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[{"key": "dc.contributor.advisor", "value": "Tuominen, Kimmo", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Koskivaara, Olli", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2015-06-25T18:45:00Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2015-06-25T18:45:00Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2013", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/46422", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4ss\u00e4 tutkielmassa tarkastellaan klassisesti skaalainvarianttia systeemi\u00e4 kvanttimekaniikan\nkeinoin. L\u00e4ht\u00f6kohdaksi otetaan klassinen mekaniikka, ja tarkastelemalla vaikutusintegraalia\ntodetaan, ett\u00e4 ainoa systeemin skaalainvarianssin takaava potentiaali\non muotoa D/r^2, miss\u00e4 D on vakio. T\u00e4m\u00e4n j\u00e4lkeen siirryt\u00e4\u00e4n ratkaisemaan Schr\u00f6dingerin\nyht\u00e4l\u00f6\u00e4 kyseiselle potentiaalille. Ensiksi huomataan, ett\u00e4 dimensioanalyysin\nperusteella sidottuja tiloja esitt\u00e4vi\u00e4 ratkaisuja ei voi olla. Lis\u00e4ksi havaitaan negatiivisia\nenergioita vastaavien tilojen olevan ongelmallisia. N\u00e4m\u00e4 ongelmat asetetaan\nkuitenkin sivuun, ja ratkaistaan ominaisarvo-ongelma suoraviivaisesti. Tuloksena saadaan\nnegatiivisille energioille matemaattisesti hyvin m\u00e4\u00e4ritellyt ratkaisut. Ongelmaksi\nmuodostuvat kuitenkin ratkaisujen raju oskillointi origon l\u00e4heisyydess\u00e4 sek\u00e4 negatiiviseen\n\u00e4\u00e4rett\u00f6myyteen jatkuvat energiat.\nKatkaisemalla potentiaali et\u00e4isyydelle epsilon origosta ja ratkaisemalla ominaisarvo-ongelma\nuudestaan saadaan alhaalta rajoitettu diskreetti energiaspektri, joka vastaa odotuksia\nfysikaalisesta tilanteesta. Seuraavaksi ongelmana on rajan epsilon -> 0 ottaminen siten, ett\u00e4\ntilanteen fysikaalinen mielekkyys s\u00e4ilyy. Ratkaisu saadaan vaatimalla, ett\u00e4 perustilan\nenergia E_1 pysyy rajaprosessissa vakiona. Muut energiatilat h\u00e4vi\u00e4v\u00e4t rajank\u00e4ynniss\u00e4,\nja potentiaalin voimakkuutta kuvaavan parametrin arvo m\u00e4\u00e4r\u00e4ytyy.\nTutkielman viimeisess\u00e4 osiossa keskustellaan klassisen ja kvanttimekaanisen analyysin\nv\u00e4lisest\u00e4 erosta. Skaalainvariantissa systeemiss\u00e4 ei voi esiinty\u00e4 energiaskaalaa, kuten\ndimensioanalyysikin jo ennakoi. Pitk\u00e4n analyysin seurauksena saatiin kuitenkin\nratkaisu, jolla on hyvin m\u00e4\u00e4ritelty perustila. N\u00e4enn\u00e4isen ristiriidan todetaan olevan\nseurausta anomaalisesta symmetriarikosta, joka tapahtuu siirrytt\u00e4ess\u00e4 klassisesta teoriasta\nkvanttiteoriaan. T\u00e4ss\u00e4 ty\u00f6ss\u00e4 tarkasteltu 1/r^2 -potentiaali on esimerkki tapauksesta,\njossa systeemill\u00e4 klassisesti ollut symmetria, t\u00e4ss\u00e4 tapauksessa skaalainvarianssi,\nrikkoutuu siirrytt\u00e4ess\u00e4 kvanttiteoriaan. Lopuksi esitell\u00e4\u00e4n lyhyesti muita symmetriarikkoja\nsek\u00e4 joitakin niihin liittyvi\u00e4 ilmi\u00f6it\u00e4.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.abstract", "value": "In this Bachelor\u2019s thesis a classically scale invariant system is studied from a quantum\nmechanical point of view. By studying the classical action integral it is first derived,\nthat the only potential that guarantees a system to be scale invariant is the infamous\ninverse square potential D/r^2 , where D is a constant. Next, the potential is inserted into\nSchr\u00f6dinger\u2019s equation. It is first noted that on the grounds of dimensional analysis\nthere cannot be bound states. Also, negative energy states are noticed to be problematic.\nThese problems are put aside, and the eigenvalue problem is solved straightforwardly,\nresulting with normalizable solutions for the negative energy states. Although\nthe solutions are well defined mathematically, they do not satisty the usual\nexpectations one has for a physical system. There is no ground state energy, and the\nsolutions oscillate infinitely rapidly near the origin.\nAs the origin is problematic, the potential is introduced again with a cutoff distance\nepsilon from the origin. The eigenvalue problem is solved again, resulting with a bounded\ndiscrete energy spectrum. The next problem is how to take the limit epsilon -> 0 without\nlosing the physical properties of the situation. The answer is to require that the ground\nstate energy remains constant during the process. When the limit is taken, all excited\nstates vanish, and the value of the coupling constant, describing the strength of the\npotential in question, is fixed.\nThe last part of this thesis deals with the differences between the classical and quantum\nmechanical analyses. A preferred energy scale cannot exist since the system is\nscale invariant, an argument supported by the dimensional analysis. Yet a solution\nwith a well defined ground state was obtained. This seeming conflict between the\nanalyses is identified to be a consequence of a phenomenon called anomalous symmetry\nbreaking. Thus, the study of the inverse square potential offers an example of\nbroken scale invariance. The symmetry of the classical situation is broken by quantization\nwhen a quantum mechanical analysis is carried out. Finally, other symmetry\nbreakings occuring in physics are shortly introduced.", "language": "en", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Olli Koskivaara (olalkosk) on 2015-06-25 18:45:00.057274. Form: Kandidaatintutkielma -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/kandin-tutkielma-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2015-06-25T18:45:00Z\nNo. of bitstreams: 2\nURN:NBN:fi:jyu-201506252455.pdf: 269388 bytes, checksum: fc639ce566fc2352225d1b6bd55c02df (MD5)\nlicense.html: 4784 bytes, checksum: 62352b74b86999553e3576c033505a26 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2015-06-25T18:45:00Z (GMT). No. of bitstreams: 2\nURN:NBN:fi:jyu-201506252455.pdf: 269388 bytes, checksum: fc639ce566fc2352225d1b6bd55c02df (MD5)\nlicense.html: 4784 bytes, checksum: 62352b74b86999553e3576c033505a26 (MD5)\n Previous issue date: 2013", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "23", "language": "", "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.subject.other", "value": "kvanttimekaniikka", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "skaalainvarianssi", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "symmetriarikko", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Skaalainvarianssi kvanttimekaniikassa", "language": "", "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "bachelor thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201506252455", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.dcmitype", "value": "Text", "language": "en", "element": "type", "qualifier": "dcmitype", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Bachelor's thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Kandidaatintutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Mathematics and Science", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Fysiikan laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teoreettinen fysiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2015-06-25T18:45:01Z", "language": null, "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_7a1f", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": null, "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "bachelorThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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