What you're looking at
+看动画时，你在看什么
+ +A quantum particle isn't a point but a complex-valued wavefunction ψ(x,t). Its modulus-squared |ψ(x,t)|² is the probability density of finding it at position x at time t. Here that particle is trapped between two infinitely tall walls at x = ±1—the canonical "particle in a box."
量子粒子不是一个点，而是一个复值的 波函数 ψ(x,t)。模平方 |ψ(x,t)|² 告诉你在时刻 t 的位置 x 上能找到它的概率密度。这里这个粒子被两堵无限高的墙关在 x = ±1 之间——也就是教科书上的"盒中粒子"。
I · The eigenstates of the box
+一、势阱的本征态
+For an empty box on (−1, 1), the Hamiltonian Ĥ = −½ ∂²/∂x² with Dirichlet boundary conditions has a complete set of energy eigenstates indexed by an integer n = 1, 2, 3, …
在 (−1, 1) 上的空盒子里，哈密顿量 Ĥ = −½ ∂²/∂x² 在 Dirichlet 边界条件下有一组完备的能量本征态，用整数 n = 1, 2, 3, … 编号：
These are standing waves that vanish at both walls. The energies grow as n²—higher modes oscillate faster and faster.
这些是在两堵墙上都为零的驻波。能量随 n² 增长——高阶模式的振荡频率越来越快。
II · The initial wavepacket as a sum of standing waves
+二、初始波包 = 驻波的叠加
+We start from a Gaussian, the same shape the Fortran's tdse.f90 writes onto the grid:
我们从一个高斯出发——形状跟 Fortran 的 tdse.f90 写到网格上的那个一致：
Three knobs: x₀ is the center, p₀ is the average momentum, α sets how sharp the packet is. Project this Gaussian onto every φₙ:
三个旋钮：x₀ 是中心位置，p₀ 是平均动量，α 决定波包多窄。然后把这个高斯投影到每一个 φₙ 上：
The Fortran computes these (numerically, via finite differences); this page computes them with the same trapezoidal rule but on the exact analytic eigenstates—so the spectrum agrees with the Fortran to leading order in 1/n_grid.
Fortran 用有限差分数值算 cₙ；这一页用同样的梯形积分，但作用在精确的解析本征态上——所以两套频谱在 1/n_grid 的领头阶上一致。
III · Each mode rotates on its own clock
+三、每个模式按自己的节奏旋转
+Because each φₙ is an energy eigenstate, the Schrödinger equation reduces to a phase rotation per mode:
每个 φₙ 都是哈密顿量本征态，所以薛定谔方程就退化成每个模式独立的相位旋转：
This is exactly what propagate_convert.f90 does in the Fortran: transform to the sine basis, multiply by phases exp(−i Eₙ t), transform back. Every frame on this page is one evaluation of this sum, on a 256-point grid, with the first 100 modes. No time-stepping, no accumulated error.
这正是 Fortran 里 propagate_convert.f90 干的事：变换到 sine 基底，乘上相位 exp(−i Eₙ t)，再变换回来。本页每一帧都是这个求和的一次求值——256 个网格点，前 100 个模式。没有时间推进、没有累积误差。
IV · Wigner phase space
+四、Wigner 相空间
+The right panel projects the wavefunction onto a (position, momentum) plane via the Wigner quasi-distribution:
+右图通过 Wigner 准分布把波函数投到 (位置, 动量) 平面上：
+Same formula as wigner.f90; this page computes it on every frame with an extended momentum range so the peak stays in view. The crucial feature: W can be negative. Classical probabilities can't—so any blue patch on the right is a quantum-only effect, the signature of two pieces of ψ interfering with each other.
公式跟 wigner.f90 一致；本页在更宽的动量范围内逐帧计算，让峰始终在视野内。关键性质：W 可以取负值。经典概率不行——所以右图里任何一块蓝色都是量子专属现象，是两段 ψ 之间在干涉。
V · What you'll see
+五、你会看到什么
+Bouncing. The Wigner blob slides toward whichever wall p₀ points at, with classical round-trip time T_cl ≈ 4/|p₀|.
