For the efficient identification of quantum states, we propose the notion of linear canonical wavelet transform in the framework of quantum mechanics. Using the machinery of Dirac representation theory and integration within an ordered product of operators, we recast the linear canonical wavelet transform to a matrix element of the squeezing–displacing operator $\mathcal{U}(\mu ,s){\mathcal{K}}_M$ between analyzing vector $⟨\psi |$ and two-mode quantum state vector $|f⟩$ to be transformed. We also derive the inner product relation and inversion formula for the linear canonical wavelet transform in the realm of quantum mechanics. Lastly, we present an explicit example for the lucid implementation of linear canonical wavelet transform in identifying the quantum states.

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关于线性正则小波变换的量子表示

为了有效识别量子态，我们在量子力学的框架下提出了线性正则小波变换的概念。使用狄拉克表示理论的机制和在算子的有序乘积中的集成，我们将线性规范小波变换重铸为挤压位移算子的矩阵元素$\mathcal{ü}(\mu ,s){\mathcal{ķ}}_{米}$分析向量之间$⟨\psi |$和二模量子态向量$|F⟩$被改造。我们还推导出了量子力学领域线性正则小波变换的内积关系和反演公式。最后，我们给出了一个明确的例子，说明线性规范小波变换在识别量子态时的清晰实现。