Schrödinger picture

In physics, the Schrödinger picture (also called the Schrödinger 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 Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. In physics, the Schrödinger picture (also called the Schrödinger 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 Schrödinger and Heisenberg pictures are related as active and passive transformations and commutation relations between operators are preserved in the passage between the two pictures. In the Schrödinger 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 | ψ ( t 0 ) ⟩ {displaystyle |psi (t_{0}) angle } at time t0 to a state vector | ψ ( t ) ⟩ {displaystyle |psi (t) angle } at time t, the time-evolution operator is commonly written U ( t , t 0 ) {displaystyle U(t,t_{0})} , and one has In the case where the Hamiltonian of the system does not vary with time, the time-evolution operator has the form where the exponent is evaluated via its Taylor series. The Schrödinger picture is useful when dealing with a time-independent Hamiltonian H; that is, ∂ t H = 0 {displaystyle partial _{t}H=0} . In elementary quantum mechanics, the state of a quantum-mechanical system is represented by a complex-valued wavefunction ψ(x, t). More abstractly, the state may be represented as a state vector, or ket, | ψ ⟩ {displaystyle |psi angle } . 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 | ψ ⟩ {displaystyle |psi angle } and returns some other ket | ψ ′ ⟩ {displaystyle |psi ' angle } . The differences between the Schrödinger 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 | ψ ⟩ {displaystyle |psi angle } for which the expectation value of the momentum, ⟨ ψ | p ^ | ψ ⟩ {displaystyle langle psi |{hat {p}}|psi angle } , oscillates sinusoidally in time. One can then ask whether this sinusoidal oscillation should be reflected in the state vector | ψ ⟩ {displaystyle |psi angle } , the momentum operator p ^ {displaystyle {hat {p}}} , or both. All three of these choices are valid; the first gives the Schrödinger picture, the second the Heisenberg picture, and the third the interaction picture. The time-evolution operator U(t, t0) is defined as the operator which acts on the ket at time t0 to produce the ket at some other time t: For bras, we instead have