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A discussion of nature of energy eigenvalues and eigenfunctions are discussed for general potentials in one dimension. General conditions when to expect the energy levels to be degenerate, continuous or form bands are given. Also the behaviour of eigenfunctions under parity and for also for large distances etc. are discussed.
An overview of three methods to compute the energies and eigenfunctions of an attractive Delta function potential are given.
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The energy eigenfunctions and eigenvalues for a particle in delta function potential are derived. It is found that, for an attractive delta function potential there is only one bound state.
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We derive the energy eigenvalues and eigenfunctions of a particle in a box of size \(L\).
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We derive the boundary condition on a rigid wall as a limit of the boundary condition on a the wave function at a point where the potential has a finite jump discontinuity. It is shown that there is no restriction on the derivative of the energy eigenfunction. The only boundary condition is that the eigenfunction must vanish.
The reflection and transmission coefficients for a potential problem in one dimension are defined. For this purpose it is sufficient to know the behavior of the wave function at large distances. To set up the problem one needs to impose a suitable boundary condition on the wave function at large distances. It is shown that, for real potentials, the probability conservation implies that the reflection and transmission coefficients add to unity.
$\newcommand{\Label}[1]{\label{#1}}\newcommand{\Prime}{{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$
$\newcommand{\Label}[1]{\label{#1}}\newcommand{\Prime}{^\prime}}\newcommand{\pp}[2][]{\frac{\partial#1}{\partial #2}}\newcommand{\PP}[2][]{\frac{\partial^2#1}{\partial #2^2}}\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$The steps for obtaining energy eigenvalues and eigenfunctions are given for a harmonic oscillator. The details can be found in most text books, e.g. Schiff,"Quantum Mechanics"
S- matrix is defined for a particle incident on a potential in one dimension. The transformation properties of the S-matrix under time reversal and parity are given.
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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.$\newcommand{\dd}[2][]{\frac{d#1}{d #2}} \newcommand{\DD}[2][]{\frac{d^2#1}{d #2^2}}$
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