Schrodinger wave equation is a fundamental equation in quantum mechanics that describes the behavior of particles in a quantum system. It was first introduced by Erwin Schrodinger in 1925. The equation is based on the wave-particle duality of matter, which states that particles can exhibit both wave-like and particle-like behavior. The Schrodinger wave equation is used to calculate the wave function of a particle in a given quantum state.

Derivation of Schrodinger wave equation

The Schrodinger wave equation is derived from the classical wave equation. The classical wave equation describes the behavior of waves in a medium, such as water waves or sound waves. The wave equation relates the second derivative of the wave function with respect to time and space.

The classical wave equation is given by:

∂²u/∂t² = c²∂²u/∂x²

where u is the wave function, t is time, x is position, and c is the speed of the wave.

However, the classical wave equation cannot be applied to describe the behavior of particles in a quantum system. This is because particles in a quantum system exhibit wave-particle duality, which means that they can exhibit both wave-like and particle-like behavior.

To derive the Schrodinger wave equation, Schrodinger made use of de Broglie's hypothesis, which states that particles can also be described by a wave function. Based on this hypothesis, Schrodinger proposed that the wave function of a particle in a quantum system can be described by a complex function Ψ(x,t).

The Schrodinger wave equation can be derived by replacing the classical wave equation with the wave function Ψ(x,t) and making use of the de Broglie relation between the momentum and wavelength of a particle.

The Schrodinger wave equation is given by:

iℏ∂Ψ/∂t = HΨ

where i is the imaginary unit, ℏ is the reduced Planck constant, H is the Hamiltonian operator, and Ψ is the wave function of the particle.

The Hamiltonian operator represents the total energy of the particle and is given by:

H = -ℏ²/2m ∂²/∂x² + V(x)

where m is the mass of the particle and V(x) is the potential energy function.

Conclusion:

In conclusion, the Schrodinger wave equation is a fundamental equation in quantum mechanics that describes the behavior of particles in a quantum system. It is derived from the classical wave equation by replacing the wave function with the particle's wave function and making use of the de Broglie relation between the momentum and wavelength of the particle. The Schrodinger wave equation is a complex equation that is used to calculate the wave function of a particle in a given quantum state.