Content page

[NOTES/QM-11003] Schrodinger Equation for a Charged Particle

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

Using the classical Hamiltonian and the correspondence rule $\vec p \to -i\hbar \nabla$, the expression for the Hamiltonian operator for a charged particle is written giving the time dependent Schr\"{o}dinger equation. The Schrodinger equation retains its form under gauge transformations if the wave function is assumed to transform as
\[ \psi(\vec{r},t) \to \psi^\prime (\vec{r},t) =
e^{-i(q/c)\Lambda(\vec{r},t)}\psi(\vec{r},t). \]

[NOTES/QM-11002] Probability Conservation

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

Starting from the time dependent Schr\"{o}dinger equation, an equation of continuity \[{\partial\rho\over\partial t} + \vec{\nabla}.\vec{j}=0\] is derived. Physical interpretation of the continuity equation is given in analogy with charge conservation in electromagnetic theory. The equation of continuity represents conservation of probability in quantum mechanics.

[NOTES/QM-11001] Time Dependent Schrodinger Equation :Solution for Wave function at time $t$

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

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.

[NOTES/QM-10004] Momentum Representation

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

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.

[NOTES/QM-10003] A Summary of Coordinate and Momentum Representation

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

[NOTES/QM-10002] Coordinate Representation

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

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 $\widehat{p} =-i\hbar \dd{x}$.

[NOTES/QM-10001] Representations in an Inner Product Space

A brief account of representations in a finite dimensional vector spaces is presented. The use of an ortho norrnal basis along with Dirac notation makes all frequently used formula very intuitive. The formulas for representing a vector by a column vector and an operator by matrices are given. The results for change of o.n. bases are summarized.

[NOTES/QM-09008] Perturbation Expansion in Interaction picture

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.

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

[NOTES/QM-09007] Interaction Picture of Quantum Mechanics

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

[NOTES/QM-09004] Stationary States and Constants of Motion