Schrödinger equation

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system.:1–2 It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. In classical mechanics, Newton's second law (F = ma) is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force F {displaystyle mathbf {F} } on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation. The concept of a wave function is a fundamental postulate of quantum mechanics; the wave function defines the state of the system at each spatial position, and time. Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian. This derivation is explained below. In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.:292ff Schrödinger's equation is central to all applications of quantum mechanics, including quantum field theory, which combines special relativity with quantum mechanics. Theories of quantum gravity, such as string theory, also do not modify Schrödinger's equation. The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation. The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation (TDSE), which gives a description of a system evolving with time::143 where i {displaystyle i} is the imaginary unit, ℏ = h 2 π {displaystyle hbar ={frac {h}{2pi }}} is the reduced Planck constant, Ψ {displaystyle Psi } (the Greek letter psi) is the state vector of the quantum system, t {displaystyle t} is time, and H ^ {displaystyle {hat {H}}} is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector | r ⟩ {displaystyle vert mathbf {r} angle } . It is a scalar function, expressed as Ψ ( r , t ) = ⟨ r | Ψ ⟩ {displaystyle Psi (mathbf {r} ,t)=langle mathbf {r} vert Psi angle } . Similarly, the momentum-space wave function can be defined as Ψ ~ ( p , t ) = ⟨ p | Ψ ⟩ {displaystyle { ilde {Psi }}(mathbf {p} ,t)=langle mathbf {p} vert Psi angle } , where | p ⟩ {displaystyle vert mathbf {p} angle } is the momentum eigenvector. The most famous example is the nonrelativistic Schrödinger equation for the wave function in position space Ψ ( r , t ) {displaystyle Psi (mathbf {r} ,t)} of a single particle subject to a potential V ( r , t ) {displaystyle V(mathbf {r} ,t)} , such as that due to an electric field.