Quantum Harmonic Oscillator

Quantum Harmonic Oscillator: Wavefunctions

量子谐振子：波函数

The Schrodinger equation for a harmonic oscillator may be solved to give the wavefunctions illustrated below.

谐振子的薛定谔方程可解得如下所示的波函数。

The solution of the Schrodinger equation for the first four energy states gives the normalized wavefunctions at left. These functions are plotted at left in the above illustration. The probability of finding the oscillator at any given value of x is the square of the wavefunction, and those squares are shown at right above. Note that the wavefunctions for higher n have more "humps" within the potential well. This corresponds to a shorter wavelength and therefore by the deBroglie relationship they may be seen to have a higher momentum and therefore higher energy.

薛定谔方程的前四个能量态的解给出了左边的归一化波函数。这些函数在上述示意图的左边被绘制出来。在任何给定的x值处，谐振子出现的概率是波函数的平方，这些平方值在右上方显示。请注意，较高n值的波函数在势阱内有更多“波峰”。这对应于更短的波长，因此根据德布罗意关系，它们可能被看出具有更高的动量和因此更高的能量。

The most probable value of position for the lower states is very different from the classical harmonic oscillator where it spends more time near the end of its motion. But as the quantum number increases, the probability distribution becomes more like that of the classical oscillator - this tendency to approach the classical behavior for high quantum numbers is called the correspondence principle.

对于低能态，位置的最可能值与经典谐振子的情况有很大不同，经典谐振子在运动终点附近停留时间更长。但随着量子数增大，概率分布逐渐接近经典谐振子的情况——这种对高量子数情况下接近经典行为的倾向被称为对应原理。

When the Schrodinger equation for the harmonic oscillator is solved by a series method, the solutions contain this set of polynomials, named the Hermite polynomials.

当用级数法求解谐振子的薛定谔方程时，解中包含这一组多项式，这些多项式称为赫尔米特多项式。

The wavefunctions for the quantum harmonic oscillator contain the Gaussian form which allows them to satisfy the necessary boundary conditions at infinity. In the wavefunction associated with a given value of the quantum number n, the Gaussian is multiplied by a polynomial of order n (the Hermite polynomials above) and the constants necessary to normalize the wavefunctions.

量子谐振子的波函数包含高斯形式，这使得它们能够满足在无穷远处所需的边界条件。在与给定量子数n相关的波函数中，高斯函数乘以一个阶数为n的多项式（即上文提到的赫尔姆特多项式），并乘以归一化波函数所需的常数。