Paka wa Schrödinger

Paka wa Schrödinger

Jamaa yangu mmoja alipenda kutukumbusha kwamba ukibobea sana kwenye electronics hapa kwetu bongo unaishia kubadili panel za computer na kuwa fundi saa...sijui hilo lina ukweli kiasi gani.

Kuna ukweli flani. Mi nilikuja kuhama kwa kina Schrondinger na kuhamia fani nyingine. Kwa sasa allahhamdulilah sipo haba!
 
Kuna ukweli flani. Mi nilikuja kuhama kwa kina Schrondinger na kuhamia fani nyingine. Kwa sasa allahhamdulilah sipo haba!

Ndio maana kila mtu bongo anataka kuwa mwanasiasa. Kuna rafiki yangu mmoja kaanzisha kombe la nini sijui na ameanza kutoa misaada huku na kule nadhani ndio strategy zenyewe hizo.
 
Somo zuri ila mpaka ushibe ukisoma na njaa unalala
 
Somo zuri ila mpaka ushibe ukisoma na njaa unalala

Nikianza kukumbuka masuala ya schrödinger eqn, hamiltonion operator, azimuthal & colatitude, behaviour of particles in an atom, hua napata kichefu chefu....

Duh,umenikumbusha Schrodinger equation wakati nipo form six. Li equation li refu hilo. Kuli kremu lazima unywe maji mengi kwanza!

Mkuu utafiti , nimecheka sana lakini kutokana na maoni ya wanaintelejensia wengine inawezekana kuna correlation fulani ya mambo haya na empty stomach! 🙂
 
Mkuu Monstgala hii mada kwangu imekuwa ngumu kumeza, ila nakubaliana na nondo zako kiongozi. Hebu rushia zile liquid kiasi japo tuende sawa

Mkuu dudupori , ni kweli hii field ni ngumu hiyo ni wazi. Lakini unapoelewa phenomenon fulani inaweza hata kusaidia katika uelewa au reasoning katika mambo mengine ambayo si lazima yahusiane moja kwa moja na field hiyo. Quantum mechanics ni inaelezea tabia zisizokuwa za kawaida za vitu vinavyounda maada ikiwemo sisi viumbe.

Ubongo wa binadamu unajenga uelewa kwa kiasi fulani kutokana na kile unachoona au kuexperince. Unapokuwa “ more informed” katika mambo mbali mbali basi uchaguzi wako na uamuzi wako utakuwa umejengwa kwenye misingi thabiti zaidi ya kupima na jinsi utakavyoleta faida au madhara kwa wewe mwenyewe au jamii yako.

Ni kweli kwamba katika quantum mechanics, kuangalia phenomenon kunaweza kubadilisha matokeo yake. Majaribio (experiments) ya kupima tabia ya electrons yatakuwa na matokeo tofauti pale yanapoangaliwa. Tabia za electrons zinakuwa kama zinabadilika pale tu mtu (observer) anapokuwa anaangalia. Usipokuwa unaangalia electrons hizi zina-behave tofauti. Hii inapelekea wataalam kufikia hitimisho kuwa hivi vitu vinaweza kuwa mawimbi (waves) na pia chembe (particles).

Ufafanuzi wa tabia hii unahusisha ukweli kwamba hatuwezi kuona bila mwanga, na pale tunapomulika hivi vitu tayari tunakuwa tumeingilia mwenendo wake. Mwanga una photons ambazo nazo ni particles hivyo zinagonga ile electron katika orbit yake na kuibadilisha uelekeo wake na speed.

Hii ni tofauti sana na tabia ya vitu katika hali ya kawaida kwani unapoangalia au usipoangalia matokeo ni sawa tu. hivyo basi wakati classical mechanics inapunguza uhusika wetu wa matokeo ya tabia ya vitu vingine quantum mechanics inaongeza uhusika wa kila kitu pale tunapojaribu hata kuangalia tu. Ndio maana ya mfano wa paka yule wa Schrödinger ambapo observer anakuwa na uhusika katika matokeo.
 
