《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ࡄ޲໼ഞ௺๡ʇখऄɸʪ‫ػ‬ʊ੆ɸʪಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʇࠏϾС‫޶ޟ‬൥৹ϊ୔੝ӌ੝ӌϙ ՂԆ๽‫ۑ‬ӌٰ֖Ѡ ୕޶‫˲˘ˋˉہ‬ӌদ۞‫߾ٰ֖઺ޗ‬‫ܩ‬π ວോ

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

෾ߣ

ਫ਼R࣍

ʎɷʠʊ

R

ਫ਼k࣍

૨С‫߲ح޶ޟ‬jkXR

GBQmpBHH2 ‫؃‬Ԩ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

jkXk૨С‫ ޶ޟ‬X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

9kXj

GBQmpBHH2@pQM L2mKMM ൣପ߲ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

8

kX9

௺଺छ‫ ؃‬X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X XeࠏϾС‫޶ޟ‬൥

3

jXR

ࠏϾС‫޶ޟ‬൥ʇ࡬ต෋୩С‫ ޶ޟ‬X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

3

jXk

গஞୈҟ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

RR

jXj

_2b2`pBQ` KQ/2H ʆʍ҈উ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

Rkಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উ

R9

9XR

ˡ˱˽˚˝ʴ̅ʇߢԨౙୈൣପ߲ X X X X X X X X X X X X X X X X X X X X X X X X X X X X

R9

9Xk

୑౬Ԫॐʇ߭‫˽˟ʺڔ‬ˀƪ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

Re

9Xj

PM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʇɮʩɲʞࣰٛ X X X X X X X X X X X X X X X X X X X X X X X X

Rd

9X9

ࠏϾС‫޶ޟ‬൥ʇʍಐӇ X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X

RN

9X8

ɮʩɲʞࣰٛʍূʒൣʊʧʪܿε X X X X X X X X X X X X X X X X X X X X X X X X X X X X

ky

ʝʇʠ

kR

ਫ਼j࣍

ਫ਼9࣍

ਫ਼8࣍
‫ޖ‬۵ഞ‫ٯ‬k9

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ R ࣍ ʎɷʠʊ

ਫ਼R࣍

ʎɷʠʊ
࣪ʍສ޶໼ʊઽࠄʉಝഥ‫۽‬๽໼ʎ‫ە‬౹Ωʊ଼жњఉʉ೩ഽ଺ʉ༄ৠʞʱ฿ɧʪʡʍʆɡʩƐɼʍ౹Ωʎഐ१ഐ๽Ɣ‫ػ࡬ד‬ഐ๽ʊʇʈʝʨɹƐয়ຑ޶໼ʣϟ઼໼Ɛж๑ഒฐʆʎສ޶‫ہ‬ӌʣສ޶˙ˢʶˋঙ‫ك‬ʝʆ֊ʕƑɼʍφൣʆƐ࣪ʍສ޶໼ʆ՝ࡲɴʫʪ‫ػ‬ʍಝഥ‫۽‬ѷପʍ๽໼ʎӂງɴʫʅɩʨɹƐٰ֖ʍ฾ણɫਵɮɡʪƑສ޶໼ʊՂʄɣɾಝഥ‫۽‬ѷପʍ҈উʇɶʅສ޶ҟൢ‫ػ‬ʊ੆ɸʪສ޶˰ˋˑƪൣପ߲ʍൣ൥ɫӂງɴʫʅɣʪƑɲʫʎਵ੄ສ޶ອӌʱՂ৛ʊɶɾൣ൥ʆƐഐ१ഐ๽ʊ‫ڌ‬ʨɹສ޶࣮൙ʉʈʍഒฐʆʡ‫ە‬ɮ๑ɣʨʫʪƑສ޶ҟൢ‫ػ‬ʆ‫׋‬ළɫɡʪʍʎԖֻ‫ػ‬ʇਂ‫ݴک‬๑ɸʪુ෾‫ػ‬ʍԝ໾ѷପʆɡʩƐɲʍ˒ʶ˜˱ˁˋʱ՝ࡲɸʪສ޶˰ˋˑƪൣପ߲ʱ੄‫଺ػ‬ʊ஡ࡰɸʪൣ൥ʍφʃʇɶʅƐLFDBK- wrMxB; ʨʧʪࠏϾС‫޶ޟ‬൥ (RĜ9) ɫɡʪƑࠏϾС‫޶ޟ‬൥ʆʎৌ੄‫ػ‬ɪʨԖֻ‫ػ‬ʍ࣮൙ʱࣁ֞ɸʪࠏϾС‫޶ޟ‬ʱ஡௬ɶƐુ෾‫ػ‬ʍ࡬ต෋୩С‫޶ޟ‬ʍߢԨౙୈʱࠪʩࡰɸɲʇʆສ޶˰ˋˑƪൣପ߲ʱமʪɲʇɫʆɬʪƑਵ੄ສ޶ອӌʆʎʉɮ࣪ʍສ޶໼ʊՂʄɣɾߒʞʇɶʅʎƐE2H/vb?@a+?rBM;2` ൥ʇʡ࣌ɴʫʪ

*HQb2/ iBK2 Ti? ൥ U*hSV (8Ĝd) ʣಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b Uಝഥ‫ ۽‬h6.V (3) ɫɡʪƑ*hSʇಝഥ‫ ۽‬h6. ʎʇʡʊ௺ഥ‫ػ۽‬ʊ੆ɸʪ࣪ʍສ޶໼ U௺࣪ʍສ޶໼V ʱՂಁʇɶƐಝഥ‫۽‬ѷପʝʆ՝ࡲ
ʆɬʪʧɥʊҼ૗ʱɶɾʡʍʆɡʪɫƐຜࠖʍಝഥ‫۽‬ʍ՝ࡲʍީൣʊʎνɣɫɡʪƑ*hS ʆʎ෋୩С‫޶ޟ‬ʍߢԨౙୈʱ๛ʊΑʮɹʊਂ‫؃‬ԨࣣʍഒೣԪॐʍߢԨౙୈʱ‫ޟك‬ɸʪƑɲʍʇɬƐ࣪ʍສ޶໼ʆච
ࠃ଺ʉดӘʆɡʪʎɹʍࢀຑ޶ʍʺ˟˽ˀƪʎ෢ӂʊଜձɴʫʉɣƑφൣʆಝഥ‫ ۽‬h6. ʎ෋୩С‫޶ޟ‬ʍߢԨౙୈʱߢԨΧਮ௺଺ "Q;QHBm#Qp ഷԋʊʧʂʅࠪʩΑɥ‫߲ح‬ʇʉʂʅɣʅƐಝഥ‫ׅࣳ۽‬ђʆɡʂʅʡࢀຑ޶ʺ˟˽ˀƪஉɫ෢ӂʊଜձɴʫʅɣʪƑɼʍற૙ɪʨຑ޶೉ਔʍഷѓʱ౦ɥਂ୎ζʱ՝ࡲʆɬʪʇՎ੊ɴʫʅɩʩƐɲʫʝʆʊಝഥ‫ ۽‬h6. ɫ๑ɣʨʫɾແʇɶʅƐໂց‫ػ޶ڀ‬ʆ "Qb2@1BMbi2BM‫࡬ד‬ɫ՟ɬɾʇɬʍ‫ਂ࡬ד‬ʇಝ‫࡬ד‬ໃ՟ʍߢԨഷѓʇສ޶ยਏൣପ߲ʍ஍φ଺ʉ஡ࡰ (N- Ry) ɫɡʪƑɲʍʧɥʉ๸୐ʱߡʃಝഥ‫ ۽‬h6. ʆɡʪɫƐಝ߭෢ʉ೼ഒɫިɴʫʅɣʪƑಝഥ‫ ۽‬h6. ʆʎഥ‫ػ۽‬ɪʨҼ૗ʊʧʩຑ޶ॐʍߢԨഷѓഒஉʊΧਮɶɾʽʸ̅ˑƪ‫܈‬ɫಝগஞˡ˱˽˚˝ʴ̅ʊ‫ڇ‬ʫʪƑɼʍʽʸ̅ˑƪ‫܈‬ʎ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʊ੆ɸʪɮʩɲʞࣰٛʱѳɸɲʇʆٔଜɴʫʪɫƐɮʩɲʞࣰٛ
ʍଜʠൣʎφίʆʎʉɮƐ଼ঔʉɮʩɲʞࣰٛʱূʒࡰɸ಴๗ɫɡʪƑɼʍূʒൣʎෆɿʊٔહɶʅɩ
ʨɹ๽໼ʱ‫ۥ‬२ɸʪࣣʆʍѳ੠ʇʉʂʅɣʪƑறʊສ޶ҟൢ‫ػ‬ʊ੆ɶʅʈʍʧɥʉɮʩɲʞࣰٛʱূʕ
ʘɬɪʎ෢ʨɪʆʎʉɣƑறʊ‫ׯ‬௻Ɛࢀຑ޶ʺ˟˽ˀƪʱഉয়ॐʊࠪʪʧɥʉɮʩɲʞࣰٛɫଥΠɴʫʅɣʪ (RR)Ƒສ޶໼ʆʎ

