%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: quantum Hamiltonian operator
%%% Primary Category Code: 03.
%%% Filename: QuantumHamiltonianOperator.tex
%%% Version: 6
%%% Owner: bci1
%%% Author(s): bci1
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}
\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}
\usepackage{html}
%fancy typeface/symbols
%\usepackage{beton}
%\usepackage{concmath}
%\usepackage{stmaryrd}
%margins
%\usepackage{simplemargins}
%\usepackage{multicol}
%\setleftmargin{1in}
%\setrightmargin{1in}
%\settopmargin{1in}
%\setbottommargin{1in}
% symbols
\usepackage{amssymb}
\usepackage{amsfonts}
\usepackage[mathscr]{euscript}
\usepackage[T1]{fontenc}
% graphics
%\usepackage{pstricks}
% math
\usepackage{amsmath}
\usepackage{amsopn}
\usepackage{amstext}
\usepackage{amsthm}
% only one theoremstyle
%\theoremstyle{definition}
%\newtheorem{exercise}{}[subsection]
% math operators
\DeclareMathOperator{\pipe}{{\big |} \hspace{-2.85pt} {\big |}}
\DeclareMathOperator{\et}{\&}
% misc math stuff
\newcommand{\vol}{\mathrm{vol}}
\newcommand{\id}{\mathrm{id}}
\newcommand{\domain}{\mathrm{domain}}
\newcommand{\supp}{\mathrm{supp}}
\newcommand{\diam}{\mathrm{diameter}}
\newcommand{\incl}{\mathrm{incl}}
\newcommand{\interior}{\mathrm{interior}}
\newcommand{\triv}{\mathrm{triv}}
\newcommand{\image}{\mathrm{image}}
\newcommand{\closure}{\mathrm{closure}}
\newcommand{\degree}{\mathrm{degree}}
\newcommand{\im}{\mathrm{im}}
\newcommand{\esssup}{\mathrm{ess\ sup}}
\newcommand{\openset}{\mathrel{{\mathchoice{\rlap{$\subset$}{\;\circ}}%
{\rlap{$\subset$}{\;\circ}}%
{\rlap{$\scriptstyle\subset$}{\;\circ}}%
{\rlap{$\scriptscriptstyle\subset$}{\;\circ}}}}}
% foreignisms
\newcommand{\ie}{\emph{i.e.},}
\newcommand{\Ie}{\emph{I.e.},}
\newcommand{\cf}{\emph{cf.}}
\newcommand{\eg}{\emph{e.g.},}
\newcommand{\Eg}{\emph{E.g.},}
\newcommand{\ala}{\emph{\'a la}}
% labels for spaces
\newcommand{\N}{\mathbf{N}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\Q}{\mathbf{Q}}
\newcommand{\C}{\mathbf{C}}
\newcommand{\CP}{\mathbf{CP}}
\newcommand{\RP}{\mathbf{RP}}
\newcommand{\R}{\mathbf{R}}
\newcommand{\Rtw}{\mathbf{R}^2}
\newcommand{\Rth}{\mathbf{R}^3}
\newcommand{\Rfo}{\mathbf{R}^4}
\newcommand{\Rfi}{\mathbf{R}^5}
\newcommand{\Rp}{\mathbf{R}^p}
\newcommand{\Rn}{\mathbf{R}^n}
\newcommand{\Rnmon}{\mathbf{R}^{n-1}}
\newcommand{\Rnpon}{\mathbf{R}^{n+1}}
\newcommand{\Rmpon}{\mathbf{R}^{m+1}}
\newcommand{\RNpon}{\mathbf{R}^{N+1}}
\newcommand{\Rtwn}{\mathbf{R}^{2n}}
\newcommand{\RN}{\mathbf{R}^N}
\newcommand{\Rm}{\mathbf{R}^m}
\newcommand{\Hon}{\mathbf{H}^1}
\newcommand{\Htw}{\mathbf{H}^2}
\newcommand{\Hth}{\mathbf{H}^3}
\newcommand{\Hn}{\mathbf{H}^n}
\newcommand{\Torus}{\mathbf{T}}
\newcommand{\Ttw}{\mathbf{T}^2}
\newcommand{\Tth}{\mathbf{T}^3}
\newcommand{\Tn}{\mathbf{T}^n}
\newcommand{\Son}{\mathbf{S}^1}
\newcommand{\Stw}{\mathbf{S}^2}
\newcommand{\Sth}{\mathbf{S}^3}
\newcommand{\Sfo}{\mathbf{S}^4}
\newcommand{\Sfi}{\mathbf{S}^5}
\newcommand{\SN}{\mathbf{S}^N}
\newcommand{\Sn}{\mathbf{S}^n}
\newcommand{\Sm}{\mathbf{S}^m}
\newcommand{\Smmon}{\mathbf{S}^{m-1}}
\newcommand{\Snmon}{\mathbf{S}^{n-1}}
\newcommand{\Snmtw}{\mathbf{S}^{n-2}}
\newcommand{\Hnmon}{\mathbf{H}^{n-1}}
\newcommand{\Ton}{\mathbf{T}^1}
\newcommand{\T}{\mathbf{T}}
