
kathor rotter

rotordynamics mein, kathor rotter ghoornan pranaliyon kaa ack yaantrik model haye. ack manmana kathor rotter ack 3-ayami kathor vastu haye, jaise kii ack sheersh. antariksh mein aisee vastu koo unmukh karane key liye teen konon kii aavashyakta hoti haye, jinhen euler konn key roop mein jaanaa jaataa haye. ack vishesh kathor rotter "raikhik rotter" haye jise varnan karane key liye kewal dou konon kii aavashyakta hoti haye, udaaharan key liye ack dwiparmanvik anu. adhik samanya anu tri-ayami hotey hain, jaise kii paani (asmamit rotter), amonia (samamit rotter), yaa meethen (golakar rotter).
raikhik rotter.
raikhik kathor rotter model mein dou bindu dravyamaan hotey hain joe unake dravyamaan key kendra seey nishchit doori parr sthit hotey hain. donon dravyamanon key beach kii nishchit doori aur dravyamaan key maan hii kathor model kii ekmatra visheshataaen hain. haalaanki, kaee vaastavik dietomics key liye yeh model bahut pratibandhaatmak haye kyonki doori aamtaur parr poori tarah seey taya naheen hoti haye. doori mein chhote badlaavon kii bharpai key liye kathor model mein sudhaar kiye jaa sakate hain. aise mamle mein bhee kathor rotter model prasthaan kaa ack upayogi bindu haye (shoonya-crum model).
shaastriya raikhik kathor rotter.
shaastriya raikhik rotter mein ack dusare key doori sootra 4 parr dou bindu dravyamaan sootra 1 aur sootra 2 (cumm dravyamaan sootra 3 key saath) hotey hain. yadi sootra _ 4 samay seey swatantra haye tou rotter kathor haye. ack raikhik kathor rotter key gativigyaan koo aamtaur parr golakar dhruviya nirdeshank key madhyam seey varnit kiya jaataa haye, joe r3 kii ack nirdeshank pranali banaate hain. bhautiki parampara mein nirdeshank saha-akshansh (parakashtha) konn sootra _ 6, anudairdhya (azimuth) konn sootra _ 7 aur doori sootra _ 4 hain. konn antariksh mein rotter key abhivinyas koo nirdisht karate hain. raikhik kathor rotter kaa gatij oorjaa sootra _ 9 iss prakaar haye -
sootra _ 10
jahan sootra _ 11 aur sootra _ 12 paimaane (yaa laime) kaarak hain.
quantum yaantrik anuprayogon key liye paimaane key kaarak mahatvapurna hain kyonki way vakrarekhiya nirdeshank mein vyakt laplacian mein pravesh karate hain. haath mein mamle mein (sthir sootra _ 4)
sootra _ 14
raikhik kathor rotter kaa shaastriya hamiltonian kaary haye
sootra _ 15
quantum mechanical raikhik kathor rotter.
ack dwiaaparmaanvik anu kii ghoornan oorjaa kaa anuman lagaane key liye quantum yantriki mein raikhik kathor rotter model kaa upayog kiya jaa sakata haye. ghoornan oorjaa pranali key liye jadata key kshan parr nirbhar karati haye, sootra _ 16. dravyamaan sandarbh frame key kendra mein, jadata kaa kshan isake barabar haih
sootra _ 17
jahan sootra 18 anu kaa cumm dravyamaan haye aur sootra 4 donon parmaanuon key beach kii doori haye.
quantum yantriki key anusaar, ack pranali key oorjaa staron koo schroedinger sameekaran koo hull karke nirdhaarit kiya jaa sakata haih
sootra _ 20
jahan sootra _ 21 tarang falan haye aur sootra _ 22 oorjaa (hamiltonian) prachaalak haye. kshetra-mukt sthaan mein kathor rotter key liye, oorjaa prachaalak pranali kii gatij oorjaa seey mell khaata haih
sootra _ 23
jahan sootra _ 24 koo cumm plank sthiraank aur sootra _ 25 koo laplacian kahaa jaataa haye. laplacian koo golakar dhruviya nirdeshank key sandarbh mein oopar diyaa gayaa haye. inn nirdeshankon key sandarbh mein likhaa gayaa oorjaa prachaalak haih
sootra _ 26
yeh prachaalak radial bhaag koo alag karane key baad hydrogen paramanu key schroedinger sameekaran mein bhee dikhaai detaa haye. igen value sameekaran bunn jaataa haye
sootra _ 27
prateek sootra _ 28 golakar samanjasya key roop mein jaane jaane vaale karyon key ack samooh kaa pratinidhitva kartaa haye. dhyaan dein kii oorjaa sootra _ 29 parr nirbhar naheen karati haye. 