R

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ R ࣍ ʎɷʠʊС‫޶ޟ‬ʍՎ੊ડɫഐ๽ສʍԣਢડʊ੆жɸʪɾʠʊƐɼʍԣਢડɫࠄʆɡʪɲʇʱ౩Ͽɶʅഐ๽ສʊ੆
жɸʪС‫޶ޟ‬ɫʺ˽˱ƪ˚С‫޶ޟ‬ʆɡʪɲʇɫ๗ॻɴʫʪƑɼɲʆࢀຑ޶ʺ˟˽ˀƪʱഉয়ॐʊࠪʪʇ
ಝগஞˡ˱˽˚˝ʴ̅ɫಝʺ˽˱ƪ˚С‫޶ޟ‬ʊʉʂʅɶʝɥƑɼʍɾʠ૾࣭ʍ࣪ʍສ޶໼ʆʎʺ˟˽
ˀƪʽʸ̅ˑƪ‫܈‬ʇɶʅࠪʩ௬ʫʪࢀຑ޶ʺ˟˽ˀƪʍൊ९ʎࠄॐʊ‫ڌ‬ଜɴʫʪƑɶɪɶƐಝഥ‫ ۽‬h6.
ʆʎໂց‫ػ޶ڀ‬ʍԝ໾ѷପʍॐડ҈উʊɩɣʅࢀຑ޶ʺ˟˽ˀƪʍൊ९ʱࠄॐʊ‫ڌ‬ଜɸʪʇํƧʉ೜୤܏ɫ՟ɷʪɲʇɫ൙‫ܘ‬ɴʫʅɣʪ (RR- Rk)Ƒແʇɶʅ .Qm#H2@r2HH KQ/2H ʆഥ‫ੌࣳ۽‬ʗʍԝ໾ɫ՟ɲʨ
ʉɣɲʇʣƐ"Qb2@>m##`/ KQ/2H ʆߢԨౙୈ‫گ‬ʍຑ޶ॐɫೱʍॐʇʉʂʅɶʝɥɲʇɫ֣ɱʨʫʪƑɲʫʨʍฆ੠ʱ҈ٔɸʪɾʠʊʎƐࢀຑ޶ʺ˟˽ˀƪʍൊ९ʊ֥ॐʡ԰ʟʧɥʉɮʩɲʞʱɸʫʏʧɣ
ɲʇɫॐડ҈উʊʧʂʅӂ௳ɴʫʅɣʪ (RR- Rk)Ƒॐડ҈উʍٗѢɼʍʡʍʎࢀຑ޶ʺ˟˽ˀƪʱഉয়ॐʊࠪʪɮʩɲʞʱ޽ߡɶʅɣʪʧɥʊ޻ɧʪɫƐɼʍഐ๽଺ʉίළɫ෢ӂʆʎʉɣɾʠʊɮʩɲʞࣰ
ٛʍূʒൣʇɶʅٔહɸʪʊߎʂʅɣʉɣƑɼɲʆච໼ഞʆʎສ޶ҟൢ‫ػ‬ʱΑɥࠏϾС‫޶ޟ‬൥ʇಝഥ‫ ۽‬h6. ʱಐӇɸʪɲʇʆƐɮʩɲʞࣰٛʍ
ূʒൣʊʃɣʅ૦ʘʪƑறʊࠏϾС‫޶ޟ‬൥ʆ՝ࡲɴʫʪ˒ʶ˜˱ˁˋʇடஉʍٗѢʱ஡ɮʧɥʉɮʩɲ
ʞࣰٛʍѳɶൣɫʈʍʧɥʉʡʍɪ૦ʘƐສ޶ҟൢ‫ػ‬ʊ੆ɸʪɮʩɲʞࣰٛʱ۵ɧʪࣣʆʍ‫ޖ‬۵ʇɶɾɣƑච໼ഞʍ‫ۥ‬२ʎߣʍ૾ʩʆɡʪƑਫ਼ k ࣍ʆʎಝഥ‫ ۽‬h6. ʣࠏϾС‫޶ޟ‬൥ʍ‫଺੄׿‬ʉմ໼ʊ಴๗ʉ૨С‫߲ح޶ޟ‬ʱ஡௬ɸʪƑಝഥ‫۽‬h6. ʣࠏϾС‫޶ޟ‬൥ʆʎࠩʊܴ܏ࣳੌʱΑɥɲʇʊʉʪɫƐ૨С‫߲ح޶ޟ‬ʱ๑ɣʪʇܴ܏ࣳੌՎ੊ડ
ʱɡɾɪʡࢃॄࣳੌՎ੊ડʍʧɥΑɥɲʇɫʆɬƐ‫ࣣޟك‬ʍ‫૾ٵ‬ɶɫʧɮʉʪƑʝɾಝഥ‫ ۽‬h6. ʍଜ߲ѓʊʡ੝ɬɮԪʮʪɾʠʊࡥ๗ʆɡʪ (Rj)Ƒ
ਫ਼ j ࣍ʆʎࠏϾС‫޶ޟ‬൥ʱ๑ɣɾ `2b2`pQB` KQ/2H ʍ҈উʊʃɣʅࡲʘʪƑࠏϾС‫޶ޟ‬൥ʱ๑ɣʪʇ
ુ෾‫ػ‬ʍ࡬ต෋୩С‫޶ޟ‬ʍߢԨౙୈൣପ߲ʱமʪɲʇɫʆɬƐɼʍߢԨౙୈС‫޶ޟ‬ʆɡʪอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʱগஞ‫ޟك‬ʊʧʂʅ֑ʠʪɲʇɫʆɬʪɲʇʱজ෢ɸʪƑ਩ɣʅਫ਼ 9 ࣍ʆʎಝഥ‫ ۽‬h6. ʱ๑ɣʅ `2b2`pQB` KQ/2H ʍ҈উʱ‫ۼ‬ɥƑಝഥ‫ ۽‬h6. ʆʎɮʩɲʞ
ࣰٛʱѳɸɲʇʆɮʩɲʝʫɾಝগஞˡ˱˽˚˝ʴ̅ɫٔଜɴʫʪɲʇʱজ෢ɶƐɼʍɮʩɲʝʫɾಝ
গஞˡ˱˽˚˝ʴ̅ʇਫ਼ j ࣍ʆ֑ʠɾอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʱਣɫɪʩʇɶʅƐಝഥ‫ ۽‬h6. ʇࠏϾС‫ޟ‬޶൥ʍಐӇʱ‫ۼ‬ɥƑ
ਫ਼ 8 ࣍ʆʎɲʫʨʍٗѢʊʃɣʅʍʝʇʠʱࡲʘʪƑk《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ k ࣍ ૨С‫߲ح޶ޟ‬ਫ਼k࣍૨С‫߲ح޶ޟ‬ච࣍ʆʎಝഥ‫ ۽‬h6. ʣࠏϾС‫޶ޟ‬൥ʍ‫଺੄׿‬ʉմ໼ʊ಴๗ʉ૨С‫߲ح޶ޟ‬ʊʃɣʅʍ
஡௬ʱ a+?Kmix (R9) ʣ LFKm` 2i HX (Rj) ʍൣ൥ʊ࡞ʂʅ‫ۼ‬ɥƑঢɹƐস‫ح‬С‫޶ޟ‬ʱ๗য়ʇɸʪ GBQmpBHH2 ‫؃‬Ԩʱ‫ۥ‬२ɶƐɼɲʆʍࣳੌ˫ˁ˚˽ʍ .B`+ ՝൥ʱ஡௬ɸʪƑɴʨʊ GBQmpBHH2 ‫؃‬Ԩࣣʊ‫ݴ‬๑ɸʪ૨С‫޶ޟ‬ʱ๑ɣʪɲʇʆƐGBQmpBHH2@pQM L2mKMM ൣପ߲ɫa+?`ƺ/BM;2` ൣପ߲ʍʧɥʊ࢑ɰʪɲʇʱߪɸƑʝɾƐ௺ഥ‫ػ۽‬ʍ෋୩С‫޶ޟ‬ɪʨ௺଺छ‫؃‬ʇ௺଺छ‫؃‬ʊ੆ɸʪॲ२ࣁ෩С‫޶ޟ‬ʱଜձɶƐɼʫʨʱʡʇʊ6Q+F ‫؃‬Ԩʱ‫ۥ‬२ʆɬʪɲʇ
ʱߪɸƑ

kXR

GBQmpBHH2 ‫؃‬Ԩ†ߣʍ‫ڼ‬ԋԪ‫ؤ‬ʱෂɾɸС‫ ޶ޟ‬ai (i = 1, 2, . . . ) ʇɼʍʺ˽˱ƪ˚ִด ai ʱ஡௬ɸʪƑai , a†j = δij ,[ai , aj ] = a†i , a†j = 0

UkXRVa ʊ੆ɶʅ a |0 = 0 ʱෂɾɸࣳੌ |0 ʱ a ʍछ‫؃‬ʇʧʕƑ|0 ʊ੆ɶʅ a† ʱ‫ؗ‬ʩ഼ɶ‫ݴ‬๑ɴɺ|n1 , n2 , . . . =

i

1  †  ni√|0ani ! i

UkXkVʍʧɥʊ‫ݴ‬ʨʫʪࣳੌ |n1 , n2 , . . . ʎ९՜૰‫ڼ‬Ԃৌ‫ػ‬ʱʉɸƑm1 , m2 , . . .|n1 , n2 , . . . =δ m i ni ,|n1 , n2 , . . .n1 , n2 , . . .| = 1

UkXjV

n1 ,n2 ,...

