{"id":"https://openalex.org/W7225410772","doi":"https://doi.org/10.2139/ssrn.7682796","title":"Momentum Space Analysis of the Infinite Square Well: Operator Domains and Physical Tails","display_name":"Momentum Space Analysis of the Infinite Square Well: Operator Domains and Physical Tails","publication_year":2026,"publication_date":"2026-01-01","ids":{"openalex":"https://openalex.org/W7225410772","doi":"https://doi.org/10.2139/ssrn.7682796"},"language":"en","primary_location":{"id":"doi:10.2139/ssrn.7682796","doi":"https://doi.org/10.2139/ssrn.7682796","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.7682796","pdf_url":null,"source":{"id":"https://openalex.org/S4210172589","display_name":"SSRN Electronic Journal","issn_l":"1556-5068","issn":["1556-5068"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1318003438","host_organization_name":"RELX Group (Netherlands)","host_organization_lineage":["https://openalex.org/I1318003438"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.2139/ssrn.7682796","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5151564163","display_name":"Amir Kahrom","orcid":"https://orcid.org/0009-0008-2963-3588","observed_orcids":["https://orcid.org/0009-0008-2963-3588"]},"institutions":[{"id":"https://openalex.org/I110525433","display_name":"Islamic Azad University, Tehran","ror":"https://ror.org/01kzn7k21","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433"]},{"id":"https://openalex.org/I155419210","display_name":"Islamic Azad University, Science and Research Branch","ror":"https://ror.org/03187yj51","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433","https://openalex.org/I155419210"]},{"id":"https://openalex.org/I41775361","display_name":"Islamic Azad University Central Tehran Branch","ror":"https://ror.org/03a11m818","country_code":"IR","type":"education","lineage":["https://openalex.org/I110525433","https://openalex.org/I41775361"]}],"countries":["IR"],"is_corresponding":true,"raw_author_name":"Amir Kahrom","raw_affiliation_strings":["Science and Research Branch, Islamic Azad University (Tehran), Department of Physics, Faculty of Basic Sciences, Department of Physics, Faculty of Basic Sciences, Tehran, Tehran, Iran"],"raw_orcid":"https://orcid.org/0009-0008-2963-3588","affiliations":[{"raw_affiliation_string":"Science and Research Branch, Islamic Azad University (Tehran), Department of Physics, Faculty of Basic Sciences, Department of Physics, Faculty of Basic Sciences, Tehran, Tehran, Iran","institution_ids":["https://openalex.org/I110525433","https://openalex.org/I155419210","https://openalex.org/I41775361"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5151564163"],"corresponding_institution_ids":["https://openalex.org/I110525433","https://openalex.org/I155419210","https://openalex.org/I41775361"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9243000149726868,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11022","display_name":"Spectral Theory in Mathematical Physics","score":0.9243000149726868,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10622","display_name":"Quantum Mechanics and Applications","score":0.06759999692440033,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11262","display_name":"Quantum Mechanics and Non-Hermitian Physics","score":0.001500000013038516,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/infinite-square-well","display_name":"infinite square well","score":1.0},{"id":"https://openalex.org/keywords/momentum-distribution","display_name":"momentum distribution","score":0.9670000076293945},{"id":"https://openalex.org/keywords/free-expansion","display_name":"free expansion","score":0.640999972820282},{"id":"https://openalex.org/keywords/operator-domains","display_name":"operator domains","score":0.492000013589859},{"id":"https://openalex.org/keywords/self-adjoint-realizations","display_name":"self-adjoint realizations","score":0.8569999933242798},{"id":"https://openalex.org/keywords/dirichlet-boundary-conditions","display_name":"Dirichlet boundary conditions","score":0.9430000185966492},{"id":"https://openalex.org/keywords/sobolev-regularity","display_name":"Sobolev regularity","score":0.9200000166893005},{"id":"https://openalex.org/keywords/fourier-sine-series","display_name":"Fourier sine series","score":0.5910000205039978},{"id":"https://openalex.org/keywords/variational-principle","display_name":"variational principle","score":0.5329999923706055},{"id":"https://openalex.org/keywords/energy-quantization","display_name":"energy quantization","score":0.5640000104904175},{"id":"https://openalex.org/keywords/orthogonality-constraints","display_name":"orthogonality