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        "quote": "对于球面 $| \\mathbf { p } | = { \\sqrt { 2 m E } }$ 上的任意函数 $\\alpha _ { E , \\pm } ( \\mathbf { p } )$，其中此语境下的 $u _ { \\pm } ( \\mathbf { p } )$ 称为“[<u class=\"concept-term\" data-concept-reference=\"concepts/quantum-mechanics/spin-polarization-vector\" data-concept-identity=\"urn:uuid:7e32fa09-632f-47ec-a8be-e0c229e727da\" data-concept-route=\"/concept-identities#concept-7e32fa09-632f-47ec-a8be-e0c229e727da\" id=\"first-78139beaae7ab48617ac-zh\">自旋极化向量(spin polarization vectors)</u>](../../../../../concepts/quantum-mechanics/spin-polarization-vector)”。然而，以这种方式表示解涉及一个微妙之处。回顾<a href=\"../chapter-07/05-the-bloch-sphere#section-7-5\" class=\"cross-reference\" id=\"xref-02-solutions-of-the-pauli-equation-and-representations-of-e-3-4\" data-reference=\"section-7-5\">第 7.5 节</a>，$u _ { + } ( \\mathbf { p } )$ 在 $p _ { 3 } = - 1$ 处不连续（对于 $u _ { - } ( \\mathbf { p } ) )$ 同样如此，并且 $\\boldsymbol { \\sigma } \\cdot \\mathbf { p }$ 的任何单位长度特征向量都必须在某处具有这样的不连续性。如果 $\\alpha _ { E , \\pm } ( \\mathbf { p } )$ 在 $p _ { 3 } ~ = ~ - 1$ 处有一个零点，则乘积 $\\widetilde { \\psi } _ { E , \\pm } ( \\mathbf { p } )$ 可以是连续的。一个基本的拓扑事实仍然是，组合 $\\widetilde { \\psi } _ { E , \\pm } ( \\mathbf { p } )$ 必须有一个零点，否则它将不得不不连续。如果这个零点位于 $p _ { 3 } = - 1$，我们对 $u _ { \\pm } ( \\mathbf { p } )$ 的选择效果很好，但如果它位于别处，人们可能想要做出不同的选择。最终，需要检查所计算出的物理量是否独立于这类选择。",
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