tappa-jaaga rachana

quantum yantrikiche fages-space formulation fages spacemadhye sthitee "aanee" momentum variables "samaan payarivar thevate. yaulat, schroedinger chitra sthitee kinwa gatiche pratinidhitva vaparte (sthitee aanee gatichi jaaga dekhil pahaa). fages-space formulation done pramukh vaishishtye mhanje quantum sthitiche varnan sambhavyata vitaranadvare kele jaate (wave function, steet vector kinwa density matrixaivaji) aanee operator gunakar starr productdware badalala jaato.
hillbrand groenvoldne 1946 madhye tyachya ph. dii. prabandhat aanee swatantrapane joou moyalane haah siddhant purnapane vikasit keela hota, pratyek rachana herman well aanee yujeen wigner yanchya purvichya kalpananvar aadharit hotee.
fages-space formulation mukhya fayda assa aahe qee yamule operatorchi aupcharikta talhun quantum mechanics heey hamiltonian mechanixsarkhech disate, jyamule hilbert spacechya 'oze' chey pariman 'mukt' hote. heey sootra sankhyikeey swarupache aahe aanee quantum yantriki aanee shastriy sankhyikeey yantriki yanchyatil tarkik sambandh pradan karate, jyamule doghanmadhil naisargik tulana shakya hote (shastriy maryada pahaa). tappyatil avkashatil quantum yantrikila anekada vishisht quantum prakashiki anuprayoganmadhye (optical tappyatil jaaga pahaa), kinwa visangati aanee vishesh tantrik samasyanchya shrenichya abhyasat pasanti dilee jaate, jaree anyatha vyavharik paristhitit aupcharikta qamee samanyatah vaparali jaate.
fages spacemadhil quantum mechanixchya vikasat antarbhut asalelya sankalpanatmak kalpana kontsevichya vikrutee-parimanikaran (kontsevic parimanikaran sootra pahaa) aanee gair-sampreshaniy bhoomiti yasarkhya ganiteey shakhanmadhye shakha jhalya aahet.
tappya-jaageche vitaran.
quantum sthitiche fages-space vitaran heey eka ardh-sambhavyata vitaran aahe. fages-space formulationmadhye, fages-space distributionla wave functions kinwa density matrixcha konatahi sandarbh naan deta, quantum pranaliche moolbhut, adim varnan manale jaau shakate.
vitaranache pratinidhitva karnyache anek vegvegale marg aahet, jeey serve ekmekanshi sambandhit aahet. sarvat ullekhaneey mhanje wignerche pratinidhitva, jeey pratham shodhale gele. itar sadarikarananmadhye (sahityatil prachalitatechya andaje utaratya kramane) globber-sudarshan pii, husimi kyu, kirkwood-rehazek, mehata, revear aanee borne-jorden pratinidhitvancha samavesh hoto. jevha hamiltonian vishisht swaroop dharan karato, jase qee globber-sudarshan pii-pratinidhitvasathi samanya kram, tevha heey paryay sarvat upayukt tharatat. wignerche pratinidhitva sarvat samanya asalyane, anyatha nirdisht kelyashivay haah lekh sahasa tyala chikatoon raheel.
fages-space vitarnamadhye 2 'aine'-aayami fages spacemadhil sambhavyata ghanatesarkhe gunadharm asatat. udaharanarth, samanyatah jatil-moolya asalelya wave functionschya ulat, teey "vastavik-moolya" aahe. aapan sthitichya madhyantarat padanyachi sambhavyata samajoo shakato, udaharanarth, serve kshanant aanee sthitichya madhyantarat wigner karya samakalit karoonah
jarr ekhada nirikshak nirikshanksham darshavat asel, tarr too 'wigner transform' dware fages spacemadhye mapp keela jaau shakato. yaulat, haah operator "weil transform" dware punarprapt keela jaau shakato.
fages-space vitaranachya sandarbhat nirikshanyogyache apekshit moolya aahe
tathapi, savadhgiricha muddah disanyat samanata asoonahi, heey vastavik sanyukt sambhavyata vitaran nahee, kaaran sambhavyata siddhantachya siddhantachya svayamsiddhatemadhye aavashyak asalelya, tyachyakhalil pradesh paraspar ananya avasthanche pratinidhitva karat naheet. shivay, pahilya swatvache ullanghan karoon, (paryayipane dabalelya) susangat avasthancha apavad vagalata, teey sarvasadharanpane, agadi shuddha avasthansathihi "nakaratmak moolye" gheu shakate.
