Browse & Filter

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.

Remember: We are using MathJax for mathematical expressions. Text formatting commands of TeX/LaTeX are not supported in MathJax.

Mathematical expressions can be inserted within HTML text of all content types (e.g., page, book page, blog, forum) edited using the available online WYSIWYG HTML editor. The mathematical expressions must be typed using the syntax of TeX/LaTeX.

The easiest way to start is to author a page with some LaTeX code for mathematical expressions given here (based on the code snippets from an online cookbook).

[For a more comprehensive documentation of the TeX commands supported in MathJax, see this document.]

An overview of three methods to compute the energies  and eigenfunctions of an attractive Delta function potential are given.

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

The 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.

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

We derive the energy eigenvalues and eigenfunctions of a particle in a box of size \(L\).

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.

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$