<description/> <author>frank rioux</author> <company/> <keywords/> <revisedBy>frank rioux</revisedBy> </userData> <identityInfo> <revision>13</revision> <documentID>13f8d0ff-bb44-4835-bc95-446bcfa38d99</documentID> <versionID>12ff467c-3f31-42a8-a455-4d13335683f9</versionID> <parentVersionID>00000000-0000-0000-0000-000000000000</parentVersionID> <branchID>00000000-0000-0000-0000-000000000000</branchID> </identityInfo> </metadata> <settings> <presentation> <textRendering> <textStyles> <textStyle name="Normal"> <blockAttr margin-left="0" margin-right="0" text-indent="0" text-align="left" list-style-type="none" tabs="inherit"/> <inlineAttr font-family="Times New Roman" font-charset="0" font-size="10" font-weight="normal" font-style="normal" underline="false" line-through="false" vertical-align="baseline" color="#00f"/> </textStyle> <textStyle name="Heading 1"> <blockAttr margin-left="0" margin-right="0" text-indent="inherit" text-align="left" list-style-type="inherit" tabs="inherit"/> 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</results> </mathRendering> <pageModel show-page-frame="false" show-header-frame="false" show-footer-frame="false" header-footer-start-page="1" paper-code="1" orientation="portrait" print-single-page-width="false" page-width="612" page-height="792"> <margins left="54" right="36" top="36" bottom="49.68"/> <header use-full-page-width="false"/> <footer use-full-page-width="false"/> </pageModel> <colorModel background-color="#fff" default-highlight-color="#ffff80"/> <language math="EN" UI="EN"/> </presentation> <calculation> <builtInVariables array-origin="0" convergence-tolerance="0.001" constraint-tolerance="0.001" random-seed="1" prn-precision="4" prn-col-width="8"/> <calculationBehavior automatic-recalculation="true" matrix-strict-singularity-check="true" optimize-expressions="false" exact-boolean="false" strings-use-origin="false" zero-over-zero="0"> <compatibility multiple-assignment="MC11" local-assignment="MC11"/> </calculationBehavior> <units> <currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"> <tabs> <tab show-guide="true" position="9"/> </tabs> </ruler> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="376" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="339" left="114" top="6" width="298.5" height="15" align-x="133.5" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="14">Numerical Solutions for a Modified Harmonic Potential</f> </text> </region> <region region-id="330" left="168" top="33" width="179.25" height="33.75" align-x="180" align-y="42" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="27"/>Frank Rioux <sp count="15"/>Department of Chemistry College of St. Benedict | St. John's University </text> </region> <region region-id="276" left="12" top="75" width="222" height="11.25" align-x="20.25" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> This tutorial deals with the following potential function: </text> </region> <region region-id="277" left="162" top="104.25" width="174.75" height="45" align-x="194.25" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> <ml:program> <ml:ifThen> <ml:apply> <ml:greaterOrEqual/> <ml:apply> <ml:plus/> <ml:apply> <ml:greaterOrEqual/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">d</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:id xml:space="preserve">∞</ml:id> </ml:otherwise> </ml:program> </ml:apply> </math> <rendering item-idref="2"/> </region> <region region-id="279" left="12" top="158.25" width="504.75" height="35.25" align-x="15" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> If d <sp/> = 0 we have the harmonic oscillator on the half-line with eigenvalues 1.5, 3.5, 5.5, ... for k = <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f">m</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> = 1. For large values of </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#00f">d</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> we have the full harmonic oscillator problem displaced in the x-direction by </c> </inlineAttr> </f> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#00f"> <f family="Times New Roman" charset="0" size="10">d</f> </c> </inlineAttr> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> with eigenvalues 0.5, 1.5, 2.5, ... </c> </inlineAttr> </f>for k = <f family="Symbol" charset="2">m</f> = 1. For small to intermediate values of d the potential can be used to model the interaction of an atom or molecule with a surface. </text> </region> <region region-id="291" left="12" top="207" width="65.25" height="11.25" align-x="33.75" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Integration limit: </text> </region> <region region-id="315" left="84" top="207" width="41.25" height="16.5" align-x="108.75" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="max">x</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="316" left="144" top="207" width="60" height="11.25" align-x="161.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Effective mass: </text> </region> <region region-id="317" left="216" top="206.25" width="24" height="13.5" align-x="228" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="310" left="258" top="207" width="62.25" height="11.25" align-x="268.5" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Force constant: </text> </region> <region region-id="311" left="330" top="207" width="22.5" height="12.75" align-x="340.5" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="314" left="372" top="207" width="105" height="11.25" align-x="389.