含义Meaning
量子力学的基本方程（1926）。含时形式描述波函数的演化；对定态，分离时间后得到定态（不含时）方程，它是一个本征值问题：允许的能量 E 是哈密顿算符的本征值。
The fundamental equation of quantum mechanics (1926). For stationary states, separating time gives an eigenvalue problem: allowed energies are eigenvalues of the Hamiltonian.
符号说明Symbols
| 符号Symbol | 含义 Meaning | 单位 Unit |
|---|---|---|
| \(\Psi, \psi\) | （含时 / 定态）波函数wavefunction (time-dependent / stationary) | m⁻³ᐟ² |
| \(\hat H\) | 哈密顿算符（总能量算符）Hamiltonian (energy operator) | J |
| \(\hbar\) | 约化普朗克常数 h/2πreduced Planck constant | J·s |
| \(\nabla^2\) | 拉普拉斯算符Laplacian | m⁻² |
| \(V\) | 势能potential energy | J |
推导Derivation
德布罗意：粒子具有波长 \(\lambda = h/p\)，平面波 \(\Psi = e^{i(kx - \omega t)}\)，\(p = \hbar k\)，\(E = \hbar\omega\)。
de Broglie: \(\lambda = h/p\); for a plane wave \(p = \hbar k\), \(E = \hbar\omega\).
于是 \(-i\hbar\,\partial/\partial x\) 作用给出 p，\(i\hbar\,\partial/\partial t\) 给出 E：动量、能量对应微分算符。
So \(-i\hbar\,\partial_x\) returns p and \(i\hbar\,\partial_t\) returns E — observables become operators.
把经典能量关系 \(E = p^2/2m + V\) 换成算符作用在 Ψ 上，得含时方程。
Replacing \(E = p^2/2m + V\) by operators acting on Ψ gives the time-dependent equation.
若 V 不含时，设 \(\Psi = \psi(\mathbf r)e^{-iEt/\hbar}\) 分离变量，得 \(\hat H\psi = E\psi\)。
For time-independent V, separate \(\Psi = \psi e^{-iEt/\hbar}\) to obtain \(\hat H\psi = E\psi\).
玻恩诠释：\(|\psi|^2\) 是概率密度，要求 ψ 单值、连续、平方可积并归一化——这些边界条件导致能量量子化。
Born: \(|\psi|^2\) is a probability density; requiring ψ to be single-valued, continuous and normalisable quantises the energy.
适用条件与局限Assumptions & limitations
非相对论方程；重元素内层电子、精细结构需用狄拉克方程或相对论修正。
Non-relativistic; heavy-element core electrons and fine structure need Dirac’s equation.
多电子体系无法解析求解，必须用近似（玻恩–奥本海默、哈特里–福克、DFT 等）。
Many-electron systems need approximations (Born–Oppenheimer, Hartree–Fock, DFT).