Operators play an important role in the formalism of quantum mechanics. By means of the technique of integration within an ordered product of operators and Dirac notation, we analytically prove that a new kind of asymmetric two-mode projection integral operator and another asymmetric two-mode Fock state projection summation operator are equal to the same Hermitian operator. For such Hermitian operator, we also rigorously demonstrate it corresponds to a parity measurement of a two-mode quantum state after passing through a beam splitter. The method used in this work provides a good exercise in quantum mechanics at the graduate level.

中文翻译：

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非对称二模相干态和二模Fock态的投影算子

运算符在量子力学的形式主义中起着重要作用。利用算子有序积和狄拉克表示法的积分技术，我们分析证明一种新型的非对称二模投影积分算子和另一个非对称二模福克状态投影求和算子等于同一厄米算子。对于这种Hermitian算子，我们还严格证明它对应于通过分束器后的双模量子态的奇偶性测量。在这项工作中使用的方法在研究生级别提供了很好的量子力学练习。