| Summary: | 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">U</mi><mrow><mo>(</mo><mi>μ</mi><mo>,</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msub><mi mathvariant="script">K</mi><mi>M</mi></msub></mrow></semantics></math></inline-formula> between analyzing vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">⟨</mo><mi>ψ</mi><mo stretchy="false">|</mo></mrow></semantics></math></inline-formula> and two-mode quantum state vector <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">|</mo><mi>f</mi><mo stretchy="false">⟩</mo></mrow></semantics></math></inline-formula> 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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