sootra _ 30
sootra 31 gunaa apakshayi haih nishchit sootra 32 aur sootra 33 vaale karyon mein samaan oorjaa hoti haye.
ghoornan sthiraank" sootra _ 34 kaa parichay dete hue, humm likhate hain,
sootra _ 35
parasparik lambai kii ikaaiyon mein ghoornan sthiraank haye,
sootra _ 36
see" prakaash kii gati key saath. yadi sootra 37, sootra 38 aur sootra 39 key liye see. jee. aiss. ikaaiyon kaa upayog kiya jaataa haye, tou sootra 40 koo semi-1 yaa tarang sankhyaon mein vyakt kiya jaataa haye, joe ack ikai haye jiska upayog aksar ghoornan-compan spectroscopy key liye kiya jaataa haye. ghoornan sthiraank sootra _ 41 doori sootra _ 4 parr nirbhar kartaa haye. aksar koee sootra _ 43 likhata haye jahan sootra _ 44 sootra _ 4 kaa santulan moolya haye (vaha moolya jisake liye rotter mein parmaanuon kii paraspar kriya oorjaa nyuntam haye).
ack vishisht ghoornan avashoshan varnakram mein shikharon kii ack shrinkhala hoti haye joe koniya samveg quantum sankhya (sootra _ 32) key vibhinn moolyon vaale staron key beach sankraman key anuroop hoti haye jaise kii chayan niyamon key kaaran sootra _ 47 (neeche dekhein). natijatan, ghoornan shikhar sootra _ 48 key ack purnank gunak key anuroop antar key saath urjaaon parr dikhaai dete hain.
chayan niyam.
anu kaa ghoornan sankraman taba hotaa haye jaba anu ack photon [ack matratmak vidyut chumbakiya (ee. aim.) kshetra kaa ack kann] avashoshit kartaa haye. photon kii oorjaa key aadhar parr (i.ee., ee. aim. kshetra kii tarang dairdhya) iss sankraman koo ack compansheel aur/yaa key sideband key roop mein dekhaa jaa sakata haye. 
electronic sankraman. shuddh ghoornan sankraman, jisamen compansheel (= compan aur electronic) tarang kaary naheen badalata haye, vidyut chumbakiya spectrum key microwave kshetra mein hotaa haye.
aam taur parr, ghoornan sankraman kewal tabhi dekhaa jaa sakata haye jaba koniya samveg quantum sankhya sootra _ 49 sootra _ 50 dwara badal jaati haye. yeh chayan niyam samay-nirbhar schroedinger sameekaran key pratham-crum vikshobh siddhant sanniktan seey utpann hotaa haye. iss upchaar key anusaar, ghoornan sankraman kewal tabhi dekhaa jaa sakata haye jaba dwidhruv prachaalak key ack yaa adhik ghatakon mein ack gair-gayab sankraman kshan hoe. yadi sootra _ 51 aaney vaali vidyut chumbakiya tarang key vidyut kshetra ghatak kii disha haye, tou sankraman kshan haye,
sootra _ 52
ack sankraman taba hotaa haye jaba yeh samakalan gair-shoonya hoe. aanvik tarang falan key ghoornan bhaag koo compan bhaag seey alag karke, koee yeh dikhaa sakata haye kii iska matlab haye kii anu mein ack sthaayi dvidhruviya kshan honah chaahiye. compansheel nirdeshank parr ekikaran key baad sankraman kshan kaa nimnalikhit ghoornan bhaag banaa rahata haye,
sootra _ 53
yahaan sootra _ 54 sthaayi dvidhruviya kshan kaa "z" ghatak haye. kshan sootra _ 18 dwidhruv prachaalak kaa compansheel ausat ghatak haye. ack visham paramanu anu key aksh key saath sthaayi dwidhruv kaa kewal ghatak hii gayab naheen hotaa haye.
golakar harmonics sootra _ 56 kii orthogonality key upayog seey yeh nirdhaarit karna sambhav haye kii sootra _ 57, sootra _ 58, sootra _ 59, aur sootra _ 60 key kaun seey maan dvidhruviya sankraman kshan samakalan key liye gair-shoonya maanon mein parinam denge. iss baadha key parinamswarup kathor rotter key liye dekhe gaye chayan niyam hainh
sootra _ 61
gair-kathor raikhik rotter.