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Potential energy minimum: </text> </region> <region region-id="312" left="486" top="207" width="23.25" height="12.75" align-x="497.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">d</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="285" left="12" top="243" width="67.5" height="11.25" align-x="29.25" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Potential energy: </text> </region> <region region-id="318" left="96" top="236.25" width="83.25" height="27.75" align-x="129" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">V</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="320" left="12" top="285" width="86.25" height="11.25" align-x="33.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Integration algorithm: </text> </region> <region region-id="321" left="108" top="285" width="24" height="12.75" align-x="119.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Given</ml:id> </math> <rendering item-idref="8"/> </region> <region region-id="331" left="156" top="271.5" width="159.75" height="37.5" align-x="280.5" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="332" left="342" top="284.25" width="37.5" height="13.5" align-x="368.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="10"/> </region> <region region-id="333" left="408" top="284.25" width="45.75" height="13.5" align-x="435.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Ψ'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0.1</ml:real> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="325" left="108" top="326.25" width="95.25" height="17.25" align-x="123" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ψ</ml:id> <ml:apply> <ml:Odesolve method="adaptive"/> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="326" left="228" top="327" width="98.25" height="11.25" align-x="228" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Normalize wavefunction: </text> </region> <region region-id="327" left="360" top="319.5" width="107.25" height="60" align-x="386.25" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="334" left="12" top="363" width="57" height="11.25" align-x="25.5" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Energy guess: </text> </region> <region region-id="335" left="84" top="363" width="28.5" height="12.75" align-x="94.5" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">E</ml:id> <ml:real>0.5</ml:real> </ml:globalDefine> </math> <rendering item-idref="14"/> </region> <region region-id="266" left="18" top="396" width="229.5" height="179.25" align-x="18" align-y="396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="264" left="270" top="396" width="236.25" height="179.25" align-x="270" align-y="396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="340" left="12" top="597" width="108" height="11.25" align-x="12" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculate average position: </text> </region> <region region-id="361" left="264" top="597" width="152.25" height="11.25" align-x="264" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculate potential and kinetic energy: </text> </region> <region region-id="348" left="12" top="621.75" width="125.25" height="42.75" align-x="40.5" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">X</ml:id> <ml:apply> <ml:integral 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subscript="avg">X</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>5.000000000751446</ml:real> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> <region region-id="366" left="264" top="621.75" width="147" height="42.75" align-x="291" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">V</ml:id> <ml:apply> <ml:integral auto-algorithm="false" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="367" left="432" top="639" width="54.75" height="16.5" align-x="458.25" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">V</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.25000000003135</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="368" left="264" top="681" width="72" height="16.5" align-x="291" align-y="690" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">T</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve" subscript="avg">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="369" left="432" top="681" width="54.75" height="16.5" align-x="458.25" align-y="690" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.24999999996865002</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="372" left="12" top="723" width="41.25" height="11.25" align-x="30" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Exercises: </text> </region> <region region-id="374" left="12" top="746.25" width="502.5" height="77.25" align-x="18" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> For d = 0, k = <c val="#00f"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">m</f> </inlineAttr> </c> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> = 1 confirm that the first three energy eigenvalues are 1.5, 3.5 and 5.5 E</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> h </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f">. Start with x</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> max </c> </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> <f family="Times New Roman" charset="0" size="10"> = 5, but be prepared to adjust to larger values if necessary. x</f> </c> </inlineAttr> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> max </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> is effectively infinity.