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    <title>MaplePrimes - answers and comments on Question, How do I define commutators of operators which are functions in the Maple Physics package?</title>
    <link>http://www.mapleprimes.com/questions/120244-How-Do-I-Define-Commutators-Of-Operators</link>
    <language>en-us</language>
    <copyright>2026 Maplesoft, A Division of Waterloo Maple Inc.</copyright>
    <generator>Maplesoft Document System</generator>
    <lastBuildDate>Sat, 13 Jun 2026 18:28:33 GMT</lastBuildDate>
    <pubDate>Sat, 13 Jun 2026 18:28:33 GMT</pubDate>
    <itunes:subtitle />
    <itunes:summary />
    <description>The latest answers and comments added to the Question, How do I define commutators of operators which are functions in the Maple Physics package?</description>
    <image>
      <url>http://www.mapleprimes.com/images/mapleprimeswhite.jpg</url>
      <title>MaplePrimes - answers and comments on Question, How do I define commutators of operators which are functions in the Maple Physics package?</title>
      <link>http://www.mapleprimes.com/questions/120244-How-Do-I-Define-Commutators-Of-Operators</link>
    </image>
    <item>
      <title>How to do it with current or previous Physics</title>
      <link>http://www.mapleprimes.com/questions/120244-How-Do-I-Define-Commutators-Of-Operators?ref=Feed:MaplePrimes:How do I define commutators of operators which are functions in the Maple Physics package?:Comments#answer151459</link>
      <itunes:summary>&lt;p&gt;&lt;br&gt; &lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;I just saw your post ... answering two years after is probably irrelevant, but anyway: things have changed since 2011. Using the Physics library up-to-date (you can download it from the the &lt;/span&gt;&lt;a href="http://www.maplesoft.com/products/maple/features/physicsresearch.aspx"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;"Maple Physics: Research &amp;amp; Development"&lt;/span&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;updates page), current version:&lt;/span&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(1)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;Set some commutation rules to see how it works&lt;/span&gt;&lt;/p&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -29;" src="/view.aspx?sf=151459/468808/0d641d9822249631b520597b8b6c5e82.gif" alt="[algebrarules = {%Commutator(A, B) = 1, %Commutator(A, C) = g, %Commutator(B(x), C(y)) = f(x, y)}, quantumoperators = {A, B, C, f, g}]" width="546" height="46" align="middle"&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(2)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The first commutator has a constant in the rhs: in these cases you can now index the operators &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: italic;"&gt;A &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;and &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: italic;"&gt;B&lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;or give functionality to them as desired and the result will always be the right-hand side of &lt;/span&gt;&lt;!-- HelpHyperlink topic=type,constant --&gt; &lt;span style="color: #008080; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&lt;span style="text-decoration: underline;"&gt;type constant&lt;/span&gt;&lt;/span&gt; &lt;!-- /HelpHyperlink --&gt; &lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;:&lt;/span&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(3)&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(4)&lt;/td&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(5)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;In the second commutator, the right-hand side is not constant, so this one works as in previous releases: &lt;/span&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(6)&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(7)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/0fd01219d2b448479d620b5d3079c854.gif" alt="Commutator(A[i], C[j])" width="156" height="23"&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(8)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The third commutator has functionality in the left-hand and right-hand sides, the same one, so in this case you have more flexibility&lt;/span&gt;&lt;/p&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(9)&lt;/td&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
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&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(10)&lt;/td&gt;
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&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/af552ce26c79be7b68c681492d8dda32.gif" alt="proc (x, y) options operator, arrow; Physics:-`^`(x, y) end proc" width="103" height="27"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(11)&lt;/td&gt;
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&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/54abc1c72f7fc737b5a3cd28649d6c32.gif" alt="Commutator(B(u), C(w))" width="165" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/7a53583fcc84cf274d54a1b32e2283e4.gif" alt="u^w" width="21" height="27"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(12)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/b540c8053c18c0bac3010c8f786edad2.gif" alt="f := proc (x, y) options operator, arrow; 0 end proc" width="96" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/6011b459190eb3fe235f3b6799d6ef0f.gif" alt="proc (x, y) options operator, arrow; 0 end proc" width="99" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(13)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/90c51c53ac962b2c6590dc392c733ab2.gif" alt="Commutator(B(a), C(b))" width="162" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/ace3afda8710367e2ded6d423daeb33a.gif" alt="0" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(14)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The Physics update is developed assuming the Maple 17 library, but the last approach, with &lt;/span&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/623592c9bd32f778481eb08b3e90f6ae.gif" alt="f(x, y)" width="45" height="23"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;in the right-hand side of the commutator followed by defining f as a procedure, will also work in previous Maples.&lt;/span&gt;&lt;/p&gt;
