Dynamical pictures

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Quantum mechanics
${\displaystyle i\hbar {\frac {d}{dt}}|\Psi \rangle ={\hat {H}}|\Psi \rangle }$ Schrödinger equation
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In quantum mechanics, dynamical pictures (or representations) are the multiple equivalent ways to mathematically formulate the dynamics of a quantum system.

The two most important ones are the Heisenberg picture and the Schrödinger picture. These differ only by a basis change with respect to time-dependency, analogous to the Lagrangian and Eulerian specification of the flow field: in short, time dependence is attached to quantum states in the Schrödinger picture and to operators in the Heisenberg picture.

There is also an intermediate formulation known as the interaction picture (or Dirac picture) which is useful for doing computations when a complicated Hamiltonian has a natural decomposition into a simple "free" Hamiltonian and a perturbation.

Equations that apply in one picture do not necessarily hold in the others, because time-dependent unitary transformations relate operators in one picture to the analogous operators in the others. Not all textbooks and articles make explicit which picture each operator comes from, which can lead to confusion.