i|n1 , n2 , . . . ʱ |n ʇࣈຊɶʅ࢑ɮƑ{|n} ʱՂଞʇɸʪ‫؃‬Ԩ H ʎ >BH#2`i ‫؃‬ԨʇʉʪƑH ࣣʍস‫ح‬С‫ ޶ޟ‬O ʎO=|mm|O|nn| =Omn |mn|mn(Omn = m|O|n)

UkX9V

mn

ʇ࢑ɮɲʇɫʆɬʪƑɸʪʇС‫|{ ޶ޟ‬mn|} ʱՂଞʇɶʅС‫ ޶ޟ‬O ʱୈҟɶɾ‫ح‬ʊʉʂʅɣʪƑɲʍʧɥʊ H ࣣʍস‫ح‬С‫޶ޟ‬ʍ‫؃‬Ԩ H̄ ʱ۵ɧʪɲʇɫʆɬƐH̄ ʱ GBQmpBHH2 ‫؃‬Ԩʇ‫ڐ‬ʕƑH̄ ʍ‫ ٿ‬A, B ʊj《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ k ࣍ ૨С‫߲ح޶ޟ‬੆ɶʅഉয়ॐʱଜʠʪࠎਔ


(A, B)H̄ = i` A† B

A, B ∈ H̄

UkX8V

ʱ H̄ ʍௐঋʇɶʅ஡௬ɸʪƑɲʍʇɬ H̄ ʎ >BH#2Hi ‫؃‬ԨʇʉʩƐ{|mn|} ɫ९՜૰‫ڼ‬Ԃৌ‫ػ‬ʱ२ɸɲʇɫ(|mn| , |m n |)H̄ =δmi mi δni ni ,|mn| (|mn| , •)H̄ = I • (I ʎۙஉС‫)޶ޟ‬

UkXeV

m,n

i

ʇʉʪɲʇɪʨʮɪʪƑH̄ ʍ˫ˁ˚˽ʊ੆ɶʅʡ .B`+ ՝൥ UA → |A⟫ , |mn| → ||mn|⟫ = |m, n⟫ʉʈV ʱ๑ɣʪʇƐ⟪m, n|m , n ⟫ =δmi mi δni ni ,|m, n⟫⟪m, n| = I

UkXdV

m,n

i

ʇ࢑ɮɲʇɫʆɬʪƑkXk૨С‫޶ޟ‬

GBQmpBHH2 ‫؃‬Ԩ H̄ ɪʨ H̄ ʗʍস‫ࠎح‬ਔʱ૨С‫޶ޟ‬ʇ‫ڐ‬ʕƑ˓ʹ˕ˁ૨С‫ ޶ޟ‬ǎi ʇ˓˽˒૨С‫޶ޟ‬ãi ʱǎi |m, n⟫ = |ai |mn|⟫UkX3V

ãi |m, n⟫ = |mn| a†i ⟫

UkXNV

†

†

ʇଜʠʪƑɴʨʊ ǎi ʇ ãi ʍʺ˽˱ƪ˚ִดʱ ǎi ʇ ãi ʱ⟪m, n| ǎ†i = ⟪ai |mn||

UkXRyV⟪m, n| ã†i = ⟪|mn| a†i

UkXRRVʆଜʠʪƑɸʪʇ UkX3V- UkXNV- UkXRyV- UkXRRV ʇ UkX8V ɪʨǎ†i |m, n⟫ = a†i |mn|⟫

UkXRkVã†i |m, n⟫ = ||mn| ai ⟫UkXRjV††ʇʉʪƑΤࣣʍٗѢɪʨ૨С‫ ޶ޟ‬ǎi , ǎi , ãi , ãi ʎߣʍ‫ڼ‬ԋԪ‫ؤ‬ʱෂɾɸɲʇɫߪɴʫʪƑ



 

ǎi , ǎ†j = ãi , ã†j = δij ,(Qi?2`rBb2) = 0

UkXR9V‫گ‬ʍ‫ޟك‬ʍɾʠʊA(a, a† ) =

Cmn a†

m(a)n(Cmn ʎഉয়ॐ)m,n9

UkXR8V

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ k ࣍ ૨С‫߲ح޶ޟ‬ʆ೅ɴʫʪС‫ ޶ޟ‬A ʇ௰ίʍС‫ ޶ޟ‬B ʍঋɫƐ|B⟫ ʊ੆ɸʪ૨С‫޶ޟ‬ʍ‫ݴ‬๑ʇɶʅʈʍʧɥʊ೅‫ڇ‬ɴʫʪɪʱ֑ʠʪƑঢɹঋ AB ʱଜձʊɶɾɫʂʅഷ‫ح‬ɸʪʇ

|AB⟫ =

Cmn a†mn(a) B⟫ =m,nCmn ǎ†m(ǎ) |B⟫ = A(ǎ, ǎ† ) |B⟫n

UkXReVm,nʇʉʪƑɲʍʧɥʊ A ɫܾɪʨ B ʊ‫ݴ‬๑ɸʪ࣪܏ʎƐA ʍ઺तʱ a → ǎ, a† → ǎ† ʇ‫଺߲ح‬ʊ࢑ɬԋɧʅ‫ݴ‬ʨʫʪ૨С‫޶ޟ‬ɫ |B⟫ ʊ‫ݴ‬๑ɸʪʧɥʊ೅՝ʆɬʪƑடํʊঋ BA† ʊʃɣʅ֑ʠʪʇ

BA† ⟫ = B



∗
a†
Cmn

n

m

(a) ⟫ =

m,n



∗
ã†
Cmn

m

(ã) |B⟫ = A∗ (ã, ã† ) |B⟫n

UkXRdVm,nʇʉʪƑɾɿɶ A∗ ʎ A ʱୈҟɶɾʇɬʍɸʘʅʍ‫ؤ‬ॐʊʃɣʅഉয়ִดʱʇʂɾʡʍʱ೅ɸƑUkXReVʇ UkXRdV ɪʨߣʍ˓˽˒ִดਝɫ஡ɪʫʪƑǍ1 Ǎ2∼= Ã1 Ã2

UkXR3V∼

c1 Ǎ1 + c2 Ǎ2 = c∗1 Ã1 + c∗2 Ã2 ∼= Ã†1
Ǎ†1
 ∼
= Ǎ1
Ã1

UkXRNV
UkXkyVUkXkRV(Ǎ1 , Ǎ2 ʎ௰ίʍ૨С‫޶ޟ‬Ɛc1 , c2 ʎ௰ίʍഉয়ॐ)˓˽˒ִดਝʱ๑ɣʪʇ A∗ (ã, ã† ) ʇ A(ǎ, ǎ† ) ʍԪ‫ؤ‬ʎA∗ (ã, ã† ) = A(ǎ, ǎ† )∼

UkXkkVʇ࢑ɮɲʇɫʆɬʪʍʆ

BA† ⟫ = A(ǎ, ǎ† )

∼

|B⟫

UkXkjVʇʉʪƑɲʍʧɥʊ A† ɫ B ʊϞɪʨ‫ݴ‬๑ɸʪ࣪܏ʎƐA(ǎ, ǎ† ) ʍ˓˽˒ִดɫ |B⟫ ʊ‫ݴ‬๑ɸʪʧɥʊ
†೅՝ʆɬʪƑࣣʍজ෢ʆʎС‫ ޶ޟ‬A ɫ a, a† ʍʞʆ࢑ɪʫʪ࣪܏ʱΑʂɾɫƐA ɫ ai , ai (i = 1, 2, . . .)ʆ࢑ɪʫʪ࣪܏ʊʡԜઅʊҼ૗ʆɬʪƑkXj

GBQmpBHH2@pQM L2mKMM ൣପ߲ܴ܏ࣳੌʱ԰ʠɾφ౶଺ʉࣳੌʎ H ࣣʍস‫ح‬С‫޶ޟ‬ʆɡʪ෋୩С‫޶ޟ‬ʆ೅‫ڇ‬ɸʪɲʇɫʆɬʪƑ‫׿‬੄଺ʊʎƐࢃॄࣳੌ |φ1  , |φ2  , · · · ∈ H ʇࡥʞ {pk | 0 ≤ pk ≤ 1,

k pk = 1} ʱ๑ɣʪʇ෋୩С‫޶ޟ‬

ρ(t) ʎ
ρ(t) =

pk |φk (t)φk (t)|UkXk9Vkʇ࢑ɮɲʇɫʆɬƐɲʍʇɬഐ๽ສ O(a, a† ) ʍՎ੊ડʎ

i`[Oρ] = ⟪I|Oρ⟫ = ⟪I| Ǒ |ρ⟫

UkXk8VǑ = O(ǎ, ǎ† )UkXkeV8《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ k ࣍ ૨С‫߲ح޶ޟ‬ʇ‫ޟك‬ɴʫʪƑࣳੌ |φk  ʍߢԨౙୈʎ a+?`ƺ/BM;2` ൣପ߲i/|φk (t) = H |φk (t)/t

