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 phi_1 and phi_2 the TDSE reads

Time-dependent Schroedinger equation in matrix form Explanation symbols hbar and t


Hamiltonian in matrix form Explanation of symbols in the Hamiltonian

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 T is given by

P_1 (T) = | phi_1 (T) |^2

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

Hamiltonian in terms of the momentum, the vector potential and the scalar potential Explanation of symbols in the Hamiltonian

The quantum state of the particle is characterized by the amplitude Psi (R,T) 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 T is given by

P (R,T) = | Psi(R,T) |^2

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.