Introduction to Schrödinger's Equation

Fundamental theory in physics describing the properties of nature on an atomic scale.

Schrödinger's Equation is a cornerstone of quantum mechanics. Named after the Austrian physicist Erwin Schrödinger, who developed it in 1926, this equation describes how the quantum state of a quantum system changes with time.

Role of Schrödinger's Equation in Quantum Mechanics

In classical mechanics, the state of a physical system is described by the positions and velocities of its constituent particles. However, in quantum mechanics, the state of a system is described by a wave function, which is a solution to Schrödinger's Equation. This equation, therefore, plays a central role in the theory of quantum mechanics.

Time-Dependent and Time-Independent Schrödinger's Equation

Schrödinger's Equation comes in two forms: time-dependent and time-independent. The time-dependent Schrödinger's Equation is a partial differential equation that describes how the wave function of a physical system evolves over time. It is given by:

ĤΨ = iħ ∂Ψ/∂t

where Ĥ is the Hamiltonian operator (which represents the total energy of the system), Ψ is the wave function, i is the imaginary unit, ħ is the reduced Planck's constant, and t is time.

The time-independent Schrödinger's Equation, on the other hand, is used for systems where the Hamiltonian does not depend on time. It is given by:

ĤΨ = EΨ

where E is the total energy of the system. The solutions to this equation are the possible energy levels of the system and their corresponding wave functions.

Physical Interpretation of Schrödinger's Equation

The physical interpretation of Schrödinger's Equation is one of the most profound aspects of quantum mechanics. The wave function Ψ in the equation is a mathematical representation of the quantum state of a system. The absolute square of the wave function, |Ψ|^2, gives the probability density of finding the system in a particular state.

In conclusion, Schrödinger's Equation is a fundamental equation in quantum mechanics that describes the behavior of quantum systems. It introduces the concept of a wave function to represent the state of a system and provides a way to calculate how this state evolves over time.