  in physics, the schrÃ¶dinger picture (also called the schrÃ¶dinger representation) is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators (observables and others) are constant with respect to time. this differs from the heisenberg picture which keeps the states constant while the observables evolve in time, and from the interaction picture in which both the states and the observables evolve in time. the schrÃ¶dinger and heisenberg pictures are related as active and passive transformations and have the same measurement statistics. in the schrÃ¶dinger picture, the state of a system evolves with time. the evolution for a closed quantum system is brought about by a unitary operator, the time evolution operator. for time evolution from a state vector |\psi(t_0)\rangle at time t_0 to a state vector |\psi(t)\rangle at time t, the time-evolution operator is commonly written u(t, t_0), and one has :|\psi(t)\rangle = u(t, t_0) |\psi(t_0)\rangle. in the case where the hamiltonian of the system does not vary with time, the time-evolution operator has the form : u(t, t_0) = e^{-ih(t-t_0)/\hbar}, where the exponent is evaluated via its taylor series. the schrÃ¶dinger picture is useful when dealing with a time-independent hamiltonian h; that is, \partial_th=0 . ==background== in elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction Ï(x, t). more abstractly, the state may be represented as a state vector, or ket, | \psi \rangle. this ket is an element of a hilbert space, a vector space containing all possible states of the system. a quantum-mechanical operator is a function which takes a ket | \psi \rangle and returns some other ket | \psi' \rangle. the differences between the schrÃ¶dinger and heisenberg pictures of quantum mechanics revolve around how to deal with systems that evolve in time: the time-dependent nature of the system must be carried by some combination of the state vectors and the operators. for example, a quantum harmonic oscillator may be in a state | \psi \rangle for which the [[expectation value (quantum mechanics)|expe
Erwin_SchrAPdinger Foundational_quantum_physics Quantum_mechanics