Sawa mkuu,
ukiangalia maelezo ya awali kabisa, mleta uzi amezitaja possibilities 4


  1. Paka wa kwanza na wa pili wote wako hai (mabomu yote mawili hayakulipuka)
  2. Paka wa kwanza na wa pili wote wamekufa (mabomu yote mawili yamelipuka)
  3. Paka wa kwanza yuko hai na wa pili kafa (bomu moja limelipuka lingine halikulipuka)
  4. Paka wa kwanza kafa na wa pili yuko hai (bomu moja limelipuka lingine halikulipuka)
(based on classical mechanics)
na possibilities 2


  1. Paka wa kwanza yuko hai na wa pili kafa (bomu moja limelipuka lingine halikulipuka)
  2. Paka wa kwanza kafa na wa pili yuko hai (bomu moja limelipuka lingine halikulipuka)

(based on quantum mechanics)

Sasa swali langu lilikuwa kwanini nne(4) katika classical na mbili(2) katika quantum mechanics?
Sijui kama nimeeleweka.
Naomba pia Kiranga apite huku.


Naweza kuelezea lkn kwanza ningependa kujua uzoefu wako na Quantum physics unakwenda mbali kwa kiasi gani, naomba usinielewe vibaya kwa swali langu kwani napenda kujua ni/tuanzie wapi, Je una uzoefu mzuri na Heisenberg Uncertainty principle (dxdp> h(planck contant)/2? kwa maana uelewa mzuri wa Schrödinger cat's experiment unaanzia hapo!
 
Naweza kuelezea lkn kwanza ningependa kujua uzoefu wako na Quantum physics unakwenda mbali kwa kiasi gani, naomba usinielewe vibaya kwa swali langu kwani napenda kujua ni/tuanzie wapi, Je una uzoefu mzuri na Heisenberg Uncertainty principle (dxdp> h(planck contant)/2? kwa maana uelewa mzuri wa Schrödinger cat's experiment unaanzia hapo!

Mkuu, twende kazi nikikwama nitakwambia. Ufahamu wangu ni wa kawaida tu ila Heisenberg nimemsoma.
 
Mkuu, twende kazi nikikwama nitakwambia. Ufahamu wangu ni wa kawaida tu ila Heisenberg nimemsoma.

Basi kama umeelewa uncertainty principle nafikiri unaweza kuelezea hilo kwa kutumia uncertainty principle kwamba huwezi kuwa na states mbili zinazofanana kwa wakati mmoja yaani huwezi kuwa na paka wote wawili wakiwa hai au wote wawili wakiwa wamekufa kwa maana uncertainty principle inakataza, kwenye uncertainty principal huwezi kujua position ya particle (dx) na wakati huo huo pia ukajua momentum yake (dp), kama ukiweza kumeasure momentum ya particle basi position yake inasambaa all over space kwa maana nyingine position ya particle haiwezi kuwa localised, na hii unaweza kucalculate position ya particle kwa kutumia Gausian wave function ya kuthibitisha hilo!

nafikiri hiyo ndiyo sababu, ya kwamba huwezi kuwa na states ya vitu viwili vinavyofanana kwa wakati mmoja, yaani paka wote wawe hai na au wote wawe wamekufa, ni lazima mmoja awe hai na mwingine awe amekufa ama kinyume chake kwa kutumia uncertainty principle, hivyo ndivyo ninavyofikiri kwamba ni sababu, sasa classical physics uncertainty principle haihusiki hivyo kila kitu kinawezekana kwa maana classically unadili na vitu macroscopic au vitu vyote vinavyoonekan kwa macho!
 
Mkuu Tiger technically quantum superposition inahusiana na swali lako:Why only two possibilities katika quantum state instead of many from the combination of other possibilities of classical mechanics.

Kwa two states (paka wawili wa Schödinger)

Paul Dirac katika superposition principle anasema: if two or more quantum states correspond to situations that are physically possible, so does their superposition.

"The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process.

It will never be different from both a and b [i.e, either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states."




Albert Einstein aliwahikusema kama mtu unafahamu kitu fulani, unatakiwa uwe na uwezo wa kumsimulia bibi yako na akakuelewa, otherwise hukielewi.
This implies that you have to use as simple language as your bibi can understand.
Ukiniwekea tu principles za quantum superposition hutokuwa umenisadia mkuu. Naamini kuna watu wengine wanajiuliza swali kama nililouliza na wangependa kufahamishwa.
 
