Schrodinger picture

The Schrödinger picture of quantum mechanics is a model for dealing with time-dependent problems. The following assumptions are made:

1. States are generally time-dependent :
   ${\displaystyle \vert \psi ,t\rangle _{S}=\vert \psi (t)\rangle }$
2. Operators can at most explicitly depend on time: the only exception is the time expansion operator .
   ${\displaystyle {\frac {d{\hat {A}}_{S}}{dt}}={\frac {\partial {\hat {A}}_{S}}{\partial t}}}$
   
3. The behavior of the system ( dynamics ) is described by the Schrodinger equation : , wherein the Hamiltonian is the system.
   ${\displaystyle i\hbar {\frac {d}{dt}}\vert \psi {,}t\rangle _{S}={\hat {H}}_{S}\vert \psi {,}t\rangle _{S}}$
   ${\displaystyle {\hat {H}}_{S}}$

To indicate that you are in the Schrödinger picture, states and operators are occasionally given the index "S": or ${\displaystyle |\psi (t)\rangle _{S}}$ ${\displaystyle {\hat {A}}_{S}}$

The time-dependent state is given by the state at a fixed point in time and the unitary time evolution operator : ${\displaystyle |\psi (t)\rangle _{S}}$ ${\displaystyle |\psi (t_{0})\rangle _{S}}$ ${\displaystyle t_{0}}$ ${\displaystyle {\hat {U}}(t,t_{0})}$

${\displaystyle |\psi (t)\rangle _{S}={\hat {U}}(t,t_{0})|\psi (t_{0})\rangle _{S}}$

Two other models are the Heisenberg picture and the interaction picture . All models lead to the same expected values . For the expected value of the operator we get in the Schrödinger picture: ${\displaystyle a}$ ${\displaystyle {\hat {A}}}$

${\displaystyle a=\langle \psi (t)|{\hat {A}}_{S}|\psi (t)\rangle _{S}}$