UkXkdVʆ՝ࡲɴʫʪʇɸʪƑɾɿɶƐɲʍˡ˱˽˚˝ʴ̅ H ʎС‫ ޶ޟ‬a, a† ʆ࢑ɪʫʅɣʪʇɸʪƑɸʪʇ෋୩С‫ ޶ޟ‬ρ(t) ʍߢԨౙୈʎ  ///
pk i |φk (t) φk (t)| + pk |φk (t) i φk (t)|i ρ(t) =
/t
/t
/t
k
 




pk |φk (t)φk (t)| −pk |φk (t)φk (t)| H †=Hkk= Hρ(t) − ρ(t)H†

UkXk3Vʊ࡞ɥɲʇɫʮɪʪƑɲʍߢԨౙୈൣପ߲ʎ GBQmpBHH2@pQM L2mKMM ൣପ߲ʇ‫ڐ‬ʏʫʪƑɲʍൣପ߲
ʎС‫޶ޟ‬ʍߢԨౙୈൣପ߲ʆɡʪʍʆƐ૨С‫߲ح޶ޟ‬ʱ๑ɣʪʇ‫૾ٵ‬ɶɫວɮʉʪƑɼɲʆƐUkXReV ʇ

UkXkjV ʍٗѢʱ๑ɣʪʇ
i/|ρ(t)⟫ = |Hρ(t)⟫ − ρ(t)H † ⟫/t= Ȟ |ρ(t)⟫ − H̃ |ρ(t)⟫UkXkNVʇ࢑ɮɲʇɫʆɬʪƑɲɲʆ Ȟ ʎ H ʍ઺तʱ a → ǎ, a† → ǎ† ʇ‫଺߲ح‬ʊ࢑ɬԋɧɾ૨С‫޶ޟ‬ʆɡʩƐ

H̃ ʎɼʍ˓˽˒ִดʆɡʪƑɲɲʆˡ˕˚ˡ˱˽˚˝ʴ̅ Ĥ ʱ
Ĥ ≡ Ȟ − H̃

UkXjyVʇओɾʊଜձɸʪƑɸʪʇ GBQmpBHH2@pQM L2mKMM ൣପ߲ʎi/
|ρ(t)⟫ = Ĥ |ρ(t)⟫/t

UkXjRV

ʇ a+?`ƺ/BM;2` ൣପ߲ʍʧɥʉ೅߲ʆ࢑ɮɲʇɫʆɬʪƑ

kX9௺଺छ‫؃‬

ɲɲʝʆ෋୩С‫޶ޟ‬ʍ‫଺੄׿‬ʉ‫ح‬ʎ޼ଜɺɹʊմ໼ɶʅɬɾɫƐ௺ഥ‫ػ۽‬ʍ෋୩С‫ ޶ޟ‬ρ0 ʱূʕʇɼʍ‫ۥ‬ਚʱ๑ɣʅ |ρ0 ⟫ ʱछ‫؃‬ʇɸʪ6Q+F ‫؃‬Ԩʱ‫ۥ‬२ɸʪɲʇɫʆɬʪƑɲɲʆʎজ෢ʱԜઅʊɸʪɾ
ʠʊઅφ˴ƪ˛ʍ࣪܏ʱΑɥƑഉॐ˴ƪ˛ʍ࣪܏ʊʎๆαʊҼ૗ʆɬʪƑօя୩ β ʍഥ‫ػ۽‬ʍ࣪܏Ɛ෋୩С‫ ޶ޟ‬ρ0 ʎ

ρ0 =

e−βH
,
Z


Z = i` e−βH ,H = ωa† a

UkXjkVʆ฿ɧʨʫʪƑԂৌ‫ ػ‬UkXdV ʱ๑ɣʪʇƐ|ρ0 ⟫ =

|m, n⟫⟪m, n|ρ0 ⟫ = (1 − f )

m,n


m

e

f m |m, m⟫

UkXjjV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ k ࣍ ૨С‫߲ح޶ޟ‬ʇʉʪƑ ɲɲʆ f ʎ "QHixKMM ϒ޶ f = e−βω ʆɡʪƑ೅՝ʱԜઅʊɸʪɾʠ k ࡥ‫܈‬೅՝ (R8)


O =
μ

Ǒ
Õ†

μ
,

Ōν = Ǒ†−Õν

UkXj9Vʱ஡௬ɸʪƑۡʊओɾʉ૨С‫ ޶ޟ‬ξ μ , ξ¯ν ʱ௺଺ "Q;QHBm#Qp ഷԋ (3- Re)
Bnμν =1+n−1

ξ¯ν = āμ Bn−1,μνξ μ = Bnμν aν ,
μν

μν
−n
1
n
−1,μν
,
Bn
=
1
1 1+n


n=

f
1−f



UkXj8V
UkXjeVʆ஡௬ɸʪƑԜാʉ೅՝ʍɾʠʊƐடɷ୊ߞ Uμ ʣ ν ʉʈV ɫ‫ؗ‬ʩ഼ɶ‫ڇ‬ʫɾ࣪܏ʊʎΟ෽଺ʊɼʍ୊
ߞʊʃɣʅ໾ʱࠪʪɲʇʱตਡɸʪƑɲʍ૨С‫޶ޟ‬ʎξ μ |ρ0 ⟫ = 0,
⟪I| ξ¯ν = 0 μ ν[ξ μ , ξ ν ] = ξ¯μ , ξ¯ν = 0
ξ , ξ¯ = δμν ,UkXjdVUkXj3Vʇɣɥ१ࠃʱߡʃʍʆƐ⟪I| , |ρ0 ⟫ ʱछ‫؃‬ʇɸʪ 6Q+F ‫؃‬Ԩʱ‫ۥ‬२ɸʪɲʇɫʆɬʪƑɲʫʨछ‫؃‬ʱ௺଺छ‫؃‬ʇ‫ڐ‬ʕƑɼɥɣʂɾߚ࣮ɪʨƐಝഥ‫ ۽‬h6. ʍഞැʆʎ ⟪I| , |ρ0 ⟫ ʱɼʫɽʫ 0| , |0 ʇ࢑ɮɲʇʡ
ਵɣƑච໼ഞƸறʊਫ਼ 9 ࣍V ʆʡ‫҈ڷ‬ʍʉɣ‫ڌ‬ʩ 0| , |0 ʍ՝ࡲʱ๑ɣʪƑd《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥ਫ਼j࣍ࠏϾС‫޶ޟ‬൥ච࣍ʆʎࢉʠʊࠏϾС‫޶ޟ‬൥ʍҩ๗ʊʃɣʅজ෢ɸʪƑɼɶʅુ෾‫ػ‬ʍ˒ʶ˜˱ˁˋʱ՝
ࡲɸʪiBK2 +QMpQHmiBQM H2bb ‫ ث‬Kbi2` ൣପ߲ʍ஡ࡰʱ‫ۼ‬ɥƑɲʍKbi2` ൣପ߲ʱ‫ځ‬෋ʊ҈ɮʍʎ‫ܪ‬௟ʆɡʩƐ‫଺੄׿‬ʉฆ੠ʱ҈ɮʊʎ‫ߛׯ‬ʱ๑ɣʪ಴๗ɫɡʪƑɼɲʆ‫҉ܩ‬ʎগஞୈҟɸʪࠬ൥ʱҩԣɶƐ‫଺੄׿‬ʊ `2b2`pBQ` KQ/2H ʊɩɣʅ‫ޟك‬ʱ‫ۼ‬ʂɾٗѢʱߪɸƑjXR

ࠏϾС‫޶ޟ‬൥ʇ࡬ต෋୩С‫޶ޟ‬ສ޶ҟൢ‫ػ‬ʍฆ੠ʱ۵ɧʪ࣪܏Ɛ‫ޚ‬ωʍ‫ۄ‬Ѣʱʈʍʧɥʊࠪʩ௬ʫʪɪʎ੝ɬʉฆ੠ʆɡʪƑʉɻʉʨʏƐ૾࣭ʍສ޶໼ʊɩɰʪߢԨౙୈʎƐˡ˱˽˚˝ʴ̅ʱॲ२޶ʇɸʪ˸˝ˑ˼ƪഷԋʊʧʂʅ՝ࡲ
ɴʫʪɪʨʆɡʪƑߢԨౙୈɫ˸˝ˑ˼ƪഷԋʆ՝ࡲɴʫʪʉʨʏ‫ػ‬ʎߢԨ౩୎੆࣌१ʱಡɧʪɲʇʊ
ʉʩƐ‫ޚ‬ωʍ‫ۄ‬Ѣʎ௬ʩமʉɣƑɼɲʆৌ੄‫ػ‬ʱപɷɾ‫ػ‬ʇɶʅ૾࣭ʍສ޶໼ʍൣ൥ʆΑɣƐɼʍࣣʆુ෾ɶɾɣ೼ഒ‫ػ‬ʍ˒ʶ˜˱ˁˋʱࠏϾС‫޶ޟ‬ʱެʂʅϔɬࡰɸൣ൥ɫ‫ݔ‬ʨʫʪƑɲʫɫࠏϾС‫޶ޟ‬൥ (RĜ9) ʇ‫ڐ‬ʏʫʪൣ൥ʆɡʪƑɲʍʇɬાࡰɴʫɾુ෾‫ػ‬ʍ෋୩С‫޶ޟ‬ʱ࡬ต෋୩С‫޶ޟ‬ʱ‫ڐ‬ʕƑࠏϾС‫޶ޟ‬൥ʱެɥʇƐ‫଺ࡊݍ‬ʊુ෾‫ػ‬ʍ࡬ต෋୩С‫޶ޟ‬ʆപɷɾߢԨౙୈൣପ߲ʱமʪɲʇɫʆɬƐɲʫʱສ޶ Kbi2` ൣପ߲ʇ‫ڐ‬ʕƑறʊ‫֑҉ܩ‬ʠʪ߲ʎߢԨʍࣴʞ‫ܦ‬ʞঋഒʱ԰ʝʉɣ೅߲ʆɡʪɲʇɪ
ʨ iBK2 +QMpQHmiBQM H2bb ‫ ث‬Kbi2` ൣପ߲ʇ‫ڐ‬ʕ (j- 9)Ƒ
ˡ˱˽˚˝ʴ̅ H ɫુ෾‫ػ‬ɪʨʉʪ೼ഒ Ha ƐԖֻ‫ػ‬ɪʨʉʪ೼ഒ HR Ɛુ෾‫ػ‬ʇԖֻ‫ػ‬ʍਂ‫ݴک‬๑ɪʨʉʪ೼ഒ HaR ʍ໾ʆ࢑ɰʪʇɸʪƑH = Ha + HR + λHaR

