哈密顿光学力学类比的启示

早在 19 世纪 30—40 年代，哈密顿通过光学力学类比发现了经典力学与几何光学的数学相似性。一百多年来这一重要概念几乎完全被忽略了。

一个值得注意的例外是 19 世纪末克莱因(FelixKlein)反复强调哈密顿思想的重要性，讨论了力学与光学的对应关系。薛定谔在他的论文《量子化是本征值问题》(第二部分)中，一开始就指出：“哈密顿理论和波传播过程之间的内在联系完全不是一个新的概念，这个概念不仅哈密顿本人是熟知的，而且还是他的力学理论的出发点。”他强调“这是一个十分有效的重要概念”。是形成他的波动力学概念的渊源之一。我们可用图 14-1 来表示波动力学产生的物理思想背景。

图 14-1

Ⅰ和Ⅱ之间的过渡关系早已十分清楚。Ⅰ和Ⅲ之间的对应关系，即费马原理与莫培督的最小作用量原理之间的密切相似性早在哈密顿时代已有了完善的理论。薛定谔进一步发展了哈密顿的几何光学与经典力学的类比，提出了微观力学过程是波动过程的论断，形成了波动力学的概念。他从波动光学与波动力学的物理相似性出发，实现了由波动光学向波动力学的过渡。

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TRANSLATEDFROM

THE SECOND GERMAN EDITION

BLACKIE ＆ SON LIMITED LONDON AND GLASGOW 1928

图 14-21928 年出版的薛定谔的《波动力学论文集》一书的封面

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Quantisation as a Problcm of Proper Values(Part I)

(Annalcn der I’hysik(4),vol.79,1926)

§I In this paper I wish to consider, first, the simple case of the hydrogen atom (non-relativeistieand unperturbed), and show that the customary quantum conditions can be replaced by another postulate, in which the notion of “whole numbers”, merely as such, is not introduced. Rather when integralness dose appear, it arises in the same naturnl way as it does in the ease of the node-numbers of a vibrating string. The new conception is capable of

generalissation, and strikes, I believe, very deeply at the true nature of the quantum rules.

The usul form of the true nature of the latter is conneeted with the Hamilton-Jacobi difierential equation,

(I)

H(q, ∂S ) = E.

∂q

A solution of this equation is sought as can be represented as the sum of functions, each being a runction of one only of the independent varinbles q.

Here we now put for S a new unknown Ψ such that it will appear as a produd of related functions of the single co-ordinates, i.e. we put

(2)

The constant K must be introduced from considerations of dimensions; it has those of aclion. Hence we get

(I')

 K ∂ψ 

Hq,



∂q  = E

Now we do not look for a solution of equation(I’), but proceed as follows. If we negleet the relativistie variation of mass, equation(I’) can always be transformed so as to become a quadratic form (of Ψand its first derivatives) equated to zero.(For the onc-clectron problem)

(D 804) B

图 14-3 1926 年薛定谔写的《量子化是本征值问题》(第一部分)的第一页