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  <title>Less Than, Greater Than, Bars and Quantum Mechanics</title>
  <section inlist="toc" xml:id="S1">
    <tags>
      <tag>1</tag>
      <tag role="refnum">1</tag>
      <tag role="typerefnum">§1</tag>
    </tags>
    <title><tag close=" ">1</tag>Relations</title>
    <para xml:id="S1.p1">
      <p>Relations: <Math mode="inline" tex="a&lt;b&lt;c" text="a &lt; b &lt; c" xml:id="S1.p1.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="multirelation"/>
              <XMTok font="italic" role="UNKNOWN">a</XMTok>
              <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
              <XMTok font="italic" role="UNKNOWN">b</XMTok>
              <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
              <XMTok font="italic" role="UNKNOWN">c</XMTok>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S1.p2">
      <p>Relations: <Math mode="inline" tex="&gt;a" text="absent &gt; a" xml:id="S1.p2.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
              <XMTok meaning="absent"/>
              <XMTok font="italic" role="UNKNOWN">a</XMTok>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S1.p3">
      <p>Relations: <Math mode="inline" tex="a&gt;" text="a &gt; absent" xml:id="S1.p3.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="greater-than" role="RELOP">&gt;</XMTok>
              <XMTok font="italic" role="UNKNOWN">a</XMTok>
              <XMTok meaning="absent"/>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
  </section>
  <section inlist="toc" xml:id="S2">
    <tags>
      <tag>2</tag>
      <tag role="refnum">2</tag>
      <tag role="typerefnum">§2</tag>
    </tags>
    <title><tag close=" ">2</tag>Quantum Mechanics Notations</title>
    <para xml:id="S2.p1">
      <p>Expectation: <Math mode="inline" tex="&lt;a&gt;" text="expectation@(a)" xml:id="S2.p1.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="expectation"/>
                <XMRef idref="S2.p1.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p1.m1.1">a</XMTok>
                <XMTok meaning="greater-than" role="CLOSE">&gt;</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p2">
      <p>Bra: <Math mode="inline" tex="&lt;a|" text="bra@(a)" xml:id="S2.p2.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="bra"/>
                <XMRef idref="S2.p2.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p2.m1.1">a</XMTok>
                <XMTok role="CLOSE" stretchy="false">|</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p3">
      <p>Bra: <Math mode="inline" tex="&lt;a,b,c|" text="bra@(list@(a, b, c))" xml:id="S2.p3.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="bra"/>
                <XMRef idref="S2.p3.m1.4"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMDual xml:id="S2.p3.m1.4">
                  <XMApp>
                    <XMTok meaning="list"/>
                    <XMRef idref="S2.p3.m1.1"/>
                    <XMRef idref="S2.p3.m1.2"/>
                    <XMRef idref="S2.p3.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S2.p3.m1.1">a</XMTok>
                    <XMTok role="PUNCT">,</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S2.p3.m1.2">b</XMTok>
                    <XMTok role="PUNCT">,</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S2.p3.m1.3">c</XMTok>
                  </XMWrap>
                </XMDual>
                <XMTok role="CLOSE" stretchy="false">|</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p4">
      <p>Bra: <Math mode="inline" tex="&lt;\phi=0|" text="bra@(phi = 0)" xml:id="S2.p4.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="bra"/>
                <XMRef idref="S2.p4.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMApp xml:id="S2.p4.m1.1">
                  <XMTok meaning="equals" role="RELOP">=</XMTok>
                  <XMTok font="italic" name="phi" role="UNKNOWN">ϕ</XMTok>
                  <XMTok meaning="0" role="NUMBER">0</XMTok>
                </XMApp>
                <XMTok role="CLOSE" stretchy="false">|</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p5">
      <p>Ket: <Math mode="inline" tex="|b&gt;" text="ket@(b)" xml:id="S2.p5.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="ket"/>
                <XMRef idref="S2.p5.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok role="OPEN" stretchy="false">|</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p5.m1.1">b</XMTok>
                <XMTok meaning="greater-than" role="CLOSE">&gt;</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p6">
      <p>Braket <Math mode="inline" tex="&lt;a|b&gt;" text="inner-product@(a, b)" xml:id="S2.p6.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="inner-product" role="MIDDLE"/>
                <XMRef idref="S2.p6.m1.1"/>
                <XMRef idref="S2.p6.m1.2"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p6.m1.1">a</XMTok>
                <XMTok role="MIDDLE" stretchy="false">|</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p6.m1.2">b</XMTok>
                <XMTok meaning="greater-than" role="CLOSE">&gt;</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S2.p7">
      <p>Also <Math mode="inline" tex="&lt;a|f|b&gt;" text="quantum-operator-product@(a, f, b)" xml:id="S2.p7.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="quantum-operator-product"/>
