薛定谔方程Schrödinger equation

含时形式为 iħ ∂ψ/∂t = Ĥψ。哈密顿算符 Ĥ = −(ħ²/2m)∇² + V 由动能和势能组成。对氢原子，V = −e²/(4πε₀r)，在球坐标中分离变量可精确求解，得到能量 Eₙ = −13.6 eV/n² 和由 n、l、m 三个量子数标记的原子轨道。对多电子原子和分子，由于电子间排斥项，方程无法精确求解，必须采用近似方法。

The time-dependent form is iħ ∂ψ/∂t = Ĥψ, with Hamiltonian Ĥ = −(ħ²/2m)∇² + V combining kinetic and potential energy. For hydrogen, V = −e²/(4πε₀r), and separating variables in spherical coordinates gives exact solutions, energies Eₙ = −13.6 eV/n² and orbitals labelled by quantum numbers n, l and m. For many-electron atoms and molecules, electron–electron repulsion prevents exact solution, so approximations are required.

1924 年德布罗意提出物质波假说。1925 年底至 1926 年初，奥地利物理学家埃尔温·薛定谔受此启发，在瑞士阿罗萨度假时构思出波动方程，1926 年分四篇论文发表，成功再现了氢原子光谱。同年他证明波动力学与海森堡的矩阵力学等价。玻恩随后提出波函数的概率解释。薛定谔与狄拉克共获 1933 年诺贝尔物理学奖。

De Broglie proposed matter waves in 1924. Inspired by this, the Austrian physicist Erwin Schrödinger conceived his wave equation over Christmas 1925 on holiday in Arosa, Switzerland, and published it in four papers in 1926, reproducing the hydrogen spectrum. That year he showed wave mechanics equivalent to Heisenberg's matrix mechanics, and Born then gave the probability interpretation of the wavefunction. Schrödinger shared the 1933 Nobel Prize in Physics with Dirac.

狄拉克在 1929 年写道：大部分物理学和全部化学的基本规律已经完全知道，困难只在于这些规律导出的方程太复杂而无法求解。量子化学的全部发展，就是寻找求解多电子薛定谔方程的有效近似。一维势箱、谐振子和刚性转子等模型解释了共轭分子的颜色、振动光谱和转动光谱。

Dirac wrote in 1929 that the underlying laws for much of physics and all of chemistry were completely known, the difficulty being only that the resulting equations are too complicated to solve. Quantum chemistry is essentially the search for workable approximations to the many-electron Schrödinger equation. Simple models such as the particle in a box, the harmonic oscillator and the rigid rotor explain the colours of conjugated molecules and their vibrational and rotational spectra.