</c> </inlineAttr> </f> For d = 5, k = <f family="Symbol" charset="2">m</f> = 1 confirm that the first three energy eigenvalues are 0.5, 1.5 and 2.5 Eh. Start with xmax = 10, but be prepared to adjust to larger values if necessary. Determine and compare the virial theorem for the exercises above. Calculate the probability that tunneling is occuring for the ground state for the first two exercises. 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<currentUnitSystem name="si" customized="false"/> </units> </calculation> <editor protection-level="none" view-annotations="false" view-regions="false"> <ruler is-visible="false" ruler-unit="in"> <tabs> <tab show-guide="true" position="9"/> </tabs> </ruler> <plotTemplate> <xy item-idref="1"/> </plotTemplate> <grid granularity-x="6" granularity-y="6"/> </editor> <fileFormat image-type="image/png" image-quality="75" save-numeric-results="true" exclude-large-results="false" save-text-images="false" screen-dpi="96"/> <miscellaneous> <handbook handbook-region-tag-ub="376" can-delete-original-handbook-regions="true" can-delete-user-regions="true" can-print="true" can-copy="true" can-save="true" file-permission-mask="4294967295"/> </miscellaneous> </settings> <regions> <region region-id="339" left="114" top="6" width="298.5" height="15" align-x="133.5" align-y="18" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <f size="14">Numerical Solutions for a Modified Harmonic Potential</f> </text> </region> <region region-id="330" left="168" top="33" width="179.25" height="33.75" align-x="180" align-y="42" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> <sp count="27"/>Frank Rioux <sp count="15"/>Department of Chemistry College of St. Benedict | St. John's University </text> </region> <region region-id="276" left="12" top="75" width="222" height="11.25" align-x="20.25" align-y="84" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> This tutorial deals with the following potential function: </text> </region> <region region-id="277" left="162" top="104.25" width="174.75" height="45" align-x="194.25" align-y="120" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> <ml:program> <ml:ifThen> <ml:apply> <ml:greaterOrEqual/> <ml:apply> <ml:plus/> <ml:apply> <ml:greaterOrEqual/> <ml:id xml:space="preserve">x</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:id xml:space="preserve">d</ml:id> </ml:apply> <ml:real>0</ml:real> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>1</ml:real> <ml:real>2</ml:real> </ml:apply> <ml:id xml:space="preserve">k</ml:id> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:ifThen> <ml:otherwise> <ml:id xml:space="preserve">∞</ml:id> </ml:otherwise> </ml:program> </ml:apply> </math> <rendering item-idref="2"/> </region> <region region-id="279" left="12" top="158.25" width="504.75" height="35.25" align-x="15" align-y="168" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> If d <sp/> = 0 we have the harmonic oscillator on the half-line with eigenvalues 1.5, 3.5, 5.5, ... for k = <f family="Symbol" charset="2" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f">m</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> = 1. For large values of </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#00f">d</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> we have the full harmonic oscillator problem displaced in the x-direction by </c> </inlineAttr> </f> <inlineAttr font-style="normal" underline="false" line-through="false"> <c val="#00f"> <f family="Times New Roman" charset="0" size="10">d</f> </c> </inlineAttr> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> with eigenvalues 0.5, 1.5, 2.5, ... </c> </inlineAttr> </f>for k = <f family="Symbol" charset="2">m</f> = 1. For small to intermediate values of d the potential can be used to model the interaction of an atom or molecule with a surface. </text> </region> <region region-id="291" left="12" top="207" width="65.25" height="11.25" align-x="33.75" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Integration limit: </text> </region> <region region-id="315" left="84" top="207" width="41.25" height="16.5" align-x="108.75" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="max">x</ml:id> <ml:real>10</ml:real> </ml:define> </math> <rendering item-idref="3"/> </region> <region region-id="316" left="144" top="207" width="60" height="11.25" align-x="161.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Effective mass: </text> </region> <region region-id="317" left="216" top="206.25" width="24" height="13.5" align-x="228" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">μ</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="4"/> </region> <region region-id="310" left="258" top="207" width="62.25" height="11.25" align-x="268.5" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Force constant: </text> </region> <region region-id="311" left="330" top="207" width="22.5" height="12.75" align-x="340.5" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">k</ml:id> <ml:real>1</ml:real> </ml:define> </math> <rendering item-idref="5"/> </region> <region region-id="314" left="372" top="207" width="105" height="11.25" align-x="389.