波函数Wave function

合格的波函数必须单值、连续、平方可积，并通常归一化：∫|ψ|²dτ = 1。可观测量对应线性厄米算符，其期望值 ⟨A⟩ = ∫ψ*Âψ dτ。波函数本身可正可负甚至为复数，没有直接物理意义；正负号（相位）决定轨道重叠是成键还是反键。波函数为零的地方称为节点或节面，2s 轨道有一个球形节面，2p 轨道有一个节平面。

An acceptable wavefunction must be single-valued, continuous and square-integrable, and is usually normalised, ∫|ψ|²dτ = 1. Observables correspond to linear Hermitian operators, with expectation values ⟨A⟩ = ∫ψ*Âψ dτ. The wavefunction itself can be positive, negative or complex and has no direct physical meaning, but its sign (phase) decides whether orbital overlap is bonding or antibonding. Where it is zero there is a node: a 2s orbital has a spherical nodal surface and a 2p orbital a nodal plane.

电子是费米子，多电子波函数在交换任意两个电子时必须变号（反对称性），这就是泡利不相容原理的本质。斯莱特在 1929 年提出用行列式构造反对称波函数（斯莱特行列式），是哈特里-福克方法的基础。电子密度 ρ(r) 由波函数积分得到，是 X 射线衍射可测的量，也是密度泛函理论的基本变量。

Electrons are fermions, so a many-electron wavefunction must change sign when any two electrons are exchanged (antisymmetry), which is the essence of the Pauli exclusion principle. Slater proposed in 1929 building antisymmetric wavefunctions as determinants (Slater determinants), the basis of the Hartree–Fock method. Integrating the wavefunction gives the electron density ρ(r), measurable by X-ray diffraction and the basic variable of density functional theory.

1926 年玻恩提出概率解释，获 1954 年诺贝尔物理学奖。哥本哈根诠释认为测量使波函数“坍缩”；爱因斯坦对此不满，说“上帝不掷骰子”，并与波多尔斯基、罗森提出 EPR 佯谬。薛定谔的猫思想实验也针对这一问题。贝尔不等式实验（2022 年阿斯佩、克劳泽、蔡林格获诺贝尔物理学奖）证实了量子纠缠的非定域性。

Born proposed the probability interpretation in 1926 and received the 1954 Nobel Prize in Physics. The Copenhagen interpretation holds that measurement 'collapses' the wavefunction; Einstein objected that 'God does not play dice' and with Podolsky and Rosen posed the EPR paradox, and Schrödinger's cat thought experiment addressed the same issue. Bell-inequality experiments, recognised by the 2022 Nobel Prize in Physics to Aspect, Clauser and Zeilinger, confirmed the non-locality of quantum entanglement.