反弹。Wigner 斑点朝 p₀ 指向的那堵墙滑过去，经典往返周期约为 T_cl ≈ 4/|p₀|。
Reflection flip. When the packet hits a wall, the Wigner blob splits and the momentum sign flips—you'll literally see it jump from +p₀ to −p₀ on the vertical axis.
反弹翻转。波包撞墙的瞬间，Wigner 斑点会"裂"开，动量符号翻转——你会直观看到它从纵轴的 +p₀ 跳到 −p₀。
Interference fringes. Near a wall, the incoming and reflected pieces overlap. The position panel develops vertical stripes with spatial period π / |p₀| (the de Broglie wavelength), and the phase-space picture sprouts a band of blue between the two streams—the W-function admitting that interference is happening.
干涉条纹。墙附近入射和反射波同时存在，位置面板会出现竖条纹，空间周期约为 π / |p₀|（德布罗意波长）；相空间则会在两股之间长出一条蓝带——W 函数直接写明了"这里有干涉"。
Scrambling. Different modes accumulate phases at different rates (∝ n²). After enough bounces the wavefunction looks like noise. It isn't random—it's a coherent superposition with too many components for the eye to parse.
"打乱"。不同模式按不同的速率累积相位（∝ n²）。反弹几次后波函数看起来像噪声——其实并不是随机，只是叠加的成分太多、肉眼分不清而已。
Revivals. Because every Eₙ is an integer multiple of π²/8, all phases re-align at T_rev = 16/π ≈ 5.09. Even partial revivals at T_rev / q are pretty—several little copies of the packet appear at once.
复活。因为所有 Eₙ 都是 π²/8 的整数倍，所有相位会在 T_rev = 16/π ≈ 5.09 重新对齐。在分数时刻 T_rev/q 还有漂亮的"部分复活"——同时出现几个小副本。
Energy and norm are conserved. ⟨E⟩ = Σ |cₙ|² Eₙ and ‖ψ‖² both stay constant in the readout below.
能量与归一化都守恒。下方读数里 ⟨E⟩ = Σ |cₙ|² Eₙ 和 ‖ψ‖² 都不会变。
VI · How this aligns with the Fortran code
+六、跟 Fortran 求解器对得上吗
+Same domain x ∈ (−1, 1), same boundary condition ψ = 0 at the walls, same initial Gaussian formula (matching tdse.f90:50–55), same eigenstate-expansion method (matching propagate_convert.f90), same Wigner kernel (matching wigner.f90). The one deliberate difference: this page uses the exact analytic eigenfunctions sin(nπ(x+1)/2) while the Fortran uses the finite-difference eigenvectors of its kinetic matrix—the two agree as n_grid → ∞, and for the low modes that dominate the dynamics on screen they're visually indistinguishable. Run the Fortran with the same (x₀, p₀, α) and any of its psi_⟨step⟩ output files will match the curve here at t = step × τ.
同一区域 x ∈ (−1, 1)、同一边界条件（墙上 ψ = 0）、同一初始高斯公式（对应 tdse.f90:50–55）、同一种本征态展开（对应 propagate_convert.f90）、同一 Wigner 核（对应 wigner.f90）。唯一刻意的差别：本页用解析本征函数 sin(nπ(x+1)/2)，Fortran 用其动能矩阵的有限差分本征矢——两者在 n_grid → ∞ 时一致，对屏幕上主导的低阶模式肉眼看不出差别。用同样的 (x₀, p₀, α) 跑 Fortran，它任何 psi_⟨step⟩ 文件都能对上本页 t = step × τ 时刻的曲线。
VII · Implementation notes
+七、给教授的版本说明
+Single self-contained HTML file. No dependencies, no build step. Open viz/index.html in any browser. The position panel is SVG (right-click any frame → save as a vector graphic for slides). The phase-space panel is Canvas. Animation speed, time horizon, and grid resolution are constants at the top of the <script> block.
单文件、零依赖、不需要构建。任何浏览器里打开 viz/index.html 即可。位置面板是 SVG（右键单帧 → 另存为矢量图直接进 slides）。相空间面板是 Canvas。动画速度、时间范围、网格分辨率在 <script> 块顶部的常量里可调。