# 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}}}$

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