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什么是波函数?

Triborg的回答

姆们不科普。姆们就搬运点正确的东西。

Collapse of the Wave Function: Models, Ontology, Origin, and Implications, Cambridge University Press, 2018
  1. 波函数就是一个量子态在坐标表象下的展开系数:

begin{aligned} vert alpha rangle & = int {rm{d}}^{mathcal{D}}x vert x rangle langle x vert alpharangle \ psi_{alpha}(x) & overset{{rm{def}}}{=} langle xvert alpha rangle  end{aligned}

这里 mathcal{D} 指系统的维度。

这是量子力学的基本假设。t'Hooft的讲解如下:[1]

Born’s rule is sometimes assumed to follow from Gleason’s theorem[11], which deduces the dependence of probabilities exclusively on the square of the norm, from the absence of any non-trivial alternatives. Indeed, if probabilities do depend on wave functions only, then the Born rule is the only reasonable outcome. But this is no derivation of quantum logic itself, and, as happens more often with purely mathematical theorems, any clues it might give concerning the nature of quantum mechanics itself, are misleading.[2]

这里要说明,有人以为波函数总在 L^2 里,这是不正确的。 hat{x} 的domain小于 L^2[3] 如果考虑平面波等散射态波函数,就要在rigged Hilbert space里说事。简单复读一下L2还是不够的。

装备希尔伯特空间 - 小时百科

2. 奉劝这个问题下所有回答的人,多看点Arxiv。波函数的塌缩有Rev. Mod. Phys. 讲:

journals.aps.org/rmp/ab

免费版在这里:

Models of Wave-function Collapse, Underlying Theories, and Experimental Tests

t'Hooft也专门写过文章诠释:

It is often claimed that the collapse of the wave function and Born’s rule to interpret the square of the norm as a probability, have to be introduced as separate axioms in quantum mechanics besides the Schro ̈dinger equation. Here we show that this is not true in certain models where quantum behavior can be attributed to underlying deterministic equations. It is argued that indeed the apparent spontaneous collapse of wave functions and Born’s rule are features that strongly point towards determinism underlying quantum mechanics.

arxiv.org/abs/1112.1811

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