1D box

To explain how the Schrödinger equation works, let's consider a particle confined in a one-dimensional box (also known as the particle in a box model). This is a fundamental problem in quantum mechanics, used to understand the behavior of particles in a confined space.

The potential energy V(x) is zero inside the box and infinitely large outside the box:

    v(x) = 0, 0 ≤ x ≤ l
    V(x) = ∞, elsewhere

Solving the time-independent Schrödinger equation for this system we obtain the wave function:

    ψ_n(x) = sqrt(2/L) * sin(nπx/L)

Here, n is a positive integer indicating the quantum state of the particle, and L is the length of the box.

0 l ψ₁

The energies associated with each quantum state are given as:

    E_n = n²h²/(8mL²)

In this equation:

  • h is the Planck constant.
  • m is the mass of the particle.

This simple model shows how energy levels are quantized, meaning that the particle can only occupy certain energy levels, depending on the parameters n, h, m, and L

Applications: quantum chemistry

Quantum chemistry uses the Schrödinger equation to study the properties of atoms and molecules. Solutions to the equation help predict molecular structures, chemical reactions, and energy transitions. One of the simplest systems solved using the Schrödinger equation is the hydrogen atom.

Hydrogen atom

The hydrogen atom consists of one electron and one proton. The potential energy V(r) resulting from the electrostatic force between them is given by Coulomb's law:

    v(r) = -e²/(4πε₀r)

The time-independent Schrödinger equation for the hydrogen atom can be solved to obtain wave functions describing the state of the electron and the energy levels of the atom. The resulting energy levels are quantized, similar to the particle in a box system, and are expressed as:

    E_n = - (me⁴)/(8ε₀²h²n²)

Where:

  • m is the mass of the electron.
  • e is the elementary charge.
  • ε₀ is the vacuum permittivity.
  • The quantum number n indicates the specific energy level.

This equation accurately predicts the experimentally observed energy levels in hydrogen, and validates the use of the Schrödinger equation in describing atomic systems.

E⁻ P⁺

Analogy: Guitar string and stationary waves

A helpful analogy for understanding the concept of quantization in quantum mechanics is the standing wave pattern on a stringed instrument such as a guitar. When a guitar string is plucked, it vibrates at certain frequencies to produce sound. These are similar to the quantized energy levels seen in quantum systems.

The vibration of a guitar string can create different standing wave patterns, each of which represents a specific harmonic or overtone with its own corresponding frequency. These standing waves correspond to the wave functions in the Schrödinger equation, where each harmonic level represents a quantum state.

First harmonic

Just as a guitar string can only vibrate in certain states, the electrons in an atom can only exist in specific quantized energy states, as determined by the solutions to the Schrödinger equation.

Challenges in solving the Schrödinger equation

Solving the Schrödinger equation for simple systems such as the hydrogen atom is relatively straightforward, but for multi-electron atoms and molecules it becomes more complicated. The primary challenge is in dealing with electron-electron interactions, which require sophisticated mathematical techniques and approximations.

Some common methods and approaches are as follows:

  • Hartree-Fock method: A commonly used approximation method that simplifies the problem by assuming each electron moves in an average field due to the other electrons.
  • Density functional theory (DFT): A powerful computational technique that understands electron correlation by focusing on the electron density rather than the wave function.
  • Configuration interaction (CI): A post-Hartree–Fock method that involves linear combinations of multiple electron configurations to obtain more accurate results.

Conclusion

The Schrödinger equation plays a vital role in contemporary science by providing a framework for understanding and predicting the behaviour of quantum systems. From explaining the structure of simple atoms to unravelling the complex interactions within molecules, this equation is essential for advancing fields such as chemistry, physics and materials science.

Despite its complexity, the Schrödinger equation provides profound insights into the microscopic world, revealing the fundamental nature of matter and energy. As the basis of quantum mechanics, it continues to inspire scientific exploration and discovery.


Undergraduate → 4.1.2


U
username
0%
completed in Undergraduate

← Prev (4.1.1)
Wave–particle duality
Next (4.1.3) →
Quantum Numbers and Orbitals

Comments 



Schrödinger Equation

https://www.chemistryelite.com/ug/4.1.2?ceal=en