constraints","score":0.722000002861023}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5999000072479248},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5471000075340271},{"id":"https://openalex.org/C17137986","wikidata":"https://www.wikidata.org/wiki/Q215067","display_name":"Orthogonality","level":2,"score":0.5349000096321106},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.5313000082969666},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.4749000072479248},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.4717000126838684},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.46619999408721924},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.4361000061035156},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.42170000076293945},{"id":"https://openalex.org/C110121322","wikidata":"https://www.wikidata.org/wiki/Q865811","display_name":"Distribution (mathematics)","level":2,"score":0.4115999937057495},{"id":"https://openalex.org/C99730327","wikidata":"https://www.wikidata.org/wiki/Q1501536","display_name":"Sobolev space","level":2,"score":0.40310001373291016},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.40139999985694885},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.3837999999523163},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.3785000145435333},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.3743000030517578},{"id":"https://openalex.org/C75432250","wikidata":"https://www.wikidata.org/wiki/Q29654355","display_name":"Real line","level":2,"score":0.3693000078201294},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.35670000314712524},{"id":"https://openalex.org/C75413324","wikidata":"https://www.wikidata.org/wiki/Q1530791","display_name":"Square-integrable function","level":2,"score":0.35510000586509705},{"id":"https://openalex.org/C80845027","wikidata":"https://www.wikidata.org/wiki/Q15144062","display_name":"Continuous spectrum","level":2,"score":0.3547999858856201},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.34950000047683716},{"id":"https://openalex.org/C207864730","wikidata":"https://www.wikidata.org/wiki/Q179467","display_name":"Fourier series","level":2,"score":0.33550000190734863},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.3352000117301941},{"id":"https://openalex.org/C152991086","wikidata":"https://www.wikidata.org/wiki/Q1225362","display_name":"Position and momentum space","level":2,"score":0.3165000081062317},{"id":"https://openalex.org/C175225751","wikidata":"https://www.wikidata.org/wiki/Q15927242","display_name":"Discrete Fourier series","level":5,"score":0.3118000030517578},{"id":"https://openalex.org/C61062188","wikidata":"https://www.wikidata.org/wiki/Q835065","display_name":"Second moment of area","level":2,"score":0.30660000443458557},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.3052000105381012},{"id":"https://openalex.org/C149441793","wikidata":"https://www.wikidata.org/wiki/Q200726","display_name":"Probability distribution","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C152022596","wikidata":"https://www.wikidata.org/wiki/Q7268361","display_name":"Quadratic form (statistics)","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C2780813799","wikidata":"https://www.wikidata.org/wiki/Q3274237","display_name":"Zero (linguistics)","level":2,"score":0.2745000123977661},{"id":"https://openalex.org/C156778621","wikidata":"https://www.wikidata.org/wiki/Q1365748","display_name":"Spectrum (functional analysis)","level":2,"score":0.2743000090122223},{"id":"https://openalex.org/C137800194","wikidata":"https://www.wikidata.org/wiki/Q11713455","display_name":"Interpolation (computer graphics)","level":3,"score":0.2685000002384186},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.2623000144958496},{"id":"https://openalex.org/C89451469","wikidata":"https://www.wikidata.org/wiki/Q1443036","display_name":"Parseval's theorem","level":5,"score":0.2590000033378601},{"id":"https://openalex.org/C99844830","wikidata":"https://www.wikidata.org/wiki/Q102441924","display_name":"Scaling","level":2,"score":0.25130000710487366},{"id":"https://openalex.org/C48103436","wikidata":"https://www.wikidata.org/wiki/Q599031","display_name":"State (computer science)","level":2,"score":0.2506999969482422}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2139/ssrn.7682796","doi":"https://doi.org/10.2139/ssrn.7682796","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.7682796","pdf_url":null,"source":{"id":"https://openalex.org/S4210172589","display_name":"SSRN Electronic Journal","issn_l":"1556-5068","issn":["1556-5068"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1318003438","host_organization_name":"RELX Group (Netherlands)","host_organization_lineage":["https://openalex.org/I1318003438"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},{"id":"doi:10.35542/osf.io/xv4k5","doi":"https://doi.org/10.35542/osf.io/xv4k5","is_oa":true,"landing_page_url":"https://doi.org/10.35542/osf.io/xv4k5","pdf_url":null,"source":{"id":"https://openalex.org/S4306402530","display_name":"EdArXiv (OSF Preprints)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I2799848540","host_organization_name":"Center for Open Science","host_organization_lineage":["https://openalex.org/I2799848540"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},{"id":"pmh:oai:share.osf.io:3de153f9-36b7-43cd-9934-901db30d47b1","doi":null,"is_oa":true,"landing_page_url":"https://osf.io/xv4k5","pdf_url":null,"source":{"id":"https://openalex.org/S4306402530","display_name":"EdArXiv (OSF Preprints)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I2799848540","host_organization_name":"Center for Open Science","host_organization_lineage":["https://openalex.org/I2799848540"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.2139/ssrn.7682796","doi":"https://doi.org/10.2139/ssrn.7682796","is_oa":true,"landing_page_url":"https://doi.org/10.2139/ssrn.7682796","pdf_url":null,"source":{"id":"https://openalex.org/S4210172589","display_name":"SSRN Electronic Journal","issn_l":"1556-5068","issn":["1556-5068"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1318003438","host_organization_name":"RELX Group (Netherlands)","host_organization_lineage":["https://openalex.org/I1318003438"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"posted-content"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W3014293911","https://openalex.org/W7220158479","https://openalex.org/W1972126818","https://openalex.org/W2782586632","https://openalex.org/W1988476824","https://openalex.org/W2382132892","https://openalex.org/W3003366985","https://openalex.org/W2982449429","https://openalex.org/W7112836949","https://openalex.org/W2527643645"],"abstract_inverted_index":{"The":[0,106],"infinite":[1],"square":[2],"well":[3],"is":[4,56,71,174,189],"a":[5,27,45,72,78,93,120,131],"paradigm":[6],"of":[7,26,146,218,224],"energy":[8,171],"quantization,":[9],"but":[10,130],"defining":[11],"its":[12],"momentum":[13,47,79,104],"distribution":[14,48,55,108],"requires":[15],"specifying":[16],"an":[17,114],"operational":[18],"measurement":[19],"protocol.":[20],"We":[21],"consider":[22],"the":[23,35,41,50,60,82,86,97,143,147,167,170,177,186,216],"sudden":[24],"release":[25],"confined":[28,99],"eigenstate":[29],"followed":[30],"by":[31,38],"free":[32],"expansion:":[33],"extending":[34],"spatial":[36],"state":[37,100],"zero":[39],"onto":[40],"real":[42],"line":[43],"yields":[44],"continuous":[46,54],"via":[49],"Fourier":[51,62,74],"transform.":[52],"This":[53,202],"conceptually":[57],"distinct":[58],"from":[59],"internal":[61],"sine":[63],"series;":[64],"in":[65,221],"particular,":[66],"p":[67],"n":[68,129,151],"=":[69,127],"n\u03c0\u210f/L":[70],"characteristic":[73],"scale":[75],"rather":[76],"than":[77],"eigenvalue.":[80],"Because":[81],"first-derivative":[83],"operator":[84,187],"on":[85,176],"interval":[87],"with":[88],"Dirichlet":[89],"boundary":[90],"conditions":[91],"lacks":[92],"canonical":[94],"self-adjoint":[95],"realization,":[96],"strictly":[98],"possesses":[101],"no":[102],"intrinsic":[103],"distribution.":[105],"released":[107],"exhibits":[109],"shifted":[110],"probability":[111],"lobes":[112],"and":[113,214],"oscillatory":[115],"p-4":[116],"asymptotic":[117],"tail,":[118],"yielding":[119],"finite":[121],"second":[122],"moment":[123,134],"\u27e8p":[124,135],"2":[125,158,193],"\u27e9":[126],"2mE":[128],"divergent":[132],"fourth":[133],"4":[136],"\u27e9.":[137],"These":[138],"spectral":[139],"properties":[140],"directly":[141],"reflect":[142],"Sobolev":[144],"regularity":[145,213],"zero-extended":[148],"state,":[149],"\u03c8":[150],"\u2208":[152],"H":[153,157,180,192,197],"1":[154,181,198],"(R)":[155],"\\":[156],"(R).":[159],"Finally,":[160],"we":[161],"connect":[162],"this":[163],"mathematical":[164],"structure":[165],"to":[166,191],"variational":[168,222],"formulation:":[169],"quadratic":[172],"form":[173,178],"well-defined":[175],"domain":[179,188],"0":[182,199],"(0,":[183,194,200],"L),":[184],"whereas":[185],"restricted":[190],"L)":[195],"\u2229":[196],"L).":[201],"distinction":[203],"elucidates":[204],"why":[205],"finite-energy":[206],"trial":[207],"states":[208],"need":[209],"not":[210],"satisfy":[211],"second-derivative":[212],"clarifies":[215],"role":[217],"orthogonality":[219],"constraints":[220],"approximations":[223],"excited":[225],"states.":[226]},"counts_by_year":[],"updated_date":"2026-11-11T12:16:26.678958","created_date":"2026-11-07T00:00:00"}