Schrödinger's wave mechanics, also known as wave mechanics or quantum mechanics, is a fundamental theory in physics that describes the behavior of particles on the atomic and subatomic scale. It was formulated by the Austrian physicist Erwin Schrödinger in 1925.
Wave mechanics is based on the concept of wave-particle duality, which suggests that particles such as electrons can exhibit both wave-like and particle-like properties. In this theory, the state of a particle is described by a mathematical entity called a wave function, typically denoted by the Greek letter psi (ψ). The wave function contains all the information about the particle's possible states and the probabilities of different outcomes when measurements are made.
The time evolution of the wave function is determined by Schrödinger's equation, which is a partial differential equation. It relates the rate of change of the wave function with respect to time to the Hamiltonian operator, which represents the total energy of the system. The Hamiltonian operator is constructed based on the classical mechanics of the system, including the kinetic and potential energy terms.
The Hamiltonian formalism, on the other hand, is a mathematical framework in classical mechanics that describes the dynamics of a physical system using the concept of a Hamiltonian function. It is based on the principle of least action and Hamilton's equations of motion.
While Schrödinger's wave mechanics and the Hamiltonian formalism share some conceptual similarities, they are distinct frameworks. Schrödinger's wave mechanics is a quantum mechanical theory that deals with the probabilistic behavior of particles, while the Hamiltonian formalism is a classical mechanical framework that describes the deterministic evolution of systems.
In the context of quantum mechanics, the Schrödinger equation is the appropriate equation to describe the time evolution of the wave function. Solving the Schrödinger equation involves finding the eigenvalues and eigenfunctions of the Hamiltonian operator, which correspond to the possible energy states and the associated wave functions of the system.
In summary, while the Hamiltonian formalism is not directly applicable to solving Schrödinger's equation, the Hamiltonian operator plays a crucial role in the formulation of quantum mechanics and determining the dynamics of quantum systems.