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&lt;input type="hidden" name="sequence" value="1"&gt; &lt;input type="hidden" name="cmd" value="none"&gt;&lt;/form&gt;
&lt;p&gt;&lt;br&gt; &lt;/p&gt;
&lt;p&gt;&lt;a href="/view.aspx?sf=151459/468808/MaplePrimesCommutato.mw"&gt;Download MaplePrimesCommutato.mw&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;Edgardo S. Cheb-Terrab&amp;nbsp;&lt;br&gt;Physics, Maplesoft&lt;/p&gt;</itunes:summary>
      <description>&lt;p&gt;&lt;br&gt; &lt;/p&gt;
&lt;form name="worksheet_form"&gt;
&lt;table style="width: 576px;" align="center"&gt;
&lt;tbody&gt;
&lt;tr&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/920524f4dc24a9fa7bfe3ddc26ad1a27.gif" alt="Hi*ml76" width="54" height="23"&gt;&lt;/p&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;I just saw your post ... answering two years after is probably irrelevant, but anyway: things have changed since 2011. Using the Physics library up-to-date (you can download it from the the &lt;/span&gt;&lt;a href="http://www.maplesoft.com/products/maple/features/physicsresearch.aspx"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;"Maple Physics: Research &amp;amp; Development"&lt;/span&gt;&lt;/span&gt;&lt;/a&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;updates page), current version:&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/8546eaf8bf6d539e8eeea7a08c966cdd.gif" alt="Physics:-Version()" width="122" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/de2745a4ae25829d9a40c79469cfb17e.gif" alt="&amp;quot;/Users/ecterrab/Maple/lib/Physics.mla&amp;quot;, `2013, September 6, 13:30 hours`" width="446" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(1)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;Set some commutation rules to see how it works&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/8cd3b6c86ed25b4dd3a43a0a69843ffb.gif" alt="restart; with(Physics)" width="149" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
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&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
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&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -23;" src="/view.aspx?sf=151459/468808/b10570907ac8364155519b8ab07305f5.gif" alt="Setup(op = {A, B, C, f, g}, %Commutator(A, B) = 1, %Commutator(A, C) = g, %Commutator(B(x), C(y)) = f(x, y))" width="576" height="40" align="middle"&gt;&lt;/p&gt;
&lt;/td&gt;
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&lt;/table&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/cb6fc23c9dc57e5024f258b65a58118a.gif" alt="`* Partial match of  'op' against keyword 'quantumoperators'`" width="367" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;&amp;nbsp;&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
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&lt;table&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -29;" src="/view.aspx?sf=151459/468808/0d641d9822249631b520597b8b6c5e82.gif" alt="[algebrarules = {%Commutator(A, B) = 1, %Commutator(A, C) = g, %Commutator(B(x), C(y)) = f(x, y)}, quantumoperators = {A, B, C, f, g}]" width="546" height="46" align="middle"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(2)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The first commutator has a constant in the rhs: in these cases you can now index the operators &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: italic;"&gt;A &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;and &lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: italic;"&gt;B&lt;/span&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;or give functionality to them as desired and the result will always be the right-hand side of &lt;/span&gt;&lt;!-- HelpHyperlink topic=type,constant --&gt; &lt;span style="color: #008080; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&lt;span style="text-decoration: underline;"&gt;type constant&lt;/span&gt;&lt;/span&gt; &lt;!-- /HelpHyperlink --&gt; &lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;:&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/cd99f74006aea4bdf7fcb60e5daad2e3.gif" alt="Commutator(A, B)" width="119" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
