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| <!DOCTYPE html><html lang="en"> | ||
| <head> | ||
| <meta http-equiv="content-type" content="text/html; charset=UTF-8"> | ||
| <title>Path Integrals in Quantum Mechanics</title> | ||
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| </head> | ||
| <body> | ||
| <div class="ltx_page_main"> | ||
| <div class="ltx_page_content"> | ||
| <article class="ltx_document ltx_authors_1line"> | ||
| <h1 class="ltx_title ltx_title_document">Path Integrals in Quantum Mechanics</h1> | ||
| <div class="ltx_authors"> | ||
| <span class="ltx_creator ltx_role_author"> | ||
| <span class="ltx_personname">Raghu Mahajan | ||
| </span></span> | ||
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| <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> | ||
| <ol class="ltx_toclist"> | ||
| <li class="ltx_tocentry ltx_tocentry_section"><a href="#S1" title="In Path Integrals in Quantum Mechanics" class="ltx_ref"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Particle on a ring</span></a></li> | ||
| </ol></nav> | ||
| <section id="S1" class="ltx_section"> | ||
| <h2 class="ltx_title ltx_title_section"> | ||
| <span class="ltx_tag ltx_tag_section">1 </span>Particle on a ring</h2> | ||
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| <div id="S1.p1" class="ltx_para"> | ||
| <table id="S1.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table"> | ||
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| <tbody id="S1.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> | ||
| <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> | ||
| <td class="ltx_td ltx_align_right ltx_eqn_cell" style="padding-top:5pt;padding-bottom:5pt;"><math id="S1.E1.m1" class="ltx_Math" alttext="\displaystyle H" display="inline"><mi>H</mi></math></td> | ||
| <td class="ltx_td ltx_align_left ltx_eqn_cell" style="padding-top:5pt;padding-bottom:5pt;"><math id="S1.E1.m2" class="ltx_Math" alttext="\displaystyle=-\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}x^{2}}" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>−</mo><mrow><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo></mo><mstyle displaystyle="true"><mfrac><mi mathvariant="normal">d</mi><mrow><mi mathvariant="normal">d</mi><mo></mo><msup><mi>x</mi><mn>2</mn></msup></mrow></mfrac></mstyle></mrow></mrow></mrow></math></td> | ||
| <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> | ||
| <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td> | ||
| </tr></tbody> | ||
| <tbody id="S1.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> | ||
| <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> | ||
| <td class="ltx_td ltx_align_right ltx_eqn_cell" style="padding-top:5pt;padding-bottom:5pt;"><math id="S1.E2.m1" class="ltx_Math" alttext="\displaystyle\psi(x)" display="inline"><mrow><mi>ψ</mi><mo></mo><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow></math></td> | ||
| <td class="ltx_td ltx_align_left ltx_eqn_cell" style="padding-top:5pt;padding-bottom:5pt;"><math id="S1.E2.m2" class="ltx_Math" alttext="\displaystyle=\exp\left(\mathrm{i}\,\frac{n}{R}\,x\right)" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mi>exp</mi><mo></mo><mrow><mo>(</mo><mrow><mi mathvariant="normal">i</mi><mo lspace="0.170em"></mo><mstyle displaystyle="true"><mfrac><mi>n</mi><mi>R</mi></mfrac></mstyle><mo lspace="0.170em"></mo><mi>x</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></td> | ||
| <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> | ||
| <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td> | ||
| </tr></tbody> | ||
| </table> | ||
| <p class="ltx_p">Equation (<a href="#S1.E2" title="In 1 Particle on a ring ‣ Path Integrals in Quantum Mechanics" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>) are the wavefunctions of the particle.</p> | ||
| </div> | ||
| </section> | ||
| </article> | ||
| </div> | ||
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