$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\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}}$$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
That assumption that the superposition principle be preserved under time evolution leads to unitary nature of the them evolution operator. The state vector satisfies differential equation, the Schrodinger equation, with Hamiltonian as the generator of time evolution.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$$\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}$
Assuming time development of states to be given by \[i\hbar \dd[\ket{\psi, t}]{t} = H \ket{\psi t}, \] an equation for time variation of average value of a dynamical variable is derived. Classical correspondence is used to identify the generator of time evolution with Hamiltonian. A dynamical variable not depending explicitly on time is a constant of motion if it commutes with the Hamiltonian.
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.
The time evolution of states a quantum system is given by the time dependent Schrodinger equation. Besides this framework, called the Schr\"{o}dinger picture, other scheme are possible. In the Heisenberg picture, defined here, the observable evolve according to the equation \[\dd[X]{t} =\frac{1}{i\hbar}[F, H] \]This equation corresponds to the classical equation of motion in the Poisson bracket formalism.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\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}$
Main points of time evolution in Schrodinger picture are summarized.
The time evolution of a general quantum system is reviewed in an abstract setting. The eigen states of energy are seen to have all properties that make them qualify for being called stationary states.The stationary states have the property that all observable quantities remain constant in 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}$
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$ $\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$ $\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$ $\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
A scheme to solve the time dependent Schr\"{o}dinger equation \begin{equation} \label{eq01} i\hbar \dd{t}\ket{\psi} = \hat{H} \ket{\psi} \end{equation} is described. The final solution will be presented in the form, see \eqref{eq14} \begin{equation} \ket{\psi t} = U(t, t_0) \ket{\psi t_0} \label{eq16} \end{equation}where\begin{equation}\label{EQ16A} U(t, t_0) \ket{\psi t_0} = \exp\Big(\frac{-i H(t-t_0)}{\hbar}\Big)\end{equation}
RightClick (except on links): this draggable popup. Help
Ctrl+RightClick: default RightClick.
Ctrl+LeftClick: open a link in a new tab/window.
||Message]