Basi kama umeelewa uncertainty principle nafikiri unaweza kuelezea hilo kwa kutumia uncertainty principle kwamba huwezi kuwa na states mbili zinazofanana kwa wakati mmoja yaani huwezi kuwa na paka wote wawili wakiwa hai au wote wawili wakiwa wamekufa kwa maana uncertainty principle inakataza, kwenye uncertainty principal huwezi kujua position ya particle (dx) na wakati huo huo pia ukajua momentum yake (dp), kama ukiweza kumeasure momentum ya particle basi position yake inasambaa all over space kwa maana nyingine position ya particle haiwezi kuwa localised, na hii unaweza kucalculate position ya particle kwa kutumia Gausian wave function ya kuthibitisha hilo!

nafikiri hiyo ndiyo sababu, ya kwamba huwezi kuwa na states ya vitu viwili vinavyofanana kwa wakati mmoja, yaani paka wote wawe hai na au wote wawe wamekufa, ni lazima mmoja awe hai na mwingine awe amekufa ama kinyume chake kwa kutumia uncertainty principle, hivyo ndivyo ninavyofikiri kwamba ni sababu, sasa classical physics uncertainty principle haihusiki hivyo kila kitu kinawezekana kwa maana classically unadili na vitu macroscopic au vitu vyote vinavyoonekan kwa macho!

Bado Sijakupata mkuu.
Hawa ni paka wawili katika mabox mawili na mabomu mawili tofauti. Ninachotaka kufahamishwa ni namna gani quantum mechanics inaweza kuelezea namna paka hao wawili wanavyohusina hata tuwafananishe na wave-particle duality characteristic.
 
Bado Sijakupata mkuu.
Hawa ni paka wawili katika mabox mawili na mabomu mawili tofauti. Ninachotaka kufahamishwa ni namna gani quantum mechanics inaweza kuelezea namna paka hao wawili wanavyohusina hata tuwafananishe na wave-particle duality characteristic.

mwe! mwe!....sisemi kitu mkuu.
 
Kaka hii kitu ngumu watu kuelewa lakini kama kuna mtu anafuatilia kidogo technologia mpya anakuwa ameshasoma sehemu.

Sasa hivi teleportation inawezekana kuna research zimeweza kabisa kufanya teleportation lakini kuna vitu muhimu sana vya kuangalia kabla hawajaingiza hii technolojia sokoni
Ethics
Lazima kama binadamu tujiakikishie kuwa hakutakuwa na matatizo kwenye kutumia hii technolojia, kwa mfano unapoteport mtu teleportation ya kwanza (sender) tafanya nini yule original baada ya kusoma quatam basic zaqke na kuzituma kwenye teleporation ya pili (Receiver). Tusipoangalia tutakuwa tunazalisha watu kwa kutumia mashine, na technolojia hii ikiingia mikononi mwa wahuni dunia imeisha. Lazima mamlaka za dunia ziakikishe hakuna tatizo la ethics kwenye kudestroy original ili kuendelea kuwa na mtu mmoja badala ya badala ya kuwa na wa pili atakayekuwa anabehave sawasawa na wa kwanza maana wametengenezwa kwa quantam basic zilizo sawa mpaka DNA

Errors and Omission
Imagine mashine a quatam ya kwanza inakosea kwenye kuread basics za mtu na undestroy original, then yule mtu kule anatokea hana kiungo kimoja, this will be a disaster, so watengenezaji wanajitahi kuahikikisha kuwa hii haitatokea kwani inaweza kuleta matatizo makubwa kwenye jamii kuwa na mtu mwenye makosa kama haya anaweza kuwa ni siraha inayotembea.

vitu vidogo vidogo ambavyo havihusiani na maisha ya wanyama kwa ujumla vinasafirishwa kwa njia hii sema wanaimprove uwezo wa mzigo na najua hii inawezeana kwani ukiweza kusafilisha kitu kidogo then kuongeza uzito ni rahisi

X. Zhou, D. Leung, I. Chuang Quantum gate constructions from teleportation-like primiitve

G. Brassard Physica D 120 43-47 (1998) Teleportation as a quantum computations
 
Bado Sijakupata mkuu.
Hawa ni paka wawili katika mabox mawili na mabomu mawili tofauti. Ninachotaka kufahamishwa ni namna gani quantum mechanics inaweza kuelezea namna paka hao wawili wanavyohusina hata tuwafananishe na wave-particle duality characteristic.

