Eliminating cyclic coordinates using Routh's procedure is presented.
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.
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