力学量的平均值。如果ψ是体系可能存在的状态,则任何可观测的物理量A的平均值为
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/a8773912b31bb05181050d68307adab44aede049?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
若ψ归一化时,
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/5366d0160924ab18d9975f7733fae6cd7a890be8?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
如果ψ是力学量算符的本征函数,其本征值为a时,有![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/902397dda144ad34857239c7d6a20cf430ad85a5?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
即力学量本征函数的平均值即为本征值。
如果ψ不是力学量算符的本征函数,但可写成相互正交的本征函数ψ1, ψ2,…, ψn (本征值分别为a1, a2,…, an)的线性组合的形式,即
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/d01373f082025aaffbc6c08bfdedab64024f1afd?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
,则有
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/18d8bc3eb13533fa9c472d6eaed3fd1f40345bf8?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
若ψ为已归一化的函数,即
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/78310a55b319ebc4dcb8c8d98426cffc1f1716a9?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
则有
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/b3fb43166d224f4a8877cd530ff790529922d180?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
由于ψ是不力学量算符的本征函数,因此对该体系进行力学量A的测量时,会得到各种可能的值,这些值都在力学量算符的本征值谱中,根据态叠加原理,
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/0d338744ebf81a4ceb075589d12a6059252da6bd?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
表示ψi 状态出现的概率,则在力学量A的测量中,本征值ai出现的概率也为
。