asha nakaratmak mulyache kshetra 'lahan' asalyache siddha hoteh teey kahipeksha mothya sankshipt pradeshanparyant vistaru shakat naheet aanee mhanoon shastriy maryadet nahise hotat. teey anishchitata tattvadvare sanrakshit kele jaatat, jeey tappyachya-antaral kshetranmadhye, tyapeksha lahan bhagat achook sthanikikaranas parvanagi dett nahee aanee asha prakare asha "nakaratmak sambhavyate" laa qamee virodhabhas dete. jarr samikaranachya davya bajucha arth hilbert spacemadhil operatorchya sandarbhat apekshit moolya mhanoon lavala gela tarr quantum optixchya sandarbhat heey samikaran optical equivalence pramey mhanoon olakhale jaate. (wigner karyachya gunadharmanchya aanee vyakhyechya tapashilansathi, tyacha mukhya lekh pahaa.)
quantum yantrikisathi paryayi fages-space drushtikon fages spacever wave function (keval eka ardh-sambhavyata ghanata nahee) paribhashit karnyacha prayatna karato, samanyatah segal-bergman transformdware. anishchitata tattvanshi susangat honyasathi, fages-space wave function heey eka aniyantrit karya asoo shakat nahee, anyatha teey fages spacechya aniyantritpane lahan pradeshat sthanikikaran kele jaau shakate. tyaeivaji, segal-bargman transform heey sootra _ 3 chey eka holomorphic karya aahe. fages-space wave functionshee sambandhit eka ardh-sambhavyata ghanata aahe; heey sthitee wave functionsche husimi kyu pratinidhitva aahe.
starr utpadan.
manak operator gunakarala badalnara fages-space formulationmadhil moolbhut naun-commutative binary operator haah starr product aahe, jeey chinhadware darshavile jaate. tappya-antaral vitaranachya pratyek pratinidhitvamadhye eka "bhinn" vaishishtyapurn taara utpadan asate. thosatesathi, aamhi hii charcha wigner-velchya pratinidhitvas sambandhit starr utpadanaparyant maryadit thevato.
sanketik soyisathi, aapan davya aanee ujavya derivativeschi kalpana sadar karato. f" aanee "g" yaa functionschya jodisathi, davya aanee ujavya derivativeschi vyakhya ashee kelee jaateh
taramrug utpadanachi vibhedak vyakhya ashee aahe qee
jethe ghatankiy karyachya yuktivadacha arth power sirij mhanoon lavala jaau shakato.
atirikt vibhedak sambandh heey "f" aanee "g" chya yuktivadanmadhil badalachya sandarbhat lihayala anumati detatah
moolatah forier transformdware-productchi vyakhya convolution integral swarupat karane dekhil shakya aaheh
(tyamule, eae.jee., gausian atibadhyatene rachana karatatah
kinwa 
ityadi.)
urja eigenstate vitarananna 'stargenstate', 'genstate', 'stargenfunctions' kinwa 'genfunctions' mhanoon olakhale jaate aanee sambandhit urjanna 'stargenvalues' kinwa 'genvalues' mhanoon olakhale jaate. heey-janmulyachya samikaranadvare, vel-swatantra schroedinger samikaranashi samanpane sodavale jaatat,
hamiltonian kuthe aahe, eka plain fages-space function, jeey bahutekda shastriy hamiltniansarkhech asate.
velechi utkranti.
tappyatil avkash vitaranachi vel utkranti leoville pravahachya pariman badaladvare dilee jaate. heey sootra quantum leuville samikaranachya ghanata matricks aavruttimadhye wigner parivartan lagoo kelyane udbhavate,
wonn neuman samikaran.
tyachya sambandhit taara utpadanasah tappyatil jaaga vitaranachya konatyahi pratinidhitvat, heey aahe
kinwa, visheshtah wigner karyasathi,
moyal brackett kuthe aahe, quantum commutatorche wigner transform, tarr {,} heey shastriy poison brackett aahe.
