电子自旋 1/2：720°旋转之谜Electron spin ½: the mystery of a 720° rotation

探索费米子最神奇的性质：需要旋转两圈才能回到原始状态。拖拽可视化区域可旋转视角。Explore a remarkable property of fermions: a state requires two full rotations to return to itself. Drag the visualization to rotate the view.

测试版 / 未完成beta / unfinished交互、措辞与科学表达仍在复核；纽带会在 720° 回到平整状态。Interaction, wording, and scientific communication remain under review; the belt returns to its flat state at 720°.

360°

旋转一圈Rotate once

|ψ⟩ → −|ψ⟩

720°

旋转两圈Rotate twice

|ψ⟩ → +|ψ⟩

控制面板Control panel

实时交互Live interaction

自动演示模式Automatic demonstration mode

旋转角度 γ（0° → 720°）核心参数！Rotation angle γ (0° → 720°) · primary parameter

0°

当前相位因子 eCurrent phase factor e^-iγ/2

波函数保持原样

极角 θ（0°=|↑z⟩，180°=|↓z⟩）Polar angle θ (0°=|↑z⟩, 180°=|↓z⟩)

60°

方位角 φ（决定相对相位）Azimuthal angle φ (sets relative phase)

45°

完整演示覆盖 0°→720°：360° 时相位变为The complete demonstration spans 0°→720°: at 360° the phase becomes −1， 720° 时回到and at 720° returns to +1；干涉仪中的亮暗也随之交换并恢复。The interferometer's bright and dark outputs exchange and then recover accordingly.

Bloch 球视图Bloch-sphere view

r=态向量，n=测量轴r = state vector, n = measurement axis

Bloch 球表示量子态在三维空间中的方向。旋转 360° 时，Bloch 向量回到原位，但波函数相位变为 -1。这个变号在单路测量中不可见，但干涉实验可以检测到。The Bloch sphere represents the direction of a quantum state in three dimensions. After a 360° rotation, the Bloch vector returns to its original direction while the wavefunction acquires a phase of −1. This sign change is invisible in a single-path measurement but can be detected through interference.

Dirac Belt 拓扑演示（皮带技巧）Dirac-belt topology demonstration (belt trick)

360°扭结 × 720°解开360° twist × 720° release

视角：View: | 缩放：Zoom: 45%

0°

━━━

平整Flat

360°

∞

扭结Twisted

720°

━━━

复原Restored

核心直觉：Core intuition:皮带一端固定，另一端旋转。 One end of the belt is fixed while the other rotates.
360°：上下边缘交换，产生扭结，: the upper and lower edges exchange, producing a twist that 无法解开cannot be removed
720°：虽扭两圈，可通过同伦变换: after two twists, a homotopy can 自动复原restore it continuously

相位干涉仪Phase interferometer

360°亮暗互换 × 720°复原Bright/dark exchange at 360° × recovery at 720°

干涉仪把"整体相位变号"变成可观测的亮暗变化。I₀ = (1 + cos(γ/2))/2。绿线为端口 0 强度，红线为端口 1 强度。The interferometer converts a global phase sign change into observable intensity variation. I₀ = (1 + cos(γ/2))/2. The green line is port 0 intensity and the red line port 1 intensity.

核心物理要点Core physical points

4π

4π 周期性4π periodicity

自旋 1/2 粒子需要旋转 720° (4π) 才能真正回到原态A spin-½ state requires a 720° (4π) rotation to return to its original state

−1

相位变号Phase sign change

360° 旋转使波函数获得整体负号：|ψ⟩ → -|ψ⟩A 360° rotation gives the wavefunction a global minus sign: |ψ⟩ → −|ψ⟩

干涉可见Visible in interference

虽然单路测量|ψ|²相同，但干涉实验可检测相位差Single-path measurements give the same |ψ|², but interference can detect the phase difference

费米子特性Fermionic property

这是电子等费米子区别于玻色子的本质特征This is a defining distinction between fermions such as electrons and bosons