UjXRVλ ʎુ෾‫ػ‬ʇԖֻ‫ػ‬ʍਂ‫ݴک‬๑ʍٗ܏ʍֽɴʱ೅ɸ‫ؤ‬ॐʆɡʪƑ෋୩С‫ ޶ޟ‬ρ(t) ʍߢԨౙୈʎ
GBQmpBHH2@pQM L2mKMM ൣପ߲ UkXjRVi/|ρ(t)⟫S = Ĥ |ρ(t)⟫S/tĤ = Ȟ − H̃

3

UjXkV
UjXjV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥ʊ࡞ɥƑ୊ߞʍ S ʎ a+?`ƺ/BM;2` ೉ਔʆɡʪɲʇʱ೅ɸƑɲɲʆƐ|ρ(t)⟫S = e−iĤ0 t |ρ(t)⟫I

UjX9V

Ĥ0 = Ĥa + ĤR

UjX8VʍԪ‫ؤ‬ʱ๑ɣʅਂ‫ݴک‬๑೉ਔ U୊ߞ I ʆ೅ɸʇɸʪV ʊζʪʇƐ

i/|ρ(t)⟫I = λĤaR (t) |ρ(t)⟫I
/t

UjXeV

ʇʉʪƑຜ೉ਔʍφફɸʪߢ‫ܗ‬ʱ t0 ʇɸʪƑਂ‫ݴک‬๑೉ਔʆεʉʪߢ‫ܗ‬ʍ |ρ(t)⟫I ʎƐ|ρ(t)⟫I = U (t, t0 ) |ρ(t0 )⟫I t1
U (t1 , t2 ) = h 2tT −iλdsĤaR (s)

UjXdV
UjX3V

t2

ʆٗʏʫʪƑh ʎߢԨࢇ࢘ঋʱ೅ɸƑΤђʆʎਂ‫ݴک‬๑೉ਔʱߪɸ୊ߞ I ʱࣈຊɶʅ೅՝ɸʪƑࠏϾС‫޶ޟ‬൥ʆʎߣʍʧɥʉࠏϾС‫ ޶ޟ‬P, Q ʱ஡௬ɸʪƑP = |ρR ⟫⟪IR |

UjXNV

Q=1−P
2

P = P,UjXRyV
2Q =Q

UjXRRVɲʍ |ρR ⟫ ʎԖֻ‫ػ‬ʊɩɰʪ௺ഥ‫۽‬ʍ՜Ҿѓɴʫɾ෋୩С‫޶ޟ‬ʆɡʪʇɸʪƑʝɾƐ⟪IR | ʎ ⟪IR |A⟫ =i`R [A] ʍʧɥʊԖֻ‫ػ‬ʍ೼ഒ˚˾ƪˋʱࠪʪ৸‫ݴ‬ʊ੆жɸʪƑɶɾɫʂʅƐP ʱ |ρ(t)⟫ ʊ‫ݴ‬๑ɸʪʇP |ρ(t)⟫ = |ρR ⟫ ⟪IR |ρ(t)⟫ = |ρa (t)⟫ ⊗ |ρR ⟫UjXRkVʍʧɥʊৌ੄‫ػ‬ʍ෋୩С‫| ޶ޟ‬ρ(t)⟫ ɪʨુ෾‫ػ‬ʍ࡬ต෋୩С‫| ޶ޟ‬ρa (t)⟫ ʱமʨʫʪƑ෾ೀʎ |ρa (t)⟫ʍߢԨౙୈൣପ߲ʱமʪɲʇʆɡʪƑɼɲʆ UjXeV ʇ UjXRRV ɪʨ/
P |ρ(t)⟫ = λP ĤaR (t)P |ρ(t)⟫ + λP ĤaR (t)Q |ρ(t)⟫/t/i Q |ρ(t)⟫ = λQĤaR (t)P |ρ(t)⟫ + λQĤaR (t)Q |ρ(t)⟫/ti

UjXRjVUjXR9VʱமʪƑUjXRjV ʍਫ਼ k ‫܈‬ʱࣁ֞ɸʪɾʠʊƐߣʍʧɥʉ G(t, t0 ), |ρ (t)⟫ ʱ஡௬ɸʪƑ tG(t, t0 ) = h 2tT −iλdsQĤaR (s)Q

UjXR8Vt0

Q |ρ(t)⟫ = G(t, t0 )Q |ρ (t)⟫UjXReV//(G(t, t0 )Q |ρ (t)⟫) = λQĤaR (t)QG(t, t0 )Q |ρ (t)⟫ + G(t, t0 )i Q |ρ (t)⟫/t/t

UjXRdVɸʪʇƐiɫமʨʫʪƑφൣʆ UjXR9V ɪʨi//(G(t, t0 )Q |ρ (t)⟫) = i Q |ρ(t)⟫ = λQĤaR (t)P |ρ(t)⟫ + λQĤaR Q |ρ(t)⟫/t/tN

UjXR3V

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥ʇʉʪƑUjXRdV ʇ UjXR3V ಐӇɸʪʇƐi/
Q |ρ (t)⟫ = λG(t0 , t)QĤaR (t)P |ρ(t)⟫/t

UjXRNVʇʉʪƑࣣ߲ʱঋഒɶƐUjXReV ʱ๑ɣʅɴʨʊ४๽ɸʪʇ

Q |ρ(t)⟫ = G(t, t0 )Q |ρ(t0 )⟫ + S(t) |ρ(t)⟫UjXkyVɫமʨʫʪƑɾɿɶS(t) = −iλ tdsG(t, s)QĤaR (s)P U (s, t)

UjXkRVt0ʆɡʪƑP + Q = 1 ʱ๑ɣʅഷ‫ح‬ɸʪʇ−1

Q |ρ(t)⟫ = {1 − S(t)}{G(t, t0 )Q |ρ(t0 )⟫ + S(t)P |ρ(t)⟫}UjXkkVʇʉʩƐɲʫʱ UjXRjV ʊਜ਼௬ɸʪʇ

i

/
−1
−1
(|ρa (t)⟫ ⊗ |ρR ⟫) = λP ĤaR (t){1 − S(t)} P |ρ(t)⟫ + λP ĤaR (t){1 − S(t)} G(t, t0 )Q |ρ(t)⟫/t
UjXkjV= iK̂(t) (|ρa (t)⟫ ⊗ |ρR ⟫) + i |I(t)⟫ ⊗ |ρR ⟫
−1

|ρR ⟫

UjXk9V−1

G(t, t0 )Q |ρ(t0 )⟫UjXk8VK̂(t) ≡ −i ⟪IR | λĤaR (t){1 − S(t)}|I(t)⟫ ≡ −i ⟪IR | λĤaR (t){1 − S(t)}ʇʉʪƑɲɲʆ

Q |ρ(t0 )⟫ = (1 − P ) |ρ(t0 )⟫ = |ρ(t0 )⟫ − |ρa (t0 )⟫ ⊗ |ρR ⟫UjXkeVʆɡʪɫߢ‫ ܗ‬t = t0 ʆુ෾‫ػ‬ʇ௺๡ԨʍਂԪɫ෗ɣʇєଜɸʪʇ|ρ(t0 )⟫ = |ρa (t0 )⟫ ⊗ |ρR ⟫UjXkdVʇ࢑ɮɲʇɫʆɬʪʍʆ Q |ρ(t0 )⟫ = 0 ʇʉʪƑ࡞ʂʅ |I(t)⟫ = 0 ʇʉʩƐ|ρa (t)⟫ ʍߢԨౙୈʎ

/|ρa (t)⟫ = iK̂(t) |ρa (t)⟫/t

UjXk3V/|ρa (t)⟫S = Ĥ0 + iK̂S (t) |ρa (t)⟫S/tK̂S (t) = e−iĤ0 t K̂(t)eiĤ0 t

UjXkNViʇʉʪƑa+?`ƺ/BM;2` ೉ਔʆʎƐi

UjXjyVʇʉʪƑɲʍʧɥʊƐಝগஞˡ˱˽˚˝ʴ̅ʊ iK̂ ʇɣɥ‫܈‬ɫљʮʂʅਂ‫ݴک‬๑ʍϾ‫׏‬ʱࠪʩ௬ʫʪ‫ح‬߲ʊʉʂʅɣʪƑอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʱ Ĥe,S (t) = Ĥ0 + iK̂S (t) ʆଜձɸʪʇƐ|ρS (t)⟫S ʍߢԨౙୈൣ
ପ߲ʎ

i

/
|ρa (t)⟫S = Ĥe,S (t) |ρa (t)⟫S/tʍʧɥʊ࢑ɰʪƑRy

UjXjRV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥

jXk

গஞୈҟ

K̂(t) ʱɼʍʝʝೆїɸʪʍʎ‫ܪ‬௟ʉʍʆƐλ ɫ࡝ഒࢬɴɣʇɶʅগஞୈҟɸʪɲʇʱ۵ɧʪƑK̂(t)ʱ λ ʍߣॐɳʇʊୈҟɸʪʇ

K̂(t) = −i∞n⟪IR | λĤaR (t) {S(t)} |ρR (t0 )⟫ =

n=0

∞


(−iλ)n K̂(n) (t)UjXjkVn=1ʇʉʪƑK̂(t) ʱ λ ʆୈҟɶɾߢʍ n ߣʍ‫܈‬ʱ K̂(n) (t) ʇɶɾƑடํʊ S(t) ʱୈҟɸʪƑS(t) =∞
(−iλ)n S (n) (t)UjXjjVn=0

