2016量子-33 - 3.3 3.3.1 Dirac(x q x0 x x lim q x Q(x Q x...

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1 §3.3 函数势阱 3.3.1 函数的定义和主要性质 函数的概念是 Dirac 首先提出来的,它有直接的物理背景。 例如考虑一些电荷分布在一条直线上,可以引入线电荷密度 ) ( x 描写这个分布,它的定义是 x q x x 0 lim ) ( , 其中 q 是分布在间隔 x 中的电荷,而这条直线上的总电荷 Q ) ( x 的积分给出: dx x Q ) ( . 如果实际的电荷分布是在 a x 处有一个点电荷 Q ,其它各处都没有电荷,那么显然 ) ( x a x 地方处处是零,而在 a x 这一点变成了无穷大。 Dirac 建议:把这时的线电荷密度 ) ( x 记为 ) ( ) ( a x Q x , 其中的 ) ( a x 就称为 函数。它显然是一种非常“奇异”的函数。 可以这样来 理解 函数: 0, ( ) ( ) , ( ) x a x a x a  并且 ( ) 1. x a dx   但是严格地说 不是数,所以上面这种表达 函数的方法并不符合数学中关于函数的定义。严格一点, 可以认为 函数是一种特殊的积分核,按照下式来 定义 ), ( ) ( ) ( a f dx a x x f 其中函数 ) ( x f ) (   a a x
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