On Quantum Representation of the Linear Canonical Wavelet Transform
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 tra...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2022-09-01
|
| Series: | Universe |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2218-1997/8/9/477 |
| _version_ | 1827656131456008192 |
|---|---|
| author | H. M. Srivastava Firdous A. Shah Aajaz A. Teali |
| author_facet | H. M. Srivastava Firdous A. Shah Aajaz A. Teali |
| author_sort | H. M. Srivastava |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T22:18:15Z |
| format | Article |
| id | doaj.art-78dfd3e505124ab3b3a5b7763c692525 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-09T22:18:15Z |
| publishDate | 2022-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-78dfd3e505124ab3b3a5b7763c6925252023-11-23T19:19:10ZengMDPI AGUniverse2218-19972022-09-018947710.3390/universe8090477On Quantum Representation of the Linear Canonical Wavelet TransformH. M. Srivastava0Firdous A. Shah1Aajaz A. Teali2Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaDepartment of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, IndiaDepartment of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, IndiaFor 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.https://www.mdpi.com/2218-1997/8/9/477waveletlinear canonical transformquantum statesDirac representationinversion formula |
| spellingShingle | H. M. Srivastava Firdous A. Shah Aajaz A. Teali On Quantum Representation of the Linear Canonical Wavelet Transform Universe wavelet linear canonical transform quantum states Dirac representation inversion formula |
| title | On Quantum Representation of the Linear Canonical Wavelet Transform |
| title_full | On Quantum Representation of the Linear Canonical Wavelet Transform |
| title_fullStr | On Quantum Representation of the Linear Canonical Wavelet Transform |
| title_full_unstemmed | On Quantum Representation of the Linear Canonical Wavelet Transform |
| title_short | On Quantum Representation of the Linear Canonical Wavelet Transform |
| title_sort | on quantum representation of the linear canonical wavelet transform |
| topic | wavelet linear canonical transform quantum states Dirac representation inversion formula |
| url | https://www.mdpi.com/2218-1997/8/9/477 |
| work_keys_str_mv | AT hmsrivastava onquantumrepresentationofthelinearcanonicalwavelettransform AT firdousashah onquantumrepresentationofthelinearcanonicalwavelettransform AT aajazateali onquantumrepresentationofthelinearcanonicalwavelettransform |