$\newcommand{\Prime}{{^\prime}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\xbf}{{\mathbf x}}$
The Green function and the propagator for time dependent Schr\:{o}dinger equation are defined. The time dependent Schr\"{o}dinger equation is soled to obtain the solution for propagator.
$\newcommand{\DD}[2][]{\frac{d^2#1}{d#2^2}}\newcommand{\kbf}{{\mathbf k}}\newcommand{\xbf}{{\mathbf x}}\newcommand{\rbf}{{\mathbf r}}\newcommand{\Prime}{{^\prime}}$
The energy eigenvalues and eigenfunctions are obtained for a free particle in one dimension. Properties and delta function normalization are discussed. It is shown that the energy eigenvalue must be positive. The free particle solution in three dimension is briefly given.
$\newcommand{\xbf}{\mathbf x}\newcommand{\kbf}{\mathbf k}\newcommand{\rbf}{\mathbf r}$
The energy eigen functions of free particle are given. These are found to be eigen functions momentum also. The energy eigen functions have infinite degeneracy. There an eigen function corresponding to each momentum direction.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$
A particle with localized position is described by a wave packet in quantum mechanics. Taking a free Gaussian wave packet, its wave function at arbitrary time is computed. It is fund that the average value of position varies with time like position of a classical particle. The average vale of momentum remains constant and the uncertainty \(\Delta x\) increases with time.
RightClick (except on links): this draggable popup. Help
Ctrl+RightClick: default RightClick.
Ctrl+LeftClick: open a link in a new tab/window.
||Message]