Quantum Mechanics: Wave Functions

Quantum Mechanics: Wave Functions

• Quantum mechanics is the key to understanding the behavior of matter on the molecular, atomic, and nuclear scales.
• It replaces the classical scheme of describing the state of a particle by its coordinates and velocity components.

Wave Functions and the One-Dimensional Schrödinger Equation

• The Schrödinger equation is the fundamental equation that describes the dynamics of matter waves.
• A wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position, and spin, denoted by the Greek letter psi (Ψ or ψ).
• Uppercase Ψ denotes a function of all space coordinates and time, while lowercase ψ denotes a function of space coordinates only, not time.

Wave Functions: Particle Waves

• Considering a free particle, the potential energy Ux has the same value for all x: U = 0.
• The wave function for a free particle is described by the angular frequency and wave number.

One-Dimensional Schrödinger Equation

• Developed in 1926 by Erwin Schrödinger, the equation is used to find the quantum mechanical wave function that satisfies it for a particular situation.
• The presence of the imaginary number i means that the solutions to the Schrödinger equation are complex quantities, with a real part and an imaginary part.

Interpreting Wave Functions

• The complex nature of the wave function for a free particle makes it challenging to interpret.
• The square of the absolute value of the wave function (Ψ²) of a particle at each point tells us about the probability of finding the particle around that point.
• Ψ(x, t)² dx is the probability that the particle will be found at time t at a coordinate in the range from x to x + dx, first made by Max Born.

One-Dimensional Schrödinger Equation with Potential Energy

Time-Independent One-Dimensional Schrödinger Equation

Particle in a Box

• The idea of the particle in a box is to determine, for a given potential energy function U(x), the possible stationary state wave functions, and the corresponding energies E.
• This model might represent an electron that is free to move within a long, straight molecule or along a very thin wire.

Energy Levels for a Particle in a Box

• The energy levels for a particle in a box can be found by solving the Schrödinger equation.

Wave Function for a Particle in a Box

• The wave function for a particle in a box is a solution to the Schrödinger equation.

Potential Wells and Barriers

• A potential well is a potential-energy function U(x) that has a minimum.
• A potential barrier is the opposite of a potential well, with a maximum potential energy.
• Tunneling occurs when a quantum-mechanical particle encounters a barrier, even if it has less energy, it may appear on the other side.

Tunneling

• The tunneling probability T that the particle gets through the barrier is proportional to the square of the ratio of the amplitudes of the sinusoidal wave functions on the two sides of the barrier.
• Applications of tunneling include tunnel diodes, Josephson junctions, scanning tunneling microscopes (STM), electron tunneling in enzymes, and fusion reactions.

Understand the behavior of matter at molecular, atomic, and nuclear scales through quantum mechanics, replacing classical descriptions with wave functions and the Schrödinger equation.