kathor rotter kaa upayog aamtaur parr dwiparmanvik anuon kii ghoornan oorjaa kaa varnan karane key liye kiya jaataa haye, lekin yeh aise anuon kaa poori tarah seey sateek vivaran naheen haye. aisaa isaliye haye kyonki aanvik bandhan (aur isaliye antarparmanvik doori sootra _ 4) poori tarah seey nishchit naheen hain; parmaanuon key beach kaa bandhan failata haye kyonki anu teji seey ghoomata haye (ghoornan quantum sankhya sootra _ 57 key uchch maan). iss prabhaav kaa hisab ack sudhaar kaarak koo pesh karke kiya jaa sakata haye jise apkendrit vikriti sthir sootra _ 64 key roop mein jaanaa jaataa haye (vibhinn maatraon key sheersh parr pattiyan ingit karati hain kii yeye maatraaen semi-1 mein vyakt kii jaati hain):
sootra _ 65
kahaan
gair-kathor rotter dwiparmanvik anuon key liye ack sweekarya roop seey sateek model haye lekin phir bhee kuchh hudd taka apoorna haye. aisaa isaliye haye kyonki, halanki model ghoornan key kaaran bandhan khinchaav key liye jimmedaar haye, yeh bandhan mein compan oorjaa (kshamata mein samanjasya) key kaaran kissy bhee bandhan khinchaav kii andekhi kartaa haye.
manmaane dhang seey aakar kaa kathor rotter.
manmaane aakar kaa kathor rotter manmaane aakar kaa ack kathor nikaay haye jiska dravyamaan kaa kendra kshetra-mukt sthaan r3 mein sthir (yaa samaan aaytaakaar gati mein) hotaa haye, taaki isaki oorjaa mein kewal ghoornan gatij oorjaa (aur sambhavatah nirantar anuvaad oorjaa jise nazarandaj kiya jaa sakata haye) hoe. ack kathor pind (aanshik roop seey) apane jadata tensor key kshan key teen igen maanon dwara visheshta hoe sakata haye, joe vaastavik gair-nakaratmak maan hain jinhen "jadata key pramukh kshanon" key roop mein jaanaa jaataa haye.
microwave spectroscopy mein-ghoornan sankraman parr adharit spectroscopy-aamtaur parr anuon (kathor rotter key roop mein dekhaa jaataa haye) koo nimnanusar vargeekrit kartaa haih
yeh vargeekaran jadata key pramukh kshanon key saapeksh parimaan parr nirbhar kartaa haye.
kathor rotter key nirdeshank.
bhautiki aur engineering kii vibhinn shaakhaaen ack kathor rotter key gativigyaan key vivaran key liye vibhinn nirdeshankon kaa upayog karati hain. aanvik bhautiki mein euler konon kaa upayog lagbhag vishesh roop seey kiya jaataa haye. quantum yaantrik anuprayogon mein ack parampara mein euler konon kaa upayog karna fayademand haye joe golakar dhruviya nirdeshank key bhautik parampara kaa ack saral vistaar haye.
pahala kadam rotter (ack shareer-sthir frame) key saath daahine haath key orthonormal frame (orthogonal akshon kii 3-ayami pranali) kaa judaav haye. iss frame koo manmaane dhang seey shareer seey jodaa jaa sakata haye, lekin aksar koee bhee pramukh aksh frame kaa upayog kartaa haye-jadata tensor key saamaanyikrit igenvector, jinhen hamesha orthonormal chunaa jaa sakata haye, kyonki tensor samamit haye. jaba rotter mein samroopata-aksh hotaa haye, tou yeh aamtaur parr pramukh akshon mein seey ack key saath mell khaata haye. chunana suvidhajanak haye
shareer-sthir "z"-aksh uchchatam crum samroopata aksh haye.
