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## Questions and Answers

What is the Schrödinger equation?

What symbol is commonly used to represent the wave function in quantum mechanics?

Ψ or ψ

The presence of the imaginary number i means that the solutions to the Schrödinger equation are ______ quantities.

complex

The square of the absolute value of the wave function Ψ^2 tells us about the momentum of a particle.

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Match the following terms with their corresponding descriptions:

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## Study Notes

### 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.

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## Description

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