S (n) (t) ʱ‫଺੄׿‬ʊ֑ʠʪʇ
S (1) (t) =S (2) (t) = t

dsQĤaR (s)P

t0
 t

 t
ds

t0

UjXj9Vds QĤaR (s )QĤaR (s)P +

s

 t

 t
ds

t0

ds QĤaR (s)P ĤaR (s )UjXj8Vs

K̂(n) (t) ʱ‫଺੄׿‬ʊ֑ʠʪʇ
K̂(1) (t) = ⟪IR | ĤaR (t) |ρR (t0 )⟫K̂

(2)

UjXjeV(1)(t) = ⟪IR | ĤaR (t)S (t) |ρR (t0 )⟫ t=ds ⟪IR |ĤaR (t)QĤaR (s)P |ρR (t0 )⟫
t0

=

 t

ds ⟪IR |ĤaR (t)ĤaR (s)|ρR (t0 )⟫

t0

 t

− ⟪IR |ĤaR (t)|ρR (t0 )⟫ds ⟪IR |ĤaR (s)|ρR (t0 )⟫
t0


2
(3)
(1)
(2)
K̂ (t) = ⟪IR | ĤaR (t) S (t) + S (t) |ρR (t0 )⟫UjXjdVUjXj3VXX
Xʇ೅ɴʫʪƑλ ɫ࡝ഒࢬɴɣʇɶʅ O λ2 ʝʆʱިɶʅ‫گ‬ʎ๮ʇɸʇ

K̂(t) = −iλ ⟪IR | ĤaR (t) |ρR (t0 )⟫ − λ2 tdt ⟪IR | ĤaR (t)ĤaR (t ) |ρR (t0 )⟫t0− λ2 ⟪IR | ĤaR (t) |ρR (t0 )⟫

 t
t0

ʇʉʪƑ

RR

ds ⟪IR | ĤaR (s) |ρR (t0 )⟫ UjXjNV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥

jXj_2b2`pBQ` KQ/2H ʆʍ҈উ_2b2`pBQ` KQ/2H ʎສ޶ҟൢ‫ػ‬ʍˉ̅˩˽ʉ˴˙˽ʆɡʩƐˡ˱˽˚˝ʴ̅ʎH = ωa† a +Ωk Rk† Rk +λ(Rk† a + a† Rk )k

UjX9yVkʆ฿ɧʨʫʪ (RdĜRN)Ƒઅφ˴ƪ˛ʍુ෾‫ ػ‬a ʇ௺๡ Rk ɫٗ܏ଜॐ λ ʆਂ‫ݴک‬๑ɶʅɣʪ‫ػ‬ʆɡʩƐਂ‫ݴک‬๑ʍѷପʊɩɣʅʡ௺๡ʎ௺ഥ‫ੌࣳ۽‬ʊɡʪʇɸʪƑʝɾ k ʎ௺๡ௐ೼ʍࢀΦʆɡʪƑˡ˕˚ˡ˱˽˚˝ʴ̅ʎƐĤ = H − H̃ = ωāμ aμ +Ωk R̄kμ Rkμ +k

λ(R̄kμ aμ + āμ Rkμ )UjX9RVkʆɡʪƑɲɲʆʎˡ˕˚ˡ˱˽˚˝ʴ̅ʍಝগஞ೼Ɛগஞ೼ʇɶʅƐĤa = ωāμ aμ ,

ĤR =



Ωk R̄kμ Rkμ ,ĤaR =

k

λ(R̄kμ aμ + āμ Rkμ )UjX9kVk

Ĥ0 = Ĥa + ĤR

UjX9jVʇʇʪƑɲʍ‫଺੄׿‬ʉ ĤaR ʍଜձʱ๑ɣʅ UjXjNV ʱ‫ޟك‬ɸʪƑʝɹ a, Rk ʊ௺଺ "Q;QHBm#Qp ഷԋ UkXj8V ʱ‫ۼ‬ɣƐ௺଺छ‫؃‬ʊ੆ɸʪॲ२ࣁ෩С‫ ޶ޟ‬ξ, Ξk ʱଜձɸʪƑμν
ξ μ = Bn(t)
aν ,

−1,μν
ξ¯ν = āμ Bn(t)UjX99Vμν ν
Ξμk = BN
Rk ,
k

−1,μν
Ξ̄νk = R̄kμ BN
kUjX98Vɲʍʇɬ a ਜʍ "Q;QHBm#Qp ഷԋʍˣ˻˳ƪˑʆɡʪຑ޶ॐ n(t) ʎn(t) ≡ ⟪I|a† (t)a(t)|ρ⟫

UjX9eVʆଜձɴʫʅɣʅƐߢԨʊΧਮɶʅɣʪƑφൣʆ R ਜʍ "Q;QHBm#Qp ഷԋʍˣ˻˳ƪˑʆɡʪຑ޶ॐ

Nk ʎ
Nk ≡ ⟪IR |Rk† (t)Rk (t)|ρR ⟫UjX9dVʆଜձɴʫƐߢԨʊΧਮɺɹ "Qb2@1BMbi2BM ഒೣʊ࡞ʂʅɣʪєଜɸʪƑĤaR ʍ‫଺੄׿‬ʉଜձɪʨ ĤaRʱʑʇʃɶɪ԰ʝʉɣ

⟪IR | ĤaR (t) |ρR ⟫UjX93Vʎɸʘʅ y ʇʉʪʍʆƐٗ‫ח‬iK̂(t) = −iλ2

 tds ⟪IR | ĤaR (t)ĤaR (s) |ρR ⟫t0

Rk

UjX9NV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ j ࣍ ࠏϾС‫޶ޟ‬൥ʱ‫ޟك‬ɸʫʏʧɣƑUjX9kV ʱਜ਼௬ɸʪʇ

iK̂(t) = −iλ

2

 t
k



ds a(t) − ã† (t)t0+ a† (t) − ã(t)⟪IR | Rk† (t)Rk (s) |ρR ⟫ a† (s) − ⟪IR | R̃k (t)R̃k† (s) |ρR ⟫ ã(s)⟪IR | Rk (t)Rk† (s) |ρR ⟫ a(s) − ⟪IR | R̃k† (t)R̃k (s) |ρR ⟫ ã† (s)UjX8yVɲɲʆƐ⟪IR | Rk† (t)Rk (t ) |ρR ⟫ = Nk e−iΩk t−t

UjX8RV⟪IR | Rk (t)Rk† (t ) |ρR ⟫ = (1 + Nk )e−iΩk t−t

UjX8kVʆɡʪɲʇʇƐa(t ) = a(t)e−iω t−t

UjX8jVʍԪ‫ؤ‬ʱ๑ɣʅ a ʍߢ‫ܗ‬ʱ t ʊᵿɧʪʇƐiK̂(t) = −Fk∗ (t) a(t) − ã† (t)Nk a† (t) − (Nk + 1)ã(t)k+ Fk (t) a† (t) − ã(t)(Nk + 1)a(t) − Nk ã† (t)

UjX89Vɲɲʆ UjX99V ʍԪ‫ؤ‬ʱ๑ɣʅ ξ ʆ࢑ɬ૰ɸʇ

iK̂(t) =Fk (t)ξ † (t)ξ(t) −

k



˜ +
Fk∗ (t)ξ˜† (t)ξ(t)kFk (t) = −ig 2 t(Fk (t) − Fk∗ (t))(n(t) − Nk )ξ † (t)ξ˜† (t) UjX88Vkds ei(ω−Ωk )(t−s)

UjX8eVt0ʇʉʪƑa+?`ƺ/BM;2` ೉ਔʆʎiK̂S (t) = ie−iĤ0 t K̂(t)eiĤ0 t=Fk (t)ξ † ξ −Fk∗ (t)ξ˜† ξ˜ +(Fk (t) − Fk∗ (t))(n(t) − Nk )ξ † ξ˜†kk

UjX8dV
UjX83Vkʇ೅ɴʫʪƑɶɾɫʂʅƐ࡬ต෋୩С‫޶ޟ‬ʊ੆ɸʪอ‫ۄ‬ˡ˱˽˚˝ʴ̅ Ĥe,S (t) ʎ

UjX8NVĤe,S (t) = Ĥ0 + iK̂S (t)
Fk (t)}ξ † ξ − {ω +Fk∗ (t)}ξ˜† ξ˜ + 2iAK
Fk (t){n(t) − Nk }ξ † ξ˜† + ĤR,S= {ω +kkkʇ֑ʝʂɾƑRj