Hii imekaa kifalsafa zaidi kumbuka kwamba schödinger's cat exp ilikuwa ni thought expt!

Ninavyohisi unaweza kuilezea kwa kutumia double slit expt na uncertainty principal, yaani unapokuwa na matundu mawili na kutuma mwanga au light ktk hayo matundu (kwa wakati mmoja) halafu ukaweza kurekodi pattern za hizo particles hauwezi kujua ni particle ipi imetokea tundu gani ktk QM, lkn kwenye C.Physics hilo linawezekana unaweza kujua kwa kutumia intereference pattern utakayoipata imetokea tundu gani nafikiri kila kitu kipo hapo, kama ukiifikiria double slit expt vizuri pmj na Uncertaity principal nafikiri kuna uwezekano mkubwa wa kuelewa kinachoendelea hapo!
 
Albert Einstein aliwahikusema kama mtu unafahamu kitu fulani, unatakiwa uwe na uwezo wa kumsimulia bibi yako na akakuelewa, otherwise hukielewi.
This implies that you have to use as simple language as your bibi can understand.
Ukiniwekea tu principles za quantum superposition hutokuwa umenisadia mkuu. Naamini kuna watu wengine wanajiuliza swali kama nililouliza na wangependa kufahamishwa.

Tiger nakubaliana nawe, asante. Ufafanuzi wangu kwa bibi yangu licha ya lugha nyepesi pia utategemea pia yeye anajua nini kuhusu tunachoongelea. Kama atakuwa tayari anajua mambo mengi katika eneo hilo basi nitaenda hatua za mbele kidogo lakini kama hatakuwa hajui chochote katika eneo hilo basi kuna haja ya kufafanua kwanza misingi yake na kuuelezea kanuni ambazo kwa hizi matokeo yanakuwa hivi.

Nashukuru pia approach ya kijakazi wa II inaleta majibu muafaka. One thing for sure, lazima kuwe na effort za yule anayeelezewa otherwise ni kazi bure. Kwenye reply post ya Paul Dirac kuna link. Kama bibi akiniambia haelewi hivyo pia ninachomweleza sikielewi hiyo nayo inaweza kuwa nafuu kidogo kwangu. Niels Bohr anasema yeyote ambae hastushwi na Quantum theory basi hajaielewa.
 
Basi kama umeelewa uncertainty principle nafikiri unaweza kuelezea hilo kwa kutumia uncertainty principle kwamba huwezi kuwa na states mbili zinazofanana kwa wakati mmoja yaani huwezi kuwa na paka wote wawili wakiwa hai au wote wawili wakiwa wamekufa kwa maana uncertainty principle inakataza, kwenye uncertainty principal huwezi kujua position ya particle (dx) na wakati huo huo pia ukajua momentum yake (dp), kama ukiweza kumeasure momentum ya particle basi position yake inasambaa all over space kwa maana nyingine position ya particle haiwezi kuwa localised, na hii unaweza kucalculate position ya particle kwa kutumia Gausian wave function ya kuthibitisha hilo!

nafikiri hiyo ndiyo sababu, ya kwamba huwezi kuwa na states ya vitu viwili vinavyofanana kwa wakati mmoja, yaani paka wote wawe hai na au wote wawe wamekufa, ni lazima mmoja awe hai na mwingine awe amekufa ama kinyume chake kwa kutumia uncertainty principle, hivyo ndivyo ninavyofikiri kwamba ni sababu, sasa classical physics uncertainty principle haihusiki hivyo kila kitu kinawezekana kwa maana classically unadili na vitu macroscopic au vitu vyote vinavyoonekan kwa macho!

First off, Schrodinger's Cat experiment deals with one cat in two states, not two cats in two states. Let's get that straight right off the bat. How anybody sees two cats beats me.

Actually, Heisenberg's uncertainty principle ndiyo inayosababisha kuwepo kwa states mbili na paka akawa kwa wakati mmoja huo huo yupo mzima na kafa, kama kwenye double slit experiment.

