Eigenvalues and Eigenfunctions

The wavefunction for a given physical system contains the measurable information about the system. To obtain specific values for physical parameters, for example energy, you operate on the wavefunction with the quantum mechanical operator associated with that parameter. The operator associated with energy is the Hamiltonian, and the operation on the wavefunction is the Schrodinger equation. Solutions exist for the time independent Schrodinger equation only for certain values of energy, and these values are called "eigenvalues*" of energy.

一个给定物理系统的波函数包含关于该系统的可测量信息。要获得特定物理参数的值，例如能量，你需要将波函数与该参数的量子力学算符相作用。与能量相关的算符是哈密顿量，而对波函数的作用就是薛定谔方程。只有在某些能量值下，时间无关的薛定谔方程才有解，这些值被称为“能量本征值*”。

Corresponding to each eigenvalue is an "eigenfunction*". The solution to the Schrodinger equation for a given energy

involves also finding the specific function

which describes that energy state. The solution of the time independent Schrodinger equation takes the form

每个本征值对应一个‘本征函数’。对于给定的能量，求解薛定谔方程也涉及找到具体描述该能量状态的函数。时间无关的薛定谔方程的解形式为

The eigenvalue concept is not limited to energy. When applied to a general operator Q, it can take the form

本概念不仅限于能量。当应用于一般算符Q时，它可以表现为

if the function ψ_i is an eigenfunction for that operator. The eigenvalues q_i may be discrete, and in such cases we can say that the physical variable is "quantized" and that the index i plays the role of a "quantum number" which characterizes that state.

如果函数ψ_i是该算符的本征函数，则可以这样说。本征值q_i可能是离散的，在这种情况下，我们可以认为物理量是“量子化”的，而索引i则扮演着“量子数”的角色，以表征该状态。

*"Eigenvalue" comes from the German "Eigenwert" which means proper or characteristic value. "Eigenfunction" is from "Eigenfunktion" meaning "proper or characteristic function".

'特征值'来源于德语中的'Eigenwert'，意为'特征值'或'固有值'。'特征函数'来自'Eigenfunktion'，意为'特征函数'或'固有函数'。