% differential operators
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}}
\newcommand{\pkd}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}}
\newcommand{\dkd}[3]{\frac{d^{#3} #1}{d#2^{#3}}}
\newcommand{\td}[2]{\frac{d #1}{d #2}}
\newcommand{\pdat}[3]{\left . \frac{\partial #1}{\partial #2}\right|_{#3}}
\newcommand{\dbyd}[1]{ \frac{d}{d #1}}
% Fri Oct 3 11:00:53 2003 -- should check at some point to see whether the replaced versions are actually used at all, I don't think so
\newcommand{\ddk}[3]{\frac{d^{#3} #1}{d#2^{#3}}}
% replaces...
\newcommand{\dbydk}[2]{ \frac{d^{#2}}{d #1^{#2}}}
\newcommand{\ppk}[3]{\frac{\partial^{#3} #1}{\partial #2^{#3}}}
% replaces...
\newcommand{\pbypk}[2]{ \frac{\partial^{#2}}{\partial #1^{#2}}}
\newcommand{\pp}[2]{\frac{\partial #1}{\partial #2}}
% replaces...
\newcommand{\pbyp}[1]{ \frac{\partial}{\partial #1}}
\newcommand{\ddat}[3]{\left .\frac{d #1}{d #2}\right|_{#3}}
% replaces...
\newcommand{\dbydat}[2]{\left . \frac{d}{d #1} \right|_{#2}}
\newcommand{\ppkat}[4]{\left .\frac{\partial^{#3} #1}{\partial #2^{#3}}\right|_{#4}}
\newcommand{\ddkat}[4]{\left .\frac{d^{#3} #1}{d#2^{#3}}\right|_{#4}}
% counters for new lists
%\newcounter{alistctr}
%\newcounter{Alistctr}
\newcounter{rlistctr}
\newcounter{Rlistctr}
\newcounter{123listctr}
\newcounter{123listcolonstylectr}
% a,b,c - small latin letter list
\newenvironment{alist}{
\indent \begin{list}{(\alph{alistctr})}{\usecounter{alistctr}}}
{\end{list}\setcounter{alistctr}{0}}
% A,B,C - LARGE LATIN LETTER LIST
\newenvironment{Alist}{
\indent \begin{list}{(\Alph{Alistctr})}{\usecounter{Alistctr}}
}
{\end{list}\setcounter{Alistctr}{0}}
% i,ii,iii - small roman numeral list
\newenvironment{rlist}{
\indent \begin{list}{(\roman{rlistctr})}{\usecounter{rlistctr}}
}
{\end{list}\setcounter{rlistctr}{0}}
% I,II,III - large roman numeral list
\newenvironment{Rlist}{
\indent \begin{list}{(\Roman{Rlistctr})}{\usecounter{Rlistctr}}
}
{\end{list}\setcounter{Rlistctr}{0}}
%1,2,3 - arabic numeral list
\newenvironment{123list}{
\indent \begin{list}{(\arabic{123listctr})}{\usecounter{123listctr}}
}
{\end{list}\setcounter{123listctr}{0}}
%1:,2:,3: - arabic numeral list with colon decoration
\newenvironment{123listcolonstyle}{\indent \begin{list}{\arabic{123listcolonstylectr}:}{\usecounter{123listcolonstylectr}}}
{\end{list}\setcounter{123listcolonstylectr}{0}}
% environment for definitions
\def\defn#1{\addcontentsline{toc}{subsection}{$\ast$} {\footnotesize \noindent \begin{123listcolonstyle} \setlength{\itemsep}{0em} \setlength{\topsep}{0em} \setlength{\parsep}{0em} #1 \end{123listcolonstyle}}}
%environment for proofs
\def\proof#1{\par {\footnotesize \indent \begin{tabular}{ll} #1 \end{tabular}}}
%% \include{packages}
%% \include{renewed_commands}
%% \include{simple_theorems}
%% \include{abbreviations}
%% \include{spaces}
%% \include{new_environments}
%% \include{margins}
%% \include{mathoperators}
%% \include{differentiation}
%% \include{limited_things}
%% \include{argawarga}
\newcommand{\Rset}{\mathbb{R}}
\begin{document}
\textbf{Definition 0.1}
The Hamiltonian operator \textbf{H} introduced in \htmladdnormallink{quantum mechanics}{http://planetphysics.us/encyclopedia/QuantumParadox.html} by Schr\"odinger (and thus sometimes also called the