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&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/5b5cdb5a78528021a201c1d3b2e9e5dc.gif" alt="1" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(3)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/6ad05fdef3a69544d7475b90dc0b335c.gif" alt="Commutator(A(x), B(y))" width="160" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/d5463df620dfdfb7d3b60ca90094bf77.gif" alt="1" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(4)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/54e46ae008bcf4a363e72cd8d07a408c.gif" alt="Commutator(A[i](x), B[j](y))" width="196" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/26675935346465b9f43ef89b8e305ad7.gif" alt="1" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(5)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;In the second commutator, the right-hand side is not constant, so this one works as in previous releases: &lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/8e88f69b407c8db6d6b474d0002a0976.gif" alt="Commutator(A, C)" width="120" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/78d10080a554eb5d116722a926ee0afe.gif" alt="g" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(6)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/e2eecb579779d857a6db6d962e55ba4f.gif" alt="Commutator(A(x), C(y))" width="161" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -9;" src="/view.aspx?sf=151459/468808/713eed7c6b051354f11ab59436fcae0a.gif" alt="Physics:-Commutator(A(x), C(y))" width="96" height="26"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(7)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/0fd01219d2b448479d620b5d3079c854.gif" alt="Commutator(A[i], C[j])" width="156" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -14;" src="/view.aspx?sf=151459/468808/7aad98e4a15b2eba5f21a2c5c6326c9a.gif" alt="Physics:-Commutator(A[i], C[j])" width="62" height="31"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(8)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The third commutator has functionality in the left-hand and right-hand sides, the same one, so in this case you have more flexibility&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/e215d8f21f1e60e0d0dada15dec7fa35.gif" alt="Commutator(B(x), C(y))" width="161" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/391185df251fb34e5c849c0e7052d11d.gif" alt="f(x, y)" width="45" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(9)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/438baeaef40d88cb4912c3a59cc3bb92.gif" alt="Commutator(B(s), C(t))" width="157" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/c0c371cbc50b07938a8a9dd378877fb9.gif" alt="f(s, t)" width="42" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(10)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;You can also transform f into a procedure&lt;/span&gt;&lt;/p&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/bf7f45996c589e90a2a8c4a5c45708ca.gif" alt="f := proc (x, y) options operator, arrow; x^y end proc" width="100" height="27"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/af552ce26c79be7b68c681492d8dda32.gif" alt="proc (x, y) options operator, arrow; Physics:-`^`(x, y) end proc" width="103" height="27"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(11)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/54abc1c72f7fc737b5a3cd28649d6c32.gif" alt="Commutator(B(u), C(w))" width="165" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/7a53583fcc84cf274d54a1b32e2283e4.gif" alt="u^w" width="21" height="27"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(12)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/b540c8053c18c0bac3010c8f786edad2.gif" alt="f := proc (x, y) options operator, arrow; 0 end proc" width="96" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/6011b459190eb3fe235f3b6799d6ef0f.gif" alt="proc (x, y) options operator, arrow; 0 end proc" width="99" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(13)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table style="margin-left: 0px; margin-right: 0px;"&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;&lt;span style="color: #ff0000; font-size: 100%; font-family: Courier New,monospace; font-weight: bold; font-style: normal;"&gt; &amp;gt;&amp;nbsp;&lt;/span&gt;&lt;/td&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/90c51c53ac962b2c6590dc392c733ab2.gif" alt="Commutator(B(a), C(b))" width="162" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;table&gt;
&lt;tbody&gt;
&lt;tr valign="baseline"&gt;
&lt;td&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="center"&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/ace3afda8710367e2ded6d423daeb33a.gif" alt="0" width="13" height="23"&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;td style="color: #000000; font-family: Times, serif; font-weight: bold; font-style: normal;" align="right"&gt;(14)&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;p style="margin: 0 0 0 0; padding-top: 0px; padding-bottom: 0px;" align="left"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;The Physics update is developed assuming the Maple 17 library, but the last approach, with &lt;/span&gt;&lt;img style="vertical-align: -6;" src="/view.aspx?sf=151459/468808/623592c9bd32f778481eb08b3e90f6ae.gif" alt="f(x, y)" width="45" height="23"&gt;&lt;span style="color: #000000; font-size: 100%; font-family: Times New Roman,serif; font-weight: normal; font-style: normal;"&gt;&amp;nbsp;in the right-hand side of the commutator followed by defining f as a procedure, will also work in previous Maples.&lt;/span&gt;&lt;/p&gt;
&lt;/td&gt;
&lt;/tr&gt;
&lt;/tbody&gt;
&lt;/table&gt;
&lt;input type="hidden" name="sequence" value="1"&gt; &lt;input type="hidden" name="cmd" value="none"&gt;&lt;/form&gt;
&lt;p&gt;&lt;br&gt; &lt;/p&gt;
&lt;p&gt;&lt;a href="/view.aspx?sf=151459/468808/MaplePrimesCommutato.mw"&gt;Download MaplePrimesCommutato.mw&lt;/a&gt;&lt;/p&gt;
&lt;p&gt;Edgardo S. Cheb-Terrab&amp;nbsp;&lt;br&gt;Physics, Maplesoft&lt;/p&gt;</description>
      <guid>151459</guid>
      <pubDate>Sat, 07 Sep 2013 02:54:26 Z</pubDate>
      <itunes:author>ecterrab</itunes:author>
      <author>ecterrab</author>
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