Introduction to Quantum Mechanics

Quantum mechanics is the branch of chemistry and physics that describes the behavior of matter and energy at the atomic and molecular scale.

Classical mechanics fails to explain:

Ø  Black body radiation

Ø  Photoelectric effect

Ø  Atomic spectra

Ø  Stability of atoms

Therefore, quantum mechanics was developed to accurately describe microscopic systems.

In computational chemistry, quantum mechanics provides the theoretical foundation for:

Ø  Molecular structure prediction

Ø  Energy calculations

Ø  Reaction mechanisms

Ø  Spectroscopy simulations

Why Quantum Mechanics is Important in Computational Chemistry?

Computational chemistry uses mathematical models and computer simulations based on quantum mechanics to study molecules.

Applications:

Ø  Geometry optimization

Ø  Molecular orbital calculations

Ø  Electronic spectra prediction

Ø  Drug design

Ø  Solar cell materials modeling

Without quantum mechanics, modern computational chemistry would not exist.

Basic Concepts of Quantum Mechanics

Before the postulates, students must understand:

Wave–Particle Duality

Electrons behave as both:

·         particles

·         waves

According to de Broglie:

Heisenberg Uncertainty Principle

It is impossible to simultaneously know exact:

·         position

·         momentum

Implication: Electrons do not have fixed paths (no classical orbits).

Wave Function (Ψ)

The wave function describes the quantum state of a particle.

·         Denoted by: Ψ (psi)

·         Contains all information about the system

$$\mathrm{}$$

·         Obtained by solving Schrödinger equation

P1 — Wave Function Postulate:

Statement:
The state of a quantum system is completely described by a wave function Ψ(x, y, z, t).

Key Points:

·         Ψ contains all measurable information

·         Ψ must be:

Ø  single valued

Ø  finite

Ø  continuous

Ø  normalizable

The probability of finding an electron in a region is:

∣Ψ∣²

This is called probability density.

P2 Operator Postulate:

Every observable physical quantity (such as energy, momentum, and position) is represented by a corresponding linear operator. When an operator acts on the wave function, it provides information about the measurable property of the system.

The Hamiltonian Operator (Most Important)                   

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This is the Schrödinger equation, the heart of computational chemistry.

P3 Measurement (Eigenvalue) Postulate:

The only possible result of measuring an observable is one of the eigenvalues of its operator.

Meaning:

 HΨ=EΨ

·         eigenfunctions → allowed wave functions

·         eigenvalues → allowed energies

Energy is quantized, not continuous, and only specific allowed values can be obtained from measurements.

In molecules: explains discrete spectra.

Eigenfunctions represent specific states (eigenstates) of a system, where a measurement of an observable (like momentum or energy) yields a definite value (the eigenvalue).

P4  Expectation Value Postulate:
The average or expectation value of an observable for a system in a given state can be calculated using the wave function and the corresponding operator through an integral expression involving

Ψ and its complex conjugate.

Statement:
The average value of an observable is given by:

$d\tau$ represents the volume element (differential element of all spatial coordinates) over which the integration is performed.

Meaning:

Gives the average measurable value.

In computational chemistry:

Used to compute:

Ø  dipole moment

Ø  electron density

Ø  total energy

 

P5  Time Evolution Postulate:

Statement:
The time evolution of a quantum system is governed by the time-dependent Schrödinger equation. It describes how the wave function changes with time under the influence of the Hamiltonian operator.