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Potential energy minimum: </text> </region> <region region-id="312" left="486" top="207" width="23.25" height="12.75" align-x="497.25" align-y="216" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">d</ml:id> <ml:real>5</ml:real> </ml:define> </math> <rendering item-idref="6"/> </region> <region region-id="285" left="12" top="243" width="67.5" height="11.25" align-x="29.25" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Potential energy: </text> </region> <region region-id="318" left="96" top="236.25" width="83.25" height="27.75" align-x="129" align-y="252" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">V</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:id xml:space="preserve">k</ml:id> <ml:real>2</ml:real> </ml:apply> <ml:apply> <ml:pow/> <ml:parens> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:apply> </ml:parens> <ml:real>2</ml:real> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="7"/> </region> <region region-id="320" left="12" top="285" width="86.25" height="11.25" align-x="33.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Integration algorithm: </text> </region> <region region-id="321" left="108" top="285" width="24" height="12.75" align-x="119.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:id xml:space="preserve" xmlns:ml="http://schemas.mathsoft.com/math20">Given</ml:id> </math> <rendering item-idref="8"/> </region> <region region-id="331" left="156" top="271.5" width="159.75" height="37.5" align-x="280.5" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:plus/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:div/> <ml:real>-1</ml:real> <ml:apply> <ml:mult style="default"/> <ml:real>2</ml:real> <ml:id xml:space="preserve">μ</ml:id> </ml:apply> </ml:apply> <ml:apply> <ml:derivative/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:lambda> <ml:degree> <ml:real>2</ml:real> </ml:degree> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> <ml:apply> <ml:mult style="default"/> <ml:id xml:space="preserve">E</ml:id> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:apply> </math> <rendering item-idref="9"/> </region> <region region-id="332" left="342" top="284.25" width="37.5" height="13.5" align-x="368.25" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0</ml:real> </ml:apply> </math> <rendering item-idref="10"/> </region> <region region-id="333" left="408" top="284.25" width="45.75" height="13.5" align-x="435.75" align-y="294" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:apply xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:equal/> <ml:apply> <ml:id xml:space="preserve">Ψ'</ml:id> <ml:real>0</ml:real> </ml:apply> <ml:real>0.1</ml:real> </ml:apply> </math> <rendering item-idref="11"/> </region> <region region-id="325" left="108" top="326.25" width="95.25" height="17.25" align-x="123" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">Ψ</ml:id> <ml:apply> <ml:Odesolve method="adaptive"/> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:sequence> </ml:apply> </ml:define> </math> <rendering item-idref="12"/> </region> <region region-id="326" left="228" top="327" width="98.25" height="11.25" align-x="228" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Normalize wavefunction: </text> </region> <region region-id="327" left="360" top="319.5" width="107.25" height="60" align-x="386.25" align-y="336" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:function> <ml:id xml:space="preserve">Ψ</ml:id> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> </ml:function> <ml:apply> <ml:div/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:sqrt/> <ml:apply> <ml:integral auto-algorithm="true" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:pow/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:real>2</ml:real> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:bounds> </ml:apply> </ml:apply> </ml:apply> </ml:define> </math> <rendering item-idref="13"/> </region> <region region-id="334" left="12" top="363" width="57" height="11.25" align-x="25.5" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="false"> Energy guess: </text> </region> <region region-id="335" left="84" top="363" width="28.5" height="12.75" align-x="94.5" align-y="372" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:globalDefine xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve">E</ml:id> <ml:real>0.5</ml:real> </ml:globalDefine> </math> <rendering item-idref="14"/> </region> <region region-id="266" left="18" top="396" width="229.5" height="179.25" align-x="18" align-y="396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="15"/> <rendering item-idref="16"/> </region> <region region-id="264" left="270" top="396" width="236.25" height="179.25" align-x="270" align-y="396" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <plot disable-calc="false" item-idref="17"/> <rendering item-idref="18"/> </region> <region region-id="340" left="12" top="597" width="108" height="11.25" align-x="12" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculate average position: </text> </region> <region region-id="361" left="264" top="597" width="152.25" height="11.25" align-x="264" align-y="606" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Calculate potential and kinetic energy: </text> </region> <region