Action principle is stated; Euler Lagrange EOM are obtained from the action principle.$\newcommand{\qbf}{\mathbf{q}}$
The equation \[i\hbar\frac{d U(t,t_0)}{dt} = H'_I(t) U(t,t_0).\] obeyed by the time evolution operator in the interaction picture is converted into an integral equation. A perturbative solution is obtained from the integral equation following a standard iterative procedure.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$
The eigenstates of Hamiltonian are called stationary states.In a stationary state all observable quantities are independent of time. The dynamical variables which commute with Hamiltonian are called constants of motion. The average values of constants of motion in any state do not change with time.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$ The interaction picture, also known as Dirac picture, or the intermediate picture, is defined by splitting the Hamiltonian in two parts, the free and the interaction parts. In interaction picture equation of motion for the observables is free particle equation. The state vector satisfies Schrodinger equation with interaction Hamiltonian giving the rate of time evolution.
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