Show that conformal transformations consisting of dilations \[x^\mu \to x^\mu = e^{-\rho} x^\mu \] and special conformal transformations (SCT) \[x^\mu \to x^{\prime\,\mu}= \frac{x^\mu + c^\mu x^2}{ 1 + 2c \cdot x + c^2 x^2},\] and usual Poincaré transformations form a group. Find the commutation relations for the generators of infinitesimal transformations of this group.
$\newcommand{\DD}[2][]{\frac{d^2 #1}{d^2 #2}}$ $\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}$
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.
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.
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