                <XMRef idref="S2.p7.m1.1"/>
                <XMRef idref="S2.p7.m1.2"/>
                <XMRef idref="S2.p7.m1.3"/>
              </XMApp>
              <XMWrap>
                <XMTok meaning="less-than" role="OPEN">&lt;</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p7.m1.1">a</XMTok>
                <XMTok role="CLOSE" stretchy="false">|</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p7.m1.2">f</XMTok>
                <XMTok role="OPEN" stretchy="false">|</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S2.p7.m1.3">b</XMTok>
                <XMTok meaning="greater-than" role="CLOSE">&gt;</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
  </section>
  <section inlist="toc" xml:id="S3">
    <tags>
      <tag>3</tag>
      <tag role="refnum">3</tag>
      <tag role="typerefnum">§3</tag>
    </tags>
    <title><tag close=" ">3</tag>Absolute Values</title>
    <para xml:id="S3.p1">
      <p>abs <Math mode="inline" tex="|a|" text="absolute-value@(a)" xml:id="S3.p1.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="absolute-value"/>
                <XMRef idref="S3.p1.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S3.p1.m1.1">a</XMTok>
                <XMTok role="VERTBAR" stretchy="false">|</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S3.p2">
      <p>norm <Math mode="inline" tex="||a||" text="norm@(a)" xml:id="S3.p2.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="norm"/>
                <XMRef idref="S3.p2.m1.1"/>
              </XMApp>
              <XMWrap>
                <XMTok role="OPEN">||</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S3.p2.m1.1">a</XMTok>
                <XMTok role="CLOSE">||</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S3.p3">
      <p>expr <Math mode="inline" tex="|a|-|b|" text="absolute-value@(a) - absolute-value@(b)" xml:id="S3.p3.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="minus" role="ADDOP">-</XMTok>
              <XMDual>
                <XMApp>
                  <XMTok meaning="absolute-value"/>
                  <XMRef idref="S3.p3.m1.1"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  <XMTok font="italic" role="UNKNOWN" xml:id="S3.p3.m1.1">a</XMTok>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                </XMWrap>
              </XMDual>
              <XMDual>
                <XMApp>
                  <XMTok meaning="absolute-value"/>
                  <XMRef idref="S3.p3.m1.2"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  <XMTok font="italic" role="UNKNOWN" xml:id="S3.p3.m1.2">b</XMTok>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                </XMWrap>
              </XMDual>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S3.p4">
      <p>nested <Math mode="inline" tex="||a|-|b||" text="absolute-value@(absolute-value@(a) - absolute-value@(b))" xml:id="S3.p4.m1">
          <XMath>
            <XMDual>
              <XMApp>
                <XMTok meaning="absolute-value"/>
                <XMRef idref="S3.p4.m1.3"/>
              </XMApp>
              <XMWrap>
                <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                <XMApp xml:id="S3.p4.m1.3">
                  <XMTok meaning="minus" role="ADDOP">-</XMTok>
                  <XMDual>
                    <XMApp>
                      <XMTok meaning="absolute-value"/>
                      <XMRef idref="S3.p4.m1.1"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S3.p4.m1.1">a</XMTok>
                      <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    </XMWrap>
                  </XMDual>
                  <XMDual>
                    <XMApp>
                      <XMTok meaning="absolute-value"/>
                      <XMRef idref="S3.p4.m1.2"/>
                    </XMApp>
                    <XMWrap>
                      <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                      <XMTok font="italic" role="UNKNOWN" xml:id="S3.p4.m1.2">b</XMTok>
                      <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    </XMWrap>
                  </XMDual>
                </XMApp>
                <XMTok role="VERTBAR" stretchy="false">|</XMTok>
              </XMWrap>
            </XMDual>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S3.p5">
      <p>non-nested <Math mode="inline" tex="a|a|+b|b|+c|c|" text="a * absolute-value@(a) + b * absolute-value@(b) + c * absolute-value@(c)" xml:id="S3.p5.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="plus" role="ADDOP">+</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok font="italic" role="UNKNOWN">a</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S3.p5.m1.1"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.1">a</XMTok>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok font="italic" role="UNKNOWN">b</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S3.p5.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.2">b</XMTok>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok font="italic" role="UNKNOWN">c</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S3.p5.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.3">c</XMTok>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
  </section>
