Quantum mechanics gets real with the Schrödinger equation. It's the key to unlocking how tiny particles behave, helping us figure out their energy levels and how they change over time. This equation is like a crystal ball for the quantum world.

The Schrödinger equation comes in two flavors: time-dependent and time-independent. We use these to understand different quantum systems, from particles trapped in boxes to the hydrogen atom. It's all about finding the right wave functions and energy levels.

Schrödinger Equation

Time-Dependent and Time-Independent Forms

• The Schrödinger equation is a fundamental equation in quantum mechanics that describes the behavior and evolution of a quantum system
• It comes in two forms: time-dependent and time-independent
  • The time-dependent Schrödinger equation describes how the quantum state of a system evolves over time
  • The time-independent Schrödinger equation is used to find the allowed energy levels and stationary states of a quantum system

Hamiltonian Operator and Stationary States

• The Hamiltonian operator $\hat{H}$ represents the total energy of the quantum system, including both kinetic and potential energy
  • It is a key component of the Schrödinger equation and acts on the wave function $\Psi$ to determine the system's energy
• Stationary states are special solutions to the time-independent Schrödinger equation where the probability density does not change over time
  • They correspond to the allowed energy levels of the quantum system (eigenstates)
  • The wave functions of stationary states are eigenfunctions of the Hamiltonian operator

Quantum Systems

Particle in a Box and Harmonic Oscillator

• The particle in a box model describes a quantum particle confined to a one-dimensional box with infinite potential walls
  • It is used to understand the quantization of energy levels and the formation of standing waves (eigenfunctions)
  • The allowed energy levels are given by $E_n = \frac{n^2h^2}{8mL^2}$, where $n$ is a positive integer, $h$ is Planck's constant, $m$ is the particle's mass, and $L$ is the length of the box
• The harmonic oscillator is another important quantum system that models the behavior of a particle subject to a quadratic potential energy
  • It is used to describe vibrations in molecules and the motion of atoms in a lattice
  • The allowed energy levels are given by $E_n = (n + \frac{1}{2})\hbar\omega$, where $n$ is a non-negative integer, $\hbar$ is the reduced Planck's constant, and $\omega$ is the angular frequency of the oscillator

Hydrogen Atom and Potential Energy Wells

• The hydrogen atom is a fundamental quantum system consisting of an electron bound to a proton by the Coulomb force
  • The Schrödinger equation is used to determine the allowed energy levels and orbitals (wave functions) of the electron
  • The energy levels are given by the Rydberg formula $E_n = -\frac{13.6 eV}{n^2}$, where $n$ is a positive integer
• Potential energy wells are regions where a particle experiences a lower potential energy compared to its surroundings
  • They can be used to model various quantum systems, such as atoms, molecules, and quantum dots
  • The shape of the potential well determines the allowed energy levels and wave functions of the trapped particle

Quantum Phenomena

Tunneling Effect

• The tunneling effect is a quantum phenomenon where a particle can pass through a potential barrier that it classically could not surmount
  • It occurs because of the wave-particle duality and the probabilistic nature of quantum mechanics
  • Examples include alpha decay, scanning tunneling microscopy (STM), and the operation of tunnel diodes
• The probability of a particle tunneling through a barrier depends on the barrier's height, width, and the particle's energy
  • A higher and wider barrier leads to a lower tunneling probability
  • The tunneling probability can be calculated using the WKB approximation or by solving the Schrödinger equation for the specific potential barrier