ack sthaan-nirdhaarit frame (prayogshala aksh) key saath shareer-nirdhaarit frame koo sanrekhit karke shuru hotaa haye, taaki shareer-nirdhaarit "x", "y", aur "z" aksh sthaan-nirdhaarit "x", "y", aur "z" aksh key saath mell khaate hain. doosara, shareer aur usake dhaanche koo "z"-aksh (daahine haath key niyam dwara) key chaaron or ack sakaratmak konn sootra _ 69 parr sakriya roop seey ghumaaya jaataa haye, joe sootra _ 70-koo sootra _ 71-aksh parr leey jaataa haye. teesara, sootra _ 71-aksh key chaaron or ack sakaratmak konn sootra _ 72 parr shareer aur usake dhaanche koo ghumaata haye. shareer-nirdhaarit frame key "z"-aksh mein inn dou ghurnanon key baad anudairdhya konn sootra _ 74 (aamtaur parr sootra _ 7 dwara nirdisht) aur akshansh konn sootra _ 72 (aamtaur parr sootra _ 77 dwara nirdisht), donon sthaan-nirdhaarit frame key sambandh mein hotey hain. yadi rotter apane "z"-aksh key chaaron or belnakar samamit hotaa, raikhik kathor rotter kii tarah, tou antariksh mein iska abhivinyas iss bindu parr spasht roop seey nirdisht kiya jaayega.
yadi shareer mein cylinder (akshiya) samroopata kaa abhaav haye, tou isake "z"-aksh (jisamen dhruviya nirdeshank sootra _ 72 aur sootra _ 69 haye) key chaaron or ack antim ghoornan isake abhivinyas koo poori tarah seey nirdisht karane key liye aavashyak haye. paaramparik roop seey antim ghoornan konn koo sootra _ 80 kahaa jaataa haye.
yahaan varnit euler konon key liye parampara koo sootra _ 81 parampara key roop mein jaanaa jaataa haye; yeh dikhaaya jaa sakata haye (iss lekh mein usee tarah seey) kii yeh sootra _ 82 parampara key barabar haye jisamen ghoornan kaa crum ulat jaataa haye.
lagaatar teen ghurnanon kaa kull matricks utpaad haye
sootra _ 83
maan leejiye kii sootra _ 84 shareer-nirdhaarit frame key sambandh mein shareer mein ack manmaane bindu sootra _ 85 kaa samanvaya sadish haye. sootra _ 84 key tatva sootra _ 85 key 'shareer-nishchit nirdeshank' hain. shuru mein sootra _ 84 sootra _ 85 kaa sthaan-nirdhaarit nirdeshank sadish bhee haye. sootra key ghoornan parr, sootra _ 85 key shareer-nirdhaarit nirdeshank naheen badalate hain, lekin sootra _ 85 kaa sthaan-nirdhaarit nirdeshank sadish bunn jaataa haye,
sootra _ 92
vishesh roop seey, yadi sootra _ 85 shuru mein sthaan-nirdhaarit "z"-aksh parr haye, tou isamen sthaan-nirdhaarit nirdeshank hain.
sootra _ 94
joe golakar dhruviya nirdeshankon (bhautik parampara mein) key saath patrachar koo darshaata haye.
samay "tii" key falan key roop mein euler konon kaa gyaan aur praarambhik nirdeshank sootra _ 84 kathor rotter key gativigyaan koo nirdhaarit karate hain.
shaastriya gatij oorjaa.
yahaan seey yeh maanaa jaayega kii shareer-sthir frame ack pramukh aksh frame haye; yeh tatkalik jadata tensor sootra _ 96 (sthaan-nirdhaarit frame key sambandh mein vyakt), i koo vikarnit kartaa haye.ee.,
sootra _ 97
jahan euler konn samay-nirbhar hain aur vaastav mein iss sameekaran key vyutkram dwara sootra _ 98 kii samay nirbharta nirdhaarit karate hain. iss sanketan kaa tatparya haye
kii sootra _ 99 parr euler konn shoonya hain, taaki sootra _ 99 parr shareer-nishchit frame sthaan-nirdhaarit frame key saath mell khaata haye.
kathor rotter kii shaastriya gatij oorjaa "tii" koo vibhinn tarikon seey vyakt kiya jaa sakata haih
choonki inmein seey pratyek roop kaa apana upayog haye aur ise pathyapustakon mein paaya jaa sakata haye, humm una sabhi koo prastut karenge.
koniya wegg kaa roop.
koniya wegg "tii" key falan key roop mein,
sootra _ 101
key saath
sootra _ 102
baaen haath kii or sadish sootra _ 103 mein shareer-nirdhaarit frame key sambandh mein vyakt rotter key koniya wegg key ghatak hotey hain. koniya wegg euler key samikaranon key roop mein jaani jaane vaali gati key samikaranon koo santusht kartaa haye (shoonya lagu toqa key saath, kyonki dhaarana seey rotter kshetra-mukt sthaan mein haye). yeh dikhaaya jaa sakata haye kii sootra 104 kissy bhee sadish kaa samay vyutpann naheen haye, wegg kii samanya paribhasha key vipareet.