\emph{Schr\"odinger operator}) on the \htmladdnormallink{Hilbert space}{http://planetphysics.us/encyclopedia/NormInducedByInnerProduct.html} $L^2(\Rset^n)$ is given by the action:
\[
\psi \mapsto [-\nabla^2 +V(x)]\psi, \quad\psi\in L^2(\Rset^n),
\]
The \htmladdnormallink{operator}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} defined above $[-\nabla^2 +V(x)]$ , for a potential \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $V(x)$ specified as the real-valued function $V\colon \Rset^n \to \Rset$ is called the {\em Hamiltonian operator}, \textbf{H}, and only very rarely the {\em Schr\"odinger operator}. The \htmladdnormallink{energy}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} conservation (quantum) law written with the operator \textbf{H} as the
Schr\"odinger equation is fundamental in quantum mechanics and is perhaps the most utilized, mathematical \htmladdnormallink{computation}{http://planetphysics.us/encyclopedia/LQG2.html} device in quantum mechanics of \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} with a finite number of degrees of freedom. There is also, however, the alternative approach in the Heisenberg picture, or formulation, in which the \htmladdnormallink{observable}{http://planetphysics.us/encyclopedia/QuantumSpinNetworkFunctor2.html} and other \htmladdnormallink{operators}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra4.html} are time-dependent whereas the state vectors $\psi$ are time-independent, which reverses the time dependences betwen operators and state vectors from the more popular Schr\"odinger formulation. Other formulations of \htmladdnormallink{quantum theories}{http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} occur in
\htmladdnormallink{quantum field theories}{http://planetphysics.us/encyclopedia/SpaceTimeQuantizationInQuantumGravityTheories.html} (\htmladdnormallink{QFT}{http://planetphysics.us/encyclopedia/HotFusion.html}), such as \htmladdnormallink{QED}{http://planetphysics.us/encyclopedia/HotFusion.html} (\htmladdnormallink{quantum electrodynamics}{http://planetphysics.us/encyclopedia/QED.html}) and \htmladdnormallink{QCD}{http://planetphysics.us/encyclopedia/ExtendedQuantumSymmetries.html} (quantum chromodynamics).
Although the two formulations, or pictures, are unitarily (or mathematically) equivalent, however, sometimes the claim is made that the Heisenberg picture is ``more natural and fundamental than the Schr\"odinger'' formulation because the Lorentz invariance from \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} is also encountered in the Heisenberg picture,
and also because there is a `correspondence' between the \htmladdnormallink{commutator}{http://planetphysics.us/encyclopedia/Commutator.html} of an observable operator with the Hamiltonian operator, and the Poisson bracket formulation of \htmladdnormallink{classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html}. If the state vector $\psi$, or
$\left| \psi \right\rangle$ does not change with time as in the Heisenberg picture, then the `equation of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}' of a (quantum) observable operator is :
\[
\frac{d}{dt} A_{quantum} = (i\hbar)^{-1}[A,H] + \left(\frac {\partial A}{\partial t}\right)_{classical}
\]
\end{document}
