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‎blog.html‎

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<a href="blog/spin-representation.html">
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Spin Representation
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</p>
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<p class='post-date'>
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10 November 2025
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<p class='post-subtitle'>
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Explaining the role of group theory as it relates to the property of spin in quantum mechanics.
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</p>
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<span class='hidden'>5</span>
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<p class='post-title'>
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<a href="blog/vibecoding.html">
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Vibe Coding
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<a href="blog/my-thoughts-on-the-use-of-ai.html">
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My Thoughts on the Use of AI
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</p>
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15 May 2025
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<p class='post-subtitle'>
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My thoughts on AI assistance and the future of programming.
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Discussing my experience with AI as it has shaped my development as a programmer.
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<span class='hidden'>4</span>
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‎blog/eqTable.html‎

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<title>The Operator's Handbook</title>
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‎blog/greensfn.html‎

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‎blog/groups_in_physics.html‎

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<title>My Thoughts on the Use of AI</title>
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<meta name='author' content='Chase K Osborne'>
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Vibe Coding
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My Thoughts on the Use of AI
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My thoughts on AI assistance and the future of programming.
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Discussing my experience with AI as it has shaped my development as a programmer.
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‎blog/nim.html‎

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<li><a href='../index.html'>Home</a></li>
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<li><a href='../blog.html'>Blog</a></li>
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<div class='front-matter'>
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<h1>
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Spin Representation
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Explaining the role of group theory as it relates to the property of spin in quantum mechanics.
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</h4>
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<div class='bylines'>
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<h3>Published</h3>
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<p>10 November 2025</p>
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<div class='wrap article'>
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<h2 id="introduction">Introduction</h2>
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For most undergraduates, Griffiths Quantum Mechanics is the first
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<h2 id="so3-vs-su2">SO(3) vs SU(2)</h2>
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Rotational symmetries in physical space form the group SO(3), but quantum states transform under SU(2), its double cover.
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The correspondence between these groups ensures a \(2:1\) mapping, capturing the fact that a \(360^\circ\) rotation can
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introduce a sign change in spinor wavefunctions.
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<h2 id="representations">Representations and Spin</h2>
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Irreducible representations of SU(2) are labeled by half-integers \(j\). Each \(j\) yields a \((2j+1)\)-dimensional space
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of spin states. The familiar spin-\(\frac{1}{2}\) particles reside in the fundamental representation, while spin-1 fields
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correspond to the adjoint representation.
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<h2 id="ladder-operators">Ladder Operators</h2>
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Using the commutation relations \([J_i, J_j] = i\hbar \epsilon_{ijk} J_k\), one constructs ladder operators \(J_{\pm}\)
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to traverse the eigenbasis of \(J_z\). These operators manifest the algebraic structure that group theory predicts.
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<h2 id="physical-implications">Physical Implications</h2>
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Spin-statistics, selection rules, and coupling schemes all follow from these representation-theoretic foundations.
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Understanding how group theory constrains spin prepares us to analyze addition of angular momentum and phenomena such as
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Stern–Gerlach splitting.
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</p>
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<h2 id="conclusion">Conclusion</h2>
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Group theory does more than provide abstract structure—it directly informs the quantized nature of spin. Mastery of SU(2)
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representations gives a powerful language for reasoning about quantum systems and their symmetries.
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</p>
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</div>
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<div id='bibliography'>
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<div class='wrap'>
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<ol class="bibliography">
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<li><span id="sakurai">Sakurai, J. J., &amp; Napolitano, J. (2017). <i>Modern Quantum Mechanics</i>. Pearson.</span></li>
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<li><span id="fulton">Fulton, W., &amp; Harris, J. (2004). <i>Representation Theory</i>. Springer.</span></li>
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