daahine haath kii or samay-nirbhar euler konon parr bindu samay vyutpann koo ingit karate hain. dhyaan dein kii ack alag ghoornan matricks kaa parinam upayog kiye gaye euler konn parampara key ack alag vikalp seey hoga.
lagrange roop.
sootra _ 104 kii abhivyakti kaa "tii" mein pratisthaapan detaa haye
lagrange roop mein gatij oorjaa (euler konon key samay vyutpann key ack kaary key roop mein). matricks-vector sanketan mein,
sootra _ 106
jahan sootra _ 107 euler konon mein vyakt metrick tensor haye-vakrarekhiya nirdeshank kii ack gair-uttar-samkon pranali -
sootra _ 108
koniya samveg roop.
aksar gatij oorjaa koo kathor rotter key koniya samveg sootra 109 key ack falan key roop mein likhaa jaataa haye. shareer-nirdhaarit frame key sambandh mein isamen ghatak sootra _ 110 haye, aur ise koniya wegg seey sambandhit dikhaaya jaa sakata haye,
sootra _ 111
yadi yeh koniya samveg ack sthir sthaan-nirdhaarit frame seey dekhaa jaae tou yeh ack sanrakshit (samay-swatantra) matra haye. choonki shareer-nirdhaarit frame chalati haye (samay parr nirbhar karati haye) isaliye ghatak sootra _ 110 "samay seey swatantra naheen" hain. yadi humm sthir sthaan-nirdhaarit frame key sambandh mein sootra _ 109 kaa pratinidhitva karate hain, tou humm karenge
isake ghatakon key liye samay swatantra abhivyaktiyaan khojen.
gatij oorjaa koo koniya samveg key sandarbh mein vyakt kiya jaataa haye 
sootra _ 114
hamilton form.
gatij oorjaa kaa hamilton roop saamaanyikrit kshan key sandarbh mein likhaa gayaa haye.
sootra _ 115
jahan yeh upayog kiya jaataa haye kii sootra _ 107 samamit haye. hamilton mein gatij oorjaa haye,
sootra _ 117
dwara diye gaye vyutkram metrick tensor key saath
sootra _ 118
iss vyutkram tensor kii aavashyakta laplas-beltramy prachaalak koo praapt karane key liye hoti haye, joe (sootra _ 119 seey gunaa karke) kathor rotter key quantum yaantrik oorjaa prachaalak koo detaa haye.
oopar diye gaye shaastriya hamiltonian koo nimnalikhit abhivyakti mein phir seey likhaa jaa sakata haye, joe kathor rotaron key shaastriya saankhyikiya yantriki mein utpann honey vaale charan samakalan mein aavashyak haye.
sootra _ 120
quantum yaantrik kathor rotter.
samanya roop seey, parimanikaran una prachalakon dwara saamaanyikrit kshan key pratisthaapan dwara kiya jaataa haye joe apane vihit sanyugmit charr (sthiti) key sambandh mein pahale vyutpann dete hain. iss prakaar,
sootra _ 121
aur isi tarah sootra _ 122 aur sootra _ 123 key liye. yeh ullekhaniya haye kii yeh niyam ack saral antar prachaalak dwara teenon euler konon key kaafee jatil falan sootra _ 124, euler konon key samay vyutpann, aur jadata kshanon (kathor rotter kii visheshta) koo pratisthaapit kartaa haye joe samay yaa jadata kshanon parr nirbhar naheen kartaa haye aur kewal ack euler konn mein antar kartaa haye.
quantisation niyam una prachalakon koo praapt karane key liye paryaapt haye joe shaastriya koniya moment key anuroop hain. dou prakaar key hotey hainh space-fixed aur boddy-fixed.
koniya samveg prachaalak. donon vector operator hain, i.ee.donon key teen ghatak hain joe kramshah sthaan-nirdhaarit aur shareer-nirdhaarit frame key ghoornan parr aapas mein sadish ghatakon key roop mein parivartit hotey hain. kathor rotter koniya samveg prachaalak kaa spasht roop yahaan diyaa gayaa haye (lekin savdhan rahein, unhein sootra _ 24 key saath gunaa kiya jaanaa chaahiye). shareer-sthir koniya samveg prachaalak sootra _ 126 key roop mein likhe jaate hain. way "visangat sanchaar sambandhon" koo santusht karate hain.