Uncertainty principle inakataza kuwa na specific momentum na position at any particular time, hivyo inaruhusu kuwa na a wave of probability regarding either, hivyo inaruhusu particle moja "kuwa" sehemu mbili kwa "wakati" mmoja, hivyo, by way of analogy, inaruhusu paka kuwa mzima na kafa kwa wakati mmoja.

"Wakati mmoja" of course is an entirely fictional concept as exposed by Einstein's relativity. As relativity revealed, there is no.such thing as "wakati mmoja", time is relative. If you really think deeply about the confusion.behind Schrodinger's Cat experiment, it stems.from.our classical mechanics view of the quantum world. We think "paka hawezi kuwa mzima na kafa kwa wakati mmoja huo huo" wakati there is no such thing as "wakati huo huo". This is a mere bias of the large scale universe. To a particle moving at the speed of light, the universe is timeless, time.stands still for it.

So, to that particle for example, everything happens at once and being in two positiins at the same time is what is normal. The particle cannot be only at one point at any particular time because all of time, from its perspective, is a single instant.

Here, uncertainty principle is the cause for the cat to be both dead and alive at the same time, not the limitation towards that end.

A general understanding of quantum entanglement, the Schrodinger's Cat experiment, Einstein's Relativity, the Double Slit experiment and Heisenberg's Uncertainty Principle will correlate the above.
 
First off, Schrodinger's Cat experiment deals with one cat in two states, not two cats in two states. Let's get that straight right off the bat. How anybody sees two cats beats me.

Actually, Heisenberg's uncertainty principle ndiyo inayosababisha kuwepo kwa states mbili na paka akawa kwa wakati mmoja huo huo yupo mzima na kafa, kama kwenye double slit experiment.

Uncertainty principle inakataza kuwa na specific momentum na position at any particular time, hivyo inaruhusu kuwa na a wave of probability regarding either, hivyo inaruhusu particle moja "kuwa" sehemu mbili kwa "wakati" mmoja, hivyo, by way of analogy, inaruhusu paka kuwa mzima na kafa kwa wakati mmoja.

"Wakati mmoja" of course is an entirely fictional concept as exposed by Einstein's relativity. As relativity revealed, there is no.such thing as "wakati mmoja", time is relative. If you really think deeply about the confusion.behind Schrodinger's Cat experiment, it stems.from.our classical mechanics view of the quantum world. We think "paka hawezi kuwa mzima na kafa kwa wakati mmoja huo huo" wakati there is no such thing as "wakati huo huo". This is a mere bias of the large scale universe. To a particle moving at the speed of light, the universe is timeless, time.stands still for it.

So, to that particle for example, everything happens at once and being in two positiins at the same time is what is normal. The particle cannot be only at one point at any particular time because all of time, from its perspective, is a single instant.

Here, uncertainty principle is the cause for the cat to be both dead and alive at the same time, not the limitation towards that end.

A general understanding of the Schrodinger's Cat experiment, Einstein's Relativity, the Double.Slit experiment and Heisenberg's Uncertainty Principle will correlate the above.

Kiranga. I agree, thanks. Lakini unapotaka kuelezea quantum entanglement baada ya superposition (i.e one particle at two quantum states), hapa tayari utakuwa unaongelea two particles (i.e. two intertwined electrons). Sasa kama unataka kubaki kwenye analogy ile ile ili usiwaache sana wanaokusoma (kama intention ni consideration ya wale ambao hawaijui QM kabisa, Mkuu Tiger katoa mfano wa bibi) Inakuwa muhimu kuexpand analogy yule paka mmoja wawe wawili ili sasa "two particles in two quantum states" (entanglement) zielezeke kirahisi...

Unalionaje hilo?
 
Kiranga. I agree, thanks. Lakini unapotaka kuelezea quantum entanglement baada ya superposition (i.e one particle at two quantum states), hapa tayari utakuwa unaongelea two particles (i.e. electrons). Sasa kama unataka kubaki kwenye analogy ile ile ili usiwaache sana wanaokusoma (kama intention ni consideration ya wale ambao hawaijui QM kabisa, Mkuu Tiger katoa mfano wa bibi) Inakuwa muhimu kuexpand analogy yule paka mmoja wawe wawili ili two particles in two quantum states zielezeke kirahisi...

