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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<b<c" text="a < b < c" xml:id="S1.p1.m1">
<XMath>
<XMApp>
<XMTok meaning="multirelation"/>
<XMTok font="italic" role="UNKNOWN">a</XMTok>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">b</XMTok>
<XMTok meaning="less-than" role="RELOP"><</XMTok>
<XMTok font="italic" role="UNKNOWN">c</XMTok>
</XMApp>
</XMath>
</Math></p>
</para>
<para xml:id="S1.p2">
<p>Relations: <Math mode="inline" tex=">a" text="absent > a" xml:id="S1.p2.m1">
<XMath>
<XMApp>
<XMTok meaning="greater-than" role="RELOP">></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>" text="a > absent" xml:id="S1.p3.m1">
<XMath>
<XMApp>
<XMTok meaning="greater-than" role="RELOP">></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="<a>" 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"><</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S2.p1.m1.1">a</XMTok>
<XMTok meaning="greater-than" role="CLOSE">></XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math></p>
</para>
<para xml:id="S2.p2">
<p>Bra: <Math mode="inline" tex="<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"><</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="<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"><</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="<\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"><</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>" 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">></XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math></p>
</para>
<para xml:id="S2.p6">
<p>Braket <Math mode="inline" tex="<a|b>" 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"><</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">></XMTok>
</XMWrap>
</XMDual>
</XMath>
</Math></p>
</para>
<para xml:id="S2.p7">
<p>Also <Math mode="inline" tex="<a|f|b>" 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"><</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">></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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p1.m1.1">a</XMTok>
<XMTok role="CLOSE" 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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p3.m1.1">a</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S3.p3.m1.2"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p3.m1.2">b</XMTok>
<XMTok role="CLOSE" 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="OPEN" 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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p4.m1.1">a</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S3.p4.m1.2"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p4.m1.2">b</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
<XMTok role="CLOSE" 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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.1">a</XMTok>
<XMTok role="CLOSE" 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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.2">b</XMTok>
<XMTok role="CLOSE" 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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S3.p5.m1.3">c</XMTok>
<XMTok role="CLOSE" 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<c|c|" text="a < c * absolute-value@(c)" xml:id="S4.p1.m1">
<XMath>
<XMApp>
<XMTok meaning="less-than" role="RELOP"><</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="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S4.p1.m1.2">c</XMTok>
<XMTok role="CLOSE" stretchy="false">|</XMTok>
</XMWrap>
</XMDual>
</XMApp>
</XMApp>
</XMath>
</Math></p>
</para>
<para xml:id="S4.p2">
<p>worse <Math mode="inline" tex="|a|<\min\left|z\pm\left(z^{2}-1\right)^{{1/2}}\right|" text="absolute-value@(a) < 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"><</XMTok>
<XMDual>
<XMApp>
<XMTok meaning="absolute-value"/>
<XMRef idref="S4.p2.m1.2"/>
</XMApp>
<XMWrap>
<XMTok role="OPEN" stretchy="false">|</XMTok>
<XMTok font="italic" role="UNKNOWN" xml:id="S4.p2.m1.2">a</XMTok>
<XMTok role="CLOSE" 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="OPEN" 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="CLOSE" 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>