UjXeyV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উਫ਼9࣍ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ`2b2`pBQ` KQ/2H ʍ҈উಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b (3) ʎഥ‫ػ۽‬ʊ੆ɸʪ࣪ʍສ޶໼ʍφ‫߲ح‬ʆɡʪ h?2`KQ

6B2H/ .vMKB+b (Re) ʱಝഥ‫ػ۽‬ʊҼ૗ɶɾʡʍʆɡʪƑh6. ʎຑ޶ഒೣ n ʊΧਮɶɾ"Q;QHBm#Qp ഷԋʱ๑ɣɾ‫߲ح‬ʆƐɼʍຑ޶ഒೣɫߢԨΧਮɸʪʧɥʊҼ૗ɸʪʇಝഥ‫۽‬h6. ʇʉʪƑɼʍߢԨΧਮ१ʎʽʸ̅ˑƪ‫܈‬ʍෆઢԪॐ ṅ(t) ʇɶʅ‫ڇ‬ʫʪƑಝഥ‫ ۽‬h6.
ʊʎ 62vMKM ˒ʶʴ˂˻˲൥ʱ๑ɣɾ੄‫଺ػ‬ʉগஞ‫ޟك‬ɫࠄ‫ۼ‬ʆɬʪʇɣɥ๸୐ɫɡʩƐ

QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʱଜձʆɬʪƑʽʸ̅ˑƪ‫܈‬ʇִʊ஡௬ɴʫʪ ṅ(t) ʉʈʍෆઢԪॐʎ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʊ੆ɸʪɮʩɲʞʊʧʂʅٔଜɴʫʪƑච࣍ʆʎƐࢉʠʊʽʸ̅ˑƪ‫܈‬ʱ஡௬ɶɾʍʀƐۡʊগஞ‫ޟك‬ʊʧʩ߭‫˽˟ʺڔ‬ˀƪʱ֑ʠʪƑɼɶʅƐಝഥ‫ػ۽‬ʆ
ʍ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʱଜձɶʅɮʩɲʞࣰٛʱѳɸƑɼɥɶʅமʨʫʪɮʩɲʝʫɾ
ಝগஞˡ˱˽˚˝ʴ̅ʇࠏϾС‫޶ޟ‬൥ʇʱಐӇɶƐۡʊɮʩɲʞࣰٛʍূʒൣʍϾ‫׏‬ɫʈʍʧɥʊ‫ڇ‬ʫʪɪʊʃɣʅࡲʘʪƑ

9XRˡ˱˽˚˝ʴ̅ʇߢԨౙୈൣପ߲

ච࣍ʆʎ `2b2`pQB` KQ/2HUਫ਼ jXj ছʱ‫ࣆޖ‬V ʱಝഥ‫ ۽‬h6. ʆ҈উɸʪƑಝഥ‫ ۽‬h6. ʆʎ Ĥ ʱಝগஞ೼ ĤU ʇগஞ೼ R ĤI ʊ

Ĥ = ĤU + ĤI

U9XRVĤU = Ĥa + ĤR + δ Ĥ − Q̂U9XkV

ĤI = ĤaR − δ Ĥ + Q̂

U9XjVR ৈ࣍ʆ๑ɣɾਂ‫ݴک‬๑೉ਔʱ೅ɸ୊ߞ I ʇʎεʉʪʍʆુίR9

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উʍʧɥʊʽʸ̅ˑƪ‫ ܈‬δ Ĥ, Q̂ ʱ஡௬ɸʪ (RR)Ƒδ Ĥ ʎʺ˟˽ˀƪʽʸ̅ˑƪ‫܈‬ʆδ Ĥ = δ ĤR + iδ ĤI

U9X9V

δ ĤR (t) = δω(t)ā a = δω(t)ξ¯μ ξ μδ ĤI (t) = −γ(t) μ ξ¯μ ξ μμ μʇɸʪƑ μ ʎ

1

= 1,

2

U9X8V
U9XeV

= −1 ʆɡʪƑQ̂ ʎ௺଺ "Q;QHBm#Qp ഷԋʍˣ˻˳ƪˑʆɡʪຑ޶ॐ n ɫߢԨʊΧਮɸʪɲʇʊʧʂʅ‫ڇ‬ʫʪʽʸ̅ˑƪ‫܈‬ʆɡʩ1
1Q̂ = iṅāμ

μν

−1
−1

aν = iṅξ¯μ



0
0

μν
1
ξν
0

U9XdV

ʆ฿ɧʨʫʪƑɲʍʧɥʊ௺଺ʽʸ̅ˑƪ‫ ܈‬Q̂ ʱ԰ʠʪʇ ξ ʍߢԨౙୈС‫޶ޟ‬ʱ ξ ʊʃɣʅ੆Ӆ଺ʊʇʪɲʇʊʉʩƐຑ޶ॐ n ɫߢԨౙୈɸʪ࣪܏ʊɩɣʅʡ ξ(t) ʊ੆ɸʪΜଜɶɾछ‫| ؃‬0 ʱଜձɸʪɲ
ʇɫʆɬʪƑɲʫʨʍʽʸ̅ˑƪ‫܈‬ʊ԰ʝʫʪ ṅ(t), ω(t), γ(t) ʎɲʍઞҠʆʎෆઢʍԪॐʆɡʩƐ‫گ‬ʊজ෢ɸʪɮʩɲʞࣰٛʊʧʂʅٔଜɴʫʪƑਂ‫ݴک‬๑೉ਔʆʍ aμ (t) ʍߢԨౙୈൣପ߲ʎ

i



/ μ
a (t) = aμ (t), ĤU/t= aμ (t), Ĥa + δ ĤR + iδ ĤI − Q̂


U9X3V

ʇʉʩƐɲɲɪʨ௺଺ "Q;QHBm#Qp ഷԋ


μν
Bn(t)
=

ξ μ = Bnμν aν ,ξ¯ν = āμ Bn−1,μνμν
1 + n(t) −n(t)
1
−1,μν
,
Bn(t) =
−1
1
1

n(t)
1 + n(t)

μν

U9XNV
U9XRyVʱ૾ɷʅଜձɴʫʪ ξ(t) ʍߢԨౙୈൣପ߲ʱ֑ʠʪʇ

i



/ μ
ξ (t) = ξ μ (t), Ĥa + δ ĤR + iδ ĤI/t= (ω + δω − i μ γ) ξ μ (t)U9XRRVʱமʪƑn(t) ʍߢԨΧਮ१ʊʧʂʅ aμ (t) ʇ ξ μ (t) ʍߢԨౙୈС‫޶ޟ‬ɫ Q̂ ɿɰεʉʂʅɣʪɲʇʊુ
ίƑU9XRRV ʱ‫଺߲ح‬ʊ҈ɮɲʇʆ ξ μ (t) ʍߢԨౙୈൣପ߲ʎ

ξ μ (t) = ξ μ e−i
¯μ

¯μ i

ξ (t) = ξ ettds{ω+δω(s)−iμ γ(s)}ds{ω+δω(s)+iμ γ(s)}U9XRkVU9XRjVʇ֑ʝʪƑɶɾɫʂʅ௰ίʍߢ‫ ܗ‬t ʊɩɣʅξ μ (t) |0 = 0
0| ξ¯ν (t) = 0

U9XR9V μξ (t), ξ¯ν (t) = δμν[ξ μ (t), ξ ν (t)] = ξ¯μ (t), ξ¯ν (t) = 0

U9XR8VU9XReVU9XRdVʱ२ງɸʪɾʠƐຑ޶ॐ n(t) ɫߢԨΧਮɸʪʧɥʉࣳ‫ׅ‬ʊɩɣʅʡ ξ ຑ޶ʱࢀຑ޶ʇɸʪ೉ࣛʱଜձ
ɸʪɲʇɫʆɬʪƑR8

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উ

9Xk୑౬Ԫॐʇ߭‫˽˟ʺڔ‬ˀƪ

a, R С‫޶ޟ‬ʍৌ୑౬Ԫॐʇಝগஞ୑౬Ԫॐ
Gμν (t1 , t2 ) = −i 0| h [aμH (t1 )āνH (t2 )] |0

U9XR3VΔ (t1 , t2 ) = −i 0| h [a (t1 )ā (t2 )] |0


μ
ν
Gμν
(k
,
t
;
k
,
t
)
=
−i
0|
h
R
(t
)
R̄
(t
)
|0
1 1 2 2k2 ,H 2R
k1 ,H 1μ
νΔμν
R (k1 , t1 ; k2 , t2 ) = −i 0| h Rk1 (t1 )R̄k2 (t2 ) |0

U9XRNVμνμν

U9XkyVU9XkRVɩʧʒ ξ, Ξ С‫޶ޟ‬ʍৌ୑౬Ԫॐʇಝগஞ୑౬Ԫॐ

 μν
g μν (t1 , t2 ) = −i 0| h ξH(t1 )ξ¯H(t2 ) |0

μν
μ
ν
¯
Əd (t1 , t2 ) = −i 0| h ξ (t1 )ξ (t2 ) |0gΞμν (k1 , t1 ; k2 , t2 ) = −i 0| h Ξμk1 ,H (t1 )Ξ̄νk2 ,H (t2 ) |0
 μ
ν
dμν
Ξ (k1 , t1 ; k2 , t2 ) = −i 0| h Ξk1 (t1 )Ξ̄k2 (t2 ) |0