yamule patravyavahar tattvache sankshipt udaharan milateh heey samikaran spashtapane 'γ' → 0 yaa maryadet shastriy leuville samikaranat qamee hote. pravahachya pariman vistaramadhye, tathapi, 'tappyatil jaaget bindunchi ghanata sanrakshit keleli nahee'; sambhavyata drav 'pasarnara' aanee sankuchit karnara disato. 
tyamule quantum prakshepavakra sankalpana haah yethe eka najuk muddah aahe. quantum tappyatil pravahachya gair-sthanikateche kautuk karnyasathi, morse sambhavyatesathi khalee chitrapat pahaa.
aine.buub. sthanikikaranavaril anishchitata tattvane ghatalele nirbandh lakshat gheta, niils boharne sukshmadarshak pramanavaril asha prakshepavahinyanche bhautik astitva jordarpane nakarle. aupcharik tappyatil-antaral marganchya madhyamatoon, donhi prakarananmadhye gambhir vyavharik adthale asale taree, marg-samakalit paddhat aanee quantum vaishishtyanchi paddhat vaparoon wigner karyachi vel utkranti samasya katekorpane sodavali jaau shakate.
udaharane.
sadhe harmonic ausilator.
wigner-weilchya pratinidhitvatil eka avkashiy parimanatil sadhya harmonic ausilatorsathi hamiltonian aahe
sthir" wigner karyasathi-janmulyache samikaran nantar vachate
pratham,-janmulyachya samikaranacha kalpanik bhag vicharat ghya,
yacha arth assa aahe qee ekhadi vyakti ekach yuktivadachi karye mhanoon-junstate lihu shakateh
chalanchya yaa badalamule,-janhailyu samikaranacha vastavik bhag sudharit lagure samikaranachya swarupat lihine shakya aahe (hermiteche samikaran nahee!), jyachya dravanat lagure bahupadi samavisht aahet
groenvoldne sambandhit-janavalues sadar kele
harmonic ausilatorsathi, aniyantrit wigner vitaranachi velechi utkranti sopee aahe. prarambhik utkranti hii ausilator hamiltoniandware chalavalelya vareel utkranti samikaranadvare vikasit hote, fakt "tappyachya jaaget kathorpane firoon",
samanyatah, urjechi "takkar" (kinwa susangat sthitee) sthool praman darshavu shakate aanee tappyachya jaaget eksamanpane eksamanpane shastriy vastusarkhi disate, eka sadha yantrik ausilator (animated aakruti pahaa). asha vastoonchya serve tappyanvar ('tii' = 0 varr suruvatichya sthananvar) ektrit kelyane, eka satat 'paliced', vareel sthir-junstatesarkhech eka vel-swatantra sanrachana dete, jeey mothya-krutee pranalinsathi shastriy maryadeche antardnyani drushya aahe.
mukt kann koneey gatee.
samaja ekhada kann suruvatila kiman anishchit gossian sthiteet aahe, jyachi sthitee aanee gatee yaa donhichi apekshit moolye fages spacemadhil moodavar kendrit aahet. asha rajyacha muktapane prachar karnyasathi wigner karya aahe
jethe "α" haah eka mapdand aahe joou gossienchya prarambhik rundiche varnan karato, aanee.
suruvatila, sthitee aanee kshan parasparsambandhit nasatat. asha prakare, 3 parimananmadhye, sthitee aanee gatee vahak ekmekanna samantar asanyachi shakyata duppat asanyachi apeksha aapan karato.
tathapi, sthitee vikasit hott asatana sthitee aanee gatee yancha adhikadhik paraspar sambandh nirman hoto, kaaran vitaranachya kahee bhaganna mool sthanapasun door asalelya sthanaparyant pohochanyasathi mothya gatichi aavashyakta asateh lakshanrahitpane,
kharokhar, heey darshavine shakya aahe qee kananchi gatij urja pramananshi sahmat houn keval lakshanrahit trijya banate 
jaminichya sthititil shunyetar koneey gatichi quantum-yantrik kalpana, jeey abhimukhata swatantrya nirdisht karateh
morse kshamata.
morse potentialcha vapar dietomic renuchya compan sanrachanecha andaj lavnyasathi keela jaato.
quantum tunneling.
tunneling haah eka vaishishtyapurn quantum prabhav aahe, jyamadhye eka quantum kann, jyachya varr udanyasathi pureshi urja nasate, tarihi too adthalyatun jaato. haah prabhav shastriy yantrikimadhye astitvat nahee.

