Browse & Filter

$\newcommand{\xbf}{\mathbf x}\newcommand{\kbf}{\mathbf k}\newcommand{\rbf}{\mathbf r}$

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.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} \newcommand{\average}[2]{\langle#1|#2|#1\rangle}$                  
Making use of analogy between optics ad mechanics we motivate the introduction of the Schrodinger equation.
Here Fermi's "Lectures on Quantum Mechanics" has been followed very closely..

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$

Time reversal operation in quantum mechanics of one particle is discussed.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle} \newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}} \newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}} newcommand{\average}[2]{\langle#1|#2|#1\rangle}$

For conservative systems, we show how solution of time dependent Schrodinger equation can be found by separation of variables. Explicit expression for the wave function at arbitrary time \(t\) is obtained in terms of energy eigenfunctions and eigenvalues. 

  $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$

The momentum representation is defined and its  connection with  the coordinate  representations is discussed. The transformation bewteen the two is effected by \(\innerproduct{x}{p}\) which are just the momentum eigenfunctions in the coordinate representation. Delta function normalization and the box normalization is discussed for the momentum eigenfunctions.

 $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}} \newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}} \newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$

The choice of orthonormal basis of eigenvectors of position operator gives rise to the coordinate representation. The wave function, being the expansion coefficient of state vector in this basis, gives the probability amplitude for  outcomes of position measurements.In the coordinate representation the momentum  operator assumes a simple form $\widehat{p} =-i\hbar \dd{x}$.

$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ 
$\newcommand{\matrixelement}[3]{\langle#1|#2|#3\rangle}$
$\newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}$
$\newcommand{\dd}[2][]{\frac{d#1}{d#2}}$
$\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}\newcommand{\average}[2]{\langle#1|#2|#1\rangle}$

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.

 $\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}\newcommand{\matrixelement}[3] \langle#1|#2|#3\rangle} \newcommand{\PP}[2][]{\frac{\partial^2 #1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d#2}}\newcommand{\pp}[2][]{\frac{\partial #1}{\partial #2}}$
A tabular comparison of  coordinate and momentum representations is presented.