region-id="348" left="12" top="621.75" width="125.25" height="42.75" align-x="40.5" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">X</ml:id> <ml:apply> <ml:integral auto-algorithm="false" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="19"/> </region> <region region-id="350" left="156" top="639" width="45" height="16.5" align-x="183.75" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">X</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>5.000000000751446</ml:real> </result> </ml:eval> </math> <rendering item-idref="20"/> </region> <region region-id="366" left="264" top="621.75" width="147" height="42.75" align-x="291" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">V</ml:id> <ml:apply> <ml:integral auto-algorithm="false" algorithm="adaptive"/> <ml:lambda> <ml:boundVars> <ml:id xml:space="preserve">x</ml:id> </ml:boundVars> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:mult style="default"/> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> <ml:apply> <ml:id xml:space="preserve">V</ml:id> <ml:sequence> <ml:id xml:space="preserve">x</ml:id> <ml:id xml:space="preserve">d</ml:id> </ml:sequence> </ml:apply> </ml:apply> <ml:apply> <ml:id xml:space="preserve">Ψ</ml:id> <ml:id xml:space="preserve">x</ml:id> </ml:apply> </ml:apply> </ml:lambda> <ml:bounds> <ml:real>0</ml:real> <ml:id xml:space="preserve" subscript="max">x</ml:id> </ml:bounds> </ml:apply> </ml:define> </math> <rendering item-idref="21"/> </region> <region region-id="367" left="432" top="639" width="54.75" height="16.5" align-x="458.25" align-y="648" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">V</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.25000000003135</ml:real> </result> </ml:eval> </math> <rendering item-idref="22"/> </region> <region region-id="368" left="264" top="681" width="72" height="16.5" align-x="291" align-y="690" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:define xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">T</ml:id> <ml:apply> <ml:minus/> <ml:id xml:space="preserve">E</ml:id> <ml:id xml:space="preserve" subscript="avg">V</ml:id> </ml:apply> </ml:define> </math> <rendering item-idref="23"/> </region> <region region-id="369" left="432" top="681" width="54.75" height="16.5" align-x="458.25" align-y="690" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <math optimize="false" disable-calc="false"> <ml:eval xmlns:ml="http://schemas.mathsoft.com/math20"> <ml:id xml:space="preserve" subscript="avg">T</ml:id> <result xmlns="http://schemas.mathsoft.com/math20"> <ml:real>0.24999999996865002</ml:real> </result> </ml:eval> </math> <rendering item-idref="24"/> </region> <region region-id="372" left="12" top="723" width="41.25" height="11.25" align-x="30" align-y="732" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> Exercises: </text> </region> <region region-id="374" left="12" top="746.25" width="502.5" height="77.25" align-x="18" align-y="756" show-border="false" show-highlight="false" is-protected="true" z-order="0" background-color="inherit" tag=""> <text use-page-width="false" push-down="false" lock-width="true"> For d = 0, k = <c val="#00f"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <f family="Symbol" charset="2" size="10">m</f> </inlineAttr> </c> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> = 1 confirm that the first three energy eigenvalues are 1.5, 3.5 and 5.5 E</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> h </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f">. Start with x</c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> max </c> </inlineAttr> </f> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> <f family="Times New Roman" charset="0" size="10"> = 5, but be prepared to adjust to larger values if necessary. x</f> </c> </inlineAttr> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> max </c> </inlineAttr> </f> <f family="Times New Roman" charset="0" size="10"> <inlineAttr font-weight="normal" font-style="normal" underline="false" line-through="false"> <c val="#00f"> is effectively infinity.</c> </inlineAttr> </f> For d = 5, k = <f family="Symbol" charset="2">m</f> = 1 confirm that the first three energy eigenvalues are 0.5, 1.5 and 2.5 Eh. Start with xmax = 10, but be prepared to adjust to larger values if necessary. Determine and compare the virial theorem for the exercises above. Calculate the probability that tunneling is occuring for the ground state for the first two exercises. (Answers: 0.112, 0.157) </text> </region> </regions> <binaryContent> <item item-id="1" content-encoding="gzip">H4sIAAAAAAAA/4yNvQ6CQBCEdzlQwd/GGhMbj9InsCBWJj6CnvifEA1SWPrm5+yCvUf2GGa/ GQZExBiHSVQb3OPjcnep3PO6Oz+q0tURyZljYrVzVzu1aIHplcX2cD8VdWPtmxcFM1ym7Z9K q2r5Stz79lprtVE2w2y4ZZvj8RBn/7u/8m5dueK0zANd2RZMeUJs2Ytitlj6lAO2BtGUDdvQ ixOCIQ8RgfmssOoII6muMhA9YQSOpUfiCRiN98FofABG40P9F8RIGYixMAJ/AQAA//8DAMuS Xhh8AQAA</item> <item item-id="2">iVBORw0KGgoAAAANSUhEUgAAAOkAAAA8CAYAAACDxJ1/AAAAAXNSR0IArs4c6QAAAARnQU1B AACxjwv8YQUAAAAgY0hSTQAAeiYAAICEAAD6AAAAgOgAAHUwAADqYAAAOpgAABdwnLpRPAAA BN5JREFUeF7tXdFy2zAM2/7/o7dkF/c0nUQBMiU5JPq21KYIgCDZpPV+/3l9/dKXGBADz2Xg bdIMXy8F3s0oA1RhDMZAmqqVSYNVbiI4MmkisQX1OxmQSQvdtA5/ZxFHz1om/SicaR2+sKop fYe9ZdJkk7Q2poz6fKPKpMlMWpekTCqTPoYBZJ3NWLAZMT+mKMFENEk1ScFS0WWnGJBJE5tU U/SU7bhzZdKkJpVBOaOcvFomrT6CyVC8JcYMeE8azOPsH5PWn51Zn6V5CYu8meMB8h3D+yyW A/b6O7gtrKWuM5y0PsKZidPCx3K06/PeXl678v1vkiKfobGJ9YqtFNYrplXYXoU0Y/gd+C7s 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