Quantum Mechanics

Introduction: Schrödinger equation

The time evolution of a quantum system follows from the solution of the **TDSE**, the **Time-Dependent Schrödinger Equation**.
For simplicity consider the TDSE describing a system that can be in no more than two states. For a quantum system that has only two possible states
and
the TDSE reads

where

is the Hamiltonian describing the system. The solution of this equation gives a complete description of the time evolution of the quantum system. For instance, the probability to find the system in state 1 at time is given by

The Hamiltonian for a particle in an electromagnetic potential is given by

The quantum state of the particle is characterized by the amplitude
for any point in space and time.
This amplitude is also called the **wave function** of the particle.
As before, the TDSE governs the time evolution of the wave function.
The probability to find the particle at the position
at time
is given by

The wave function contains all the information about the quantum system. Once it is known for all points in space and time, any physical quantity can be calculated.