Schrödinger equation

Grave Schroedinger (detail)

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StationaryStatesAnimation

Infinite potential well

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Hydrogen Density Plots

Schrödinger equation is a fundamental equation in quantum mechanics that describes how the quantum state of a physical system changes over time. It was formulated in 1925 by the Austrian physicist Erwin Schrödinger, and it plays a pivotal role in the theory of quantum mechanics, similar to Newton's laws of motion in classical mechanics.

Overview

The Schrödinger equation provides a way to predict the future behavior of particles at the quantum level, such as electrons, atoms, and molecules. It is a wave equation in terms of the wave function, which is a mathematical description of the quantum state of a system. The square of the absolute value of the wave function represents the probability density of finding a particle in a given place at a given time.

Types of Schrödinger Equations

There are two main forms of the Schrödinger equation: the time-dependent Schrödinger equation and the time-independent Schrödinger equation.

Time-Dependent Schrödinger Equation

The time-dependent Schrödinger equation describes how the quantum state of a system evolves over time. It is given by: \[ i\hbar\frac{\partial}{\partial t}\Psi(\mathbf{r},t) = \hat{H}\Psi(\mathbf{r},t) \] where \(i\) is the imaginary unit, \(\hbar\) is the reduced Planck's constant, \(\Psi(\mathbf{r},t)\) is the wave function of the system, \(t\) is time, \(\mathbf{r}\) represents the position vector, and \(\hat{H}\) is the Hamiltonian operator, which represents the total energy of the system.

Time-Independent Schrödinger Equation

The time-independent Schrödinger equation is used for systems with energies that do not change over time. It is derived from the time-dependent equation for stationary states and is given by: \[ \hat{H}\psi(\mathbf{r}) = E\psi(\mathbf{r}) \] where \(\psi(\mathbf{r})\) is the time-independent wave function, and \(E\) represents the energy eigenvalues of the system.

Applications

The Schrödinger equation is used in various areas of physics and chemistry to solve problems involving the behavior of particles at the quantum level. It is essential for understanding phenomena such as quantum tunneling, atomic orbitals, and the behavior of electrons in semiconductors. It also forms the basis for advanced theories and technologies, including quantum computing and quantum cryptography.

Historical Context

The development of the Schrödinger equation marked a significant advancement in the field of quantum mechanics. Schrödinger was motivated by the earlier work of Louis de Broglie and Albert Einstein's photoelectric effect, leading him to propose the wave nature of particles. His work, along with that of Werner Heisenberg's matrix mechanics, formed the foundation of modern quantum mechanics.

Conclusion

The Schrödinger equation is a cornerstone of quantum mechanics, providing a comprehensive framework for understanding the quantum world. Its formulation not only advanced the field of physics but also opened new avenues for research and technology development in the 20th and 21st centuries.