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            </XMath>
          </Math>
        </equation>
      </para>
    </subsection>
    <subsection inlist="toc" xml:id="S1.SS6">
      <tags>
        <tag>1.6</tag>
        <tag role="refnum">1.6</tag>
        <tag role="typerefnum">§1.6</tag>
      </tags>
      <title><tag close=" ">1.6</tag>Dirac bra-ket notation</title>
      <para xml:id="S1.SS6.p1">
        <equation xml:id="S1.Ex30">
          <Math mode="display" tex="\bra{\phi}\ket{\psi}\mbox{\quad as opposed to\quad}\bra{\phi}\ket{\psi}" text="inner-product@(phi, psi) * [ as opposed to ] * bra@(phi) * ket@(psi)" xml:id="S1.Ex30.m1">
            <XMath>
              <XMApp>
                <XMTok meaning="times" role="MULOP">⁢</XMTok>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="inner-product"/>
                    <XMRef idref="S1.Ex30.m1.1"/>
                    <XMRef idref="S1.Ex30.m1.2"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                    <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex30.m1.1">Ï•</XMTok>
                    <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                    <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex30.m1.2">ψ</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                  </XMWrap>
                </XMDual>
                <XMText> as opposed to </XMText>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="bra"/>
                    <XMRef idref="S1.Ex30.m1.3"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                    <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex30.m1.3">Ï•</XMTok>
                    <XMTok role="CLOSE" stretchy="true">|</XMTok>
                  </XMWrap>
                </XMDual>
                <XMDual>
                  <XMApp>
                    <XMTok meaning="ket"/>
                    <XMRef idref="S1.Ex30.m1.4"/>
                  </XMApp>
                  <XMWrap>
                    <XMTok role="OPEN" stretchy="true">|</XMTok>
                    <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex30.m1.4">ψ</XMTok>
                    <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                  </XMWrap>
                </XMDual>
              </XMApp>
            </XMath>
          </Math>
        </equation>
        <equation xml:id="S1.Ex31">
          <Math mode="display" tex="\bra{\phi}\outerproduct{\psi}{\xi}.\mbox{\quad as opposed to\quad}\bra{\phi}%&#10;\ket{\psi}\bra{\xi}" text="formulae@(bra@(phi) * outer-product@(psi, xi), [ as opposed to ] * inner-product@(phi, psi) * bra@(xi...
            <XMath>
              <XMDual>
                <XMApp>
                  <XMTok meaning="formulae"/>
                  <XMRef idref="S1.Ex31.m1.7"/>
                  <XMRef idref="S1.Ex31.m1.8"/>
                </XMApp>
                <XMWrap>
                  <XMApp xml:id="S1.Ex31.m1.7">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="bra"/>
                        <XMRef idref="S1.Ex31.m1.1"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                        <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex31.m1.1">Ï•</XMTok>
                        <XMTok role="CLOSE" stretchy="true">|</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="outer-product"/>
                        <XMRef idref="S1.Ex31.m1.2"/>
                        <XMRef idref="S1.Ex31.m1.3"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                        <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex31.m1.2">ψ</XMTok>
                        <XMTok name="rangle" role="MIDDLE" stretchy="true">⟩</XMTok>
                        <XMWrap>
                          <XMTok name="langle" role="OPEN" stretchy="false">⟨</XMTok>
                          <XMTok font="italic" name="xi" role="UNKNOWN" xml:id="S1.Ex31.m1.3">ξ</XMTok>
                          <XMTok role="VERTBAR" stretchy="true">|</XMTok>
                        </XMWrap>
                      </XMWrap>
                    </XMDual>
                  </XMApp>
                  <XMTok role="PERIOD">.</XMTok>
                  <XMApp xml:id="S1.Ex31.m1.8">
                    <XMTok meaning="times" role="MULOP">⁢</XMTok>
                    <XMText> as opposed to </XMText>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="inner-product"/>
                        <XMRef idref="S1.Ex31.m1.4"/>
                        <XMRef idref="S1.Ex31.m1.5"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>
                        <XMTok font="italic" name="phi" role="UNKNOWN" xml:id="S1.Ex31.m1.4">Ï•</XMTok>
                        <XMTok role="MIDDLE" stretchy="true">|</XMTok>
                        <XMTok font="italic" name="psi" role="UNKNOWN" xml:id="S1.Ex31.m1.5">ψ</XMTok>
                        <XMTok name="rangle" role="CLOSE" stretchy="true">⟩</XMTok>
                      </XMWrap>
                    </XMDual>
                    <XMDual>
                      <XMApp>
                        <XMTok meaning="bra"/>
                        <XMRef idref="S1.Ex31.m1.6"/>
                      </XMApp>
                      <XMWrap>
                        <XMTok name="langle" role="OPEN" stretchy="true">⟨</XMTok>