  <section inlist="toc" xml:id="S4">
    <tags>
      <tag>4</tag>
      <tag role="refnum">4</tag>
      <tag role="typerefnum">§4</tag>
    </tags>
    <title><tag close=" ">4</tag>Mixtures that look like QM</title>
    <para xml:id="S4.p1">
      <p>abs and rel: <Math mode="inline" tex="a&lt;c|c|" text="a &lt; c * absolute-value@(c)" xml:id="S4.p1.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
              <XMTok font="italic" role="UNKNOWN">a</XMTok>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMTok font="italic" role="UNKNOWN" xml:id="S4.p1.m1.1">c</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S4.p1.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S4.p1.m1.2">c</XMTok>
                    <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
    <para xml:id="S4.p2">
      <p>worse <Math mode="inline" tex="|a|&lt;\min\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|" text="absolute-value@(a) &lt; minimum@(absolute-value@(z plus-or-minus (z ^ 2 - 1) ^ (1 / 2)))" xml:id="S4.p2.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="less-than" role="RELOP">&lt;</XMTok>
              <XMDual>
                <XMApp>
                  <XMTok meaning="absolute-value"/>
                  <XMRef idref="S4.p2.m1.2"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                  <XMTok font="italic" role="UNKNOWN" xml:id="S4.p2.m1.2">a</XMTok>
                  <XMTok role="VERTBAR" stretchy="false">|</XMTok>
                </XMWrap>
              </XMDual>
              <XMApp>
                <XMTok meaning="minimum" role="OPFUNCTION" scriptpos="post" xml:id="S4.p2.m1.1">min</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="absolute-value"/>
                    <XMRef idref="S4.p2.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                    <XMApp xml:id="S4.p2.m1.3">
                      <XMTok meaning="plus-or-minus" name="pm" role="ADDOP">±</XMTok>
                      <XMTok font="italic" role="UNKNOWN">z</XMTok>
                      <XMApp>
                        <XMTok role="SUPERSCRIPTOP" scriptpos="post2"/>
                        <XMDual>
                          <XMRef idref="S4.p2.m1.3.1"/>
                          <XMWrap>
                            <XMTok role="OPEN" stretchy="true">(</XMTok>
                            <XMApp xml:id="S4.p2.m1.3.1">
                              <XMTok meaning="minus" role="ADDOP">-</XMTok>
                              <XMApp>
                                <XMTok role="SUPERSCRIPTOP" scriptpos="post3"/>
                                <XMTok font="italic" role="UNKNOWN">z</XMTok>
                                <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                              </XMApp>
                              <XMTok meaning="1" role="NUMBER">1</XMTok>
                            </XMApp>
                            <XMTok role="CLOSE" stretchy="true">)</XMTok>
                          </XMWrap>
                        </XMDual>
                        <XMApp>
                          <XMTok fontsize="70%" meaning="divide" role="MULOP">/</XMTok>
                          <XMTok fontsize="70%" meaning="1" role="NUMBER">1</XMTok>
                          <XMTok fontsize="70%" meaning="2" role="NUMBER">2</XMTok>
                        </XMApp>
                      </XMApp>
                    </XMApp>
                    <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMApp>
          </XMath>
        </Math></p>
    </para>
  </section>
  <section inlist="toc" xml:id="S5">
    <tags>
      <tag>5</tag>
      <tag role="refnum">5</tag>
      <tag role="typerefnum">§5</tag>
    </tags>
    <title><tag close=" ">5</tag>In equation</title>
    <para xml:id="S5.p1">
      <equation xml:id="S5.Ex1">
        <Math mode="display" tex="|A\rangle=|B\rangle+|C\rangle" text="ket@(A) = ket@(B) + ket@(C)" xml:id="S5.Ex1.m1">
          <XMath>
            <XMApp>
              <XMTok meaning="equals" role="RELOP">=</XMTok>
              <XMDual>
                <XMApp>
                  <XMTok meaning="ket"/>
                  <XMRef idref="S5.Ex1.m1.1"/>
                </XMApp>
                <XMWrap>
                  <XMTok role="OPEN" stretchy="false">|</XMTok>
                  <XMTok font="italic" role="UNKNOWN" xml:id="S5.Ex1.m1.1">A</XMTok>
                  <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                </XMWrap>
              </XMDual>
              <XMApp>
                <XMTok meaning="plus" role="ADDOP">+</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="ket"/>
                    <XMRef idref="S5.Ex1.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S5.Ex1.m1.2">B</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                  </XMWrap>
                </XMDual>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="ket"/>
                    <XMRef idref="S5.Ex1.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="false">|</XMTok>
                    <XMTok font="italic" role="UNKNOWN" xml:id="S5.Ex1.m1.3">C</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="false">⟩</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMApp>
          </XMath>
        </Math>
      </equation>
    </para>
  </section>
</document>
