The energy eigenvalue problem for a particle in a square well is solved. The energy eigenvalues are solutions of a transcendental equation which can be solved graphically. $\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$
For a potential problem in one dimension there are three types of energy levels. These are (a) discrete, (b) continuous doubly degenerate energy eigenvalues, and (c) continuous and non degenerate. In this talk we explain the thumb rules to find out which of this cases apply for a given potential and a specified energy value.
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.
1. Optics Mechanics Analogy --- Road to Wave Mechanics 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..
2. Time Dependent Schrodinger Equation in Coordinate Representation
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.
The set of all rotations that can be implemented physically in three dimensions form a group. These most important and frequently used rotations are the rotations which can be implemented physically. These rotations, do not change the handedness of the coordinate axes and are called proper rotations. The improper rotations take left handed systems to right handed systems, or vice versa. The statement of Euler's theorem about rotations is given. $\newcommand{\Prime}{{^\prime}}$
Two methods of obtaining the differential equation of the orbit, in a sphericall symmetric potential, are given using the Euler Lagrange equations and conservation law.
The equation for the orbit involves two constants of integration. We determine these constants and obtain an expression for the eccentricity in terms of energy angular momentum etc.. Conditions on energy for different types of possible orbits , elliptic, parabolic and hyperbolic, are written down.
||Message]
