含义Meaning
质量 m 的粒子被限制在长度 L 的无限深势阱中。能量量子化、零点能 \(E_1 > 0\)、波函数有 n − 1 个节点——是理解量子化的最简模型。
A particle confined to an infinite well of length L. Quantised energies, non-zero zero-point energy and n − 1 nodes — the simplest model of quantisation.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(n\) | 量子数quantum number | 1 |
| \(L\) | 箱长box length | m |
| \(m\) | 粒子质量particle mass | kg |
| \(N\) | π 电子数number of π electrons | 1 |
推导Derivation
箱内 V = 0：\(-\frac{\hbar^2}{2m}\psi'' = E\psi\)，通解 \(\psi = A\sin kx + B\cos kx\)，\(k = \sqrt{2mE}/\hbar\)。
Inside, V = 0: general solution \(A\sin kx + B\cos kx\) with \(k = \sqrt{2mE}/\hbar\).
边界条件 ψ(0) = 0 ⇒ B = 0；ψ(L) = 0 ⇒ \(kL = n\pi\)。
ψ(0) = 0 gives B = 0; ψ(L) = 0 gives \(kL = n\pi\).
代回得 \(E_n = \hbar^2k^2/2m = n^2h^2/(8mL^2)\)；n = 0 使 ψ ≡ 0，不允许，所以存在零点能。
Hence \(E_n = n^2h^2/8mL^2\); n = 0 is excluded, so there is zero-point energy.
归一化 \(\int_0^L|\psi|^2dx = 1\) 得 \(A = \sqrt{2/L}\)。
Normalisation gives \(A = \sqrt{2/L}\).
应用：共轭多烯中 π 电子近似在长度 L 的箱中，HOMO→LUMO 能隙 \(\Delta E = (N+1)h^2/(8mL^2)\)，链越长吸收越红移。
Application: π electrons of a polyene in a box; the gap \((N+1)h^2/8mL^2\) shrinks as the chain lengthens (red shift).
适用条件与局限Assumptions & limitations
无限深势阱是理想化；有限深势阱中波函数渗入壁内，能级略低。
Infinite walls are idealised; finite wells let ψ penetrate and lower the levels.
自由电子模型忽略键长交替和电子间作用，只能半定量。
The free-electron model ignores bond alternation and electron repulsion.
例题Worked example
丁二烯 N = 4，取 L ≈ 5.6 Å（含两端各半个键长）：λ ≈ 207 nm（实测 217 nm）。
Butadiene, N = 4, L ≈ 5.6 Å: λ ≈ 207 nm (observed 217 nm).