shaastriya hamiltonian seey gatij oorjaa prachaalak praapt karane key liye matratmak niyam "paryaapt naheen" haye. choonki shaastriya sootra _ 122 sootra _ 128 aur sootra _ 129 key saath aur inn karyon key vyutkram key saath aataa haye, isaliye shaastriya hamiltonian mein inn trikonamitiya karyon kii sthiti manmana haye. isake baad
parimanikaran abb sanchaar naheen rahata haye aur hamiltonian (oorjaa prachaalak) mein prachaalak aur karyon kaa crum chinta kaa vishay bunn jaataa haye. podolsky ney 1928 mein prastaav diyaa kii laplas-beltramy prachaalak (samay sootra _ 130) mein quantum yaantrik gatij oorjaa prachaalak key liye upayukt roop haye. iss prachaalak kaa samanya roop haye (yog paramparaah doharaae gaye soochkaankon parr yog-iss mamle mein teen euler konn sootra _ 131 parr):
sootra _ 132
jahan sootra _ 133 jee-tensor kaa nirdhaarak haih
sootra _ 134
uparokt metrick tensor key vyutkram koo dekhate hue, euler konon key sandarbh mein gatij oorjaa prachaalak kaa spasht roop saral pratisthaapan dwara anusaran kiya jaataa haye. (notah sambandhit igenvalue sameekaran kathor rotter key liye schroedinger sameekaran koo iss roop mein detaa haye kii ise pahali baar cronig aur rubby (samamit rotter key vishesh mamle key liye) dwara hull kiya gayaa thaa. yeh una kuchh mamalon mein seey ack haye jahaan schroedinger sameekaran koo vishleshanatmak roop seey hull kiya jaa sakata haye. inn sabhi mamalon koo schroedinger sameekaran key nirmaan key ack saal key bheetar hull curr liya gayaa thaa.)
aajkal nimnalikhit roop seey aage badhana aam baat haye. yeh dikhaaya jaa sakata haye kii sootra _ 135 koo shareer-nishchit koniya samveg prachaalak mein vyakt kiya jaa sakata haye (iss pramaan mein kissy koo trikonamitiya karyon key saath antar prachaalak koo saavdhaanipoorvak pargaman karna chaahiye). parinam shareer-nishchit nirdeshank mein vyakt shaastriya sootra key samaan dikhaai detaa haye,
sootra _ 136
wigner dee-matricks parr sootra _ 126 kii kriya saral haye. vishesh roop seey
sootra _ 138
taaki golakar rotter (sootra 139) key liye schroedinger sameekaran koo sootra 141 key barabar sootra 140 apakshayi oorjaa key saath hull kiya jaa sakey.
samamit sheersh (= samamit rotter) koo sootra _ 142 dwara chihnit kiya jaataa haye. yeh ack "prolet" (sigar key aakar kaa) sheersh haye yadi sootra _ 143 haye. baad key mamle mein humm hamiltonian koo iss prakaar likhate hain
sootra _ 144
aur iska upayog karein. 
sootra _ 145
isaliye
sootra _ 146
igen value sootra _ 147 sootra _ 148-gunaa apakshayi haye, kyonki sootra _ 149 vaale sabhi igen function kaa samaan igen value haye. oorjaa key saath .key. > 0 sootra _ 150 gunaa apakshayi hain. samamit sheersh key schroedinger sameekaran kaa yeh sateek samadhan pahali baar 1927 mein paaya gayaa thaa.
asmamit sheersh samasya (sootra _ 151) vishleshanatmak roop seey ghulansheel naheen haye, lekin ise sankhyaatmak roop seey hull kiya jaa sakata haye.
aanvik ghoornan kaa pratyaksh prayogaatmak avalokan.
lambe samay taka, aanvik ghoornan koo seedhe prayogaatmak roop seey naheen dekhaa jaa sakaa. paramanu sankalp key saath kewal maap takneekon ney ekal anu key ghoornan kaa pataa lagaana sambhav banaa diyaa. cumm tapman parr anuon (yaa unake hisse) key ghoornan koo jamaaya jaa sakata haye. ise seedhe scanning tunneling microscopy i dwara dekhaa jaa sakata haye.ee.sthirikaran koo uchch tapman parr ghoornan entropy dwara samajhaaya jaa sakata haye.
ekal anu star parr ghoornan uttejana kaa pratyaksh avalokan haal hii mein scanning tunneling microscope key saath asthir electron tunneling spectroscopy kaa upayog karke praapt kiya gayaa thaa. aanvik hydrogen aur usake samsthaanikon key ghoornan uttejana kaa pataa lagaaya gayaa.