Unalionaje hilo?

Ukifikiri one particle at two quantum states ni two particles hujaanza kuielewa quantum mechanics.

There is a reason it is called superposition, superposition means one particles exhibits two states (or more than one, essentially all its possible states) apparently "simultaneously".

So this is a case of one particle having many states "at the same time".

Tofautisha a particles (which may be in many states) na states za hizo particle.

There is a reason the experiment is called "The Schrodinger's Cat Experiment" and not "The Schrodinger's Cats Experiment".

There is only one cat, even after superposition. This is why it is so puzzling to us.

Had it been two cats, nothing about one being dead and another alive is puzzling.
 
Ukifikiri one particle at two quantum states ni two particles hujaanza kuielewa quantum mechanics.

There is a reason it is called superposition, superposition means one particles exhibits two states (or more than one, essentially all its possible states) apparently "simultaneously".

So this is a case of one particle having many states "at the same time".

Tofautisha a particles (which may be in many states) na states za hizo particle.

There is a reason the experiment is called "The Schrodinger's Cat Experiment" and not "The Schrodinger's Cats Experiment".

There is only one cat, even after superposition. This is why it is so puzzling to us.

Had it been two cats, nothing about one being dead and another alive is puzzling.

Mkuu hujanipata. Baada ya superposition ambayo ndio haswa "one particle exhibits two states (or more than one, essentially all its possible states) apparently "simultaneously".

Sasa angalia Quantum entanglement alafu utajua namaanisha nini! That thought experiment is expanded(not by schrödinger) on two electrons which are entangled. Behaviors ya hizi mbili ndio mifano ya paka wawili wenye conditions sawa na ile ya mfano wa kwanza (original).
 
First off, Schrodinger's Cat experiment deals with one cat in two states, not two cats in two states. Let's get that straight right off the bat. How anybody sees two cats beats me.

Actually, Heisenberg's uncertainty principle ndiyo inayosababisha kuwepo kwa states mbili na paka akawa kwa wakati mmoja huo huo yupo mzima na kafa, kama kwenye double slit experiment.

Uncertainty principle inakataza kuwa na specific momentum na position at any particular time, hivyo inaruhusu kuwa na a wave of probability regarding either, hivyo inaruhusu particle moja "kuwa" sehemu mbili kwa "wakati" mmoja, hivyo, by way of analogy, inaruhusu paka kuwa mzima na kafa kwa wakati mmoja.

"Wakati mmoja" of course is an entirely fictional concept as exposed by Einstein's relativity. As relativity revealed, there is no.such thing as "wakati mmoja", time is relative. If you really think deeply about the confusion.behind Schrodinger's Cat experiment, it stems.from.our classical mechanics view of the quantum world. We think "paka hawezi kuwa mzima na kafa kwa wakati mmoja huo huo" wakati there is no such thing as "wakati huo huo". This is a mere bias of the large scale universe. To a particle moving at the speed of light, the universe is timeless, time.stands still for it.

So, to that particle for example, everything happens at once and being in two positiins at the same time is what is normal. The particle cannot be only at one point at any particular time because all of time, from its perspective, is a single instant.

Here, uncertainty principle is the cause for the cat to be both dead and alive at the same time, not the limitation towards that end.

A general understanding of quantum entanglement, the Schrodinger's Cat experiment, Einstein's Relativity, the Double Slit experiment and Heisenberg's Uncertainty Principle will correlate the above.

Poa kwa kuliona hilo, sikulifikiria sana kuhusu kama kulikuwa na Paka wawili ktk masanduku mawili tofauti, nilichojaribu ilikuwa tu kutafuta jinsi ya kuelezea hiyo thought experiment kwa njia rahisi, ambapo kwa upande wangu nilifikiri kama mtu akielewa vizuri uncertainty principal na Double slit expt anaweza kuwa angalau na pa kuanzia, na kama Mtu haelewi uncertainty principal basi haina hata maana kuendelea kujadili hili kwani kuna pengo kubwa sana la kuziba kwanza!

Lkn umetoa maelezo mazuri!
 
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