U9XkkVU9XkjVU9Xk9VU9Xk8VʱࣣʍʧɥʊଜձɸʪƑɾɿɶ୊ߞʍ H ʎ >2Bb2M#2`; ೉ਔʱ೅ɸƑ.vbQM ൣପ߲





 



G (t1 , t2 ) = Δ (t1 , t2 ) + ds1 ds2 Δμμ (t1 , s1 )Σμ ν (s1 , s2 )Gν ν (s2 , t2 )


 

g μν (t1 , t2 ) = dμν (t1 , t2 ) + ds1 ds2 dμμ (t1 , s1 )S μ ν (s1 , s2 )g ν ν (s2 , t2 )μνμν

U9XkeV
U9XkdVʧʩƐ߭‫˽˟ʺڔ‬ˀƪ Σμν (t1 , t2 ), S μν (t1 , t2 ) ʱଜʠʪƑΣμν (t1 , t2 ) ʇ S μν (t1 , t2 ) ʍԨʊʎ௺଺ "Q@;QHBm#Qp ഷԋ U9XRyV ʱ૾ɷʅ




 

μμ
−1,ν ν
S μν (t1 , t2 ) = Bn(t)Σμ ν (t1 , t2 )Bn(t)U9Xk3VʍԪ‫ؤ‬ɫ२ງɸʪƑʝɾ 62vMKM ˒ʶʴ˂˻˲൥ʍɾʠʍ՝൥ʱΤђʧɥʊଜձɶʅɩɮƑ

G=

,

Δ=

,

ΔR =

U9XkNV

62vMKM ˒ʶʴ˂˻˲൥ʆৌ୑౬Ԫॐ G ʱୈҟɸʪʇ
=

+
+

+ ...

U9XjyVʇʉʪƑφൣʆ .vbQM ൣପ߲

=

+

Σ

U9XjRVʇʱಐӇɸʪʇƐ߭‫˽˟ʺڔ‬ˀƪ Σ ʎΣ

=U9XjkVRe

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উʇʉʪƑɲʫʎɸʘʅʍߣॐʱࠪʩ௬ʫɾ‫ځ‬෋ʉ߭‫˽˟ʺڔ‬ˀƪʇʉʂʅɣʪɲʇɫʮɪʪƑগஞˡ˱˽˚˝ʴ̅઺ʍ‫ ܈‬ĤaR ʍ߭‫˽˟ʺڔ‬ˀƪʗʍՅ฿ʎ

ΣĤaR ,μν (t1 , t2 ) = λ2
= −iλ

2





Δμν
R (k1 , t1 ; k2 , t2 )k 1 k2e−iΩk (t1 −t2 )k μ νμ  ν 
1Nk
1 + Nk1+ θ(t2 − t1 )θ(t1 − t2 )1−Nk
1 + Nk−1

U9XjjVʇʉʪƑۡʊ "Q;QHBm#Qp ഷԋʊʧʂʅ S μν ʱ֑ʠʪʇ

S ĤaR ,μν (t1 , t2 ) = −iλ2e−iΩk (t1 −t2 )k× θ(t1 − t2 )ν μ 
μ  ν 
1
01
Nk − n(t1 )+ θ(t2 − t1 )n(t2 ) − Nk−101ʇʉʪƑ਴ʊʡʽʸ̅ˑƪ‫܈‬ɪʨʍ߭‫˽˟ʺڔ‬ˀƪʗʍՅ฿ɫɡʩƐΤђʍʧɥʊʉʪƑS −δĤ,μν (t1 , t2 ) = δ(t1 − t2 )δμν [−δω(t1 ) + i μ γ(t1 )]

μν
0 1
Q̂,μν
S
(t1 , t2 ) = −iδ(t1 − t2 )ṅ(t1 )
0 0

9Xj

U9Xj9VU9Xj8V

PM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʇɮʩɲʞࣰٛಝഥ‫ ۽‬h6. ʆʎʽʸ̅ˑƪ‫܈‬ʊ԰ʝʫʪෆઢԪॐ ṅ(t), ω(t), γ(t) ʱٔଜɸʪɾʠʊƐQM@b?2HH ߭‫ڔ‬ʺ˟˽ˀƪʊ੆ɸʪɮʩɲʞࣰٛʱѳɸ (RR- Rj)Ƒɼʍɾʠʊʝɹʎ߭‫˽˟ʺڔ‬ˀƪʊɩɰʪ QM@b?2HHʍଜձʱ‫ۼ‬ɥƑഥ‫ػ۽‬ʍ࣪܏Ɛ߭‫˽˟ʺڔ‬ˀƪʎ k ߢ‫ܗ‬ʍܿʆ࢑ɰʪɾʠ߭‫˽˟ʺڔ‬ˀƪʍ k0 ೅‫ڇ‬ʱ 6Qm`B2` ഷԋʱ૾ɷʅS μν [k0 ] = ∞−∞dτ S μν (τ )eik0 τ

U9XjeVʍʧɥʊʇʪƑQM@b?2HH ʎ k0 ʎಝগஞ୑౬Ԫॐ d(t) ʍ k0 ೅‫ ڇ‬d[k0 ] ʍ‫ י‬k0 = ω ʊʧʂʅଜʠʪƑɶɾɫʂʅƐഥ‫ػ۽‬ʆʍ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʎ

S

μν

[ω] =

 ∞
−∞

dτ S μν (τ )eiωτ

U9XjdVʇʉʪƑ
φൣʆಝഥ‫ػ۽‬ʍ࣪܏ʎ߭‫˽˟ʺڔ‬ˀƪɫ k ߢ‫ܗ‬ʍܿʆʝʇʝʨʉɣɾʠ k0 ೅‫ڇ‬ɼʍʡʍɫ߭෢ʆ
ʎʉɣƑɲɲʆʎϒѢ຅ʱࠫʪʧɥʊಝഥ‫۽‬ʊҼ૗ɶɾ k0 ೅‫( ڇ‬Rj) ʱ‫ݔ‬๑ɶSμν[t, k0 ] =

 ∞
−∞

t

dτ [ θ(τ )S μν (t, t − τ )ei t−τ ds{k0 (s)}
 t+τ+θ(−τ )S μν (t + τ, t) ei t

Rdds{k0 (s)}
U9Xj3V

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উʇɸʪƑಝഥ‫۽‬ʊɩɣʅʡഥ‫ػ۽‬ʊൕʂʅಝগஞ୑౬Ԫॐ d(t1 , t2 ) ʍ k0 ೅‫ ڇ‬d[t, k0 ] ʱౙ‫ޚ‬ɴɺʪ

k0 (t) = ω + δω(t) + iγ(t) ʱ QM@b?2HH ʇɶʅଜʠʪʇƐQM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʎaμν (t) = S μν [t, ω + δω(t) + iγ(t)] ∞t=dτ [ θ(τ )S μν (t, t − τ )ei t−τ ds{ω+δω(s)+iγ(s)}
−∞

 t+τ

+ θ(−τ )S μν (t + τ, t) ei tds{ω+δω(s)−iγ(s)}
U9XjNVʇʉʪƑPM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʱ‫଺੄׿‬ʊ֑ʠʪʇ


ĤaR ,μνa(t) =k Jk (t)

0

2i AK

 t

Jk (t) = −iλ2μν

Jk (t){n(t) − Nk }k∗k Jk (t)tds ei s dτ {ω+δω(τ )+iγ(τ )−Ωk }t0−δω(t) + iγ(t)0μν
0 −iṅ(t)aQ̂,μν (t) =00a−δĤ,μν (t) =0−δω(t) − iγ(t)μνU9X9yVʇʉʪƑɮʩɲʞࣰٛʇɶʅ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪ aμν (t) ʊ੆ɶʅa12 (t) = 0_2 a11 (t) = 0

U9X9RV11

AK a (t) = 0
ʱѳɸ (RR- Rj)ƑɸʪʇෆઢԪॐ ṅ(t), ω(t), γ(t) ɫṅ(t) = 2 AK

Jk (t){n(t) − Nk }U9X9kVJk (t)

U9X9jVkδω(t) = _2

kγ(t) = − AKJk (t)

U9X99Vkʍʧɥʊ֑ʝʪƑɶɾɫʂʅಝগஞˡ˱˽˚˝ʴ̅ ĤU (t) ʎĤu (t) = {ω +Jk (t)}ξ † ξ − {ω +k

+ 2iAKJk∗ (t)}ξ˜† ξ˜kJk (t){n(t) − Nk }ξ † ξ˜† + ĤR

U9X98VkʊଜʝʪƑδω(t) ʣ γ(t) ɫࢬɴɣʇɶʅ QM@b?2HH ʍߢԨΧਮ१ʱ෗ߏɶɾ࣪܏ʎƐઅࢃʊ ω ʱ QM@b?2HH ʇʞʉɸɲʇʊʉʪƑɲʍʇɬʍ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʎaμν (t) = S μν [t, ω] =

 ∞

−∞

dτ [ θ(τ )S μν (t, t − τ )eiωτ + θ(−τ )S μν (t + τ, t) eiωτR3

U9X9eV

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উʇʉʪƑɲʍଜձʍʡʇʆ QM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʱ‫଺੄׿‬ʉ‫ح‬ʎĤaR ,μνa(t) =k Fk (t)0

Fk (t) = −iλ

2

2i AK

 t

Fk (t){n(t) − Nk }k∗k Fk (t)μν

ds ei(ω−Ωk )(t−s)

t0−δω(t) + iγ(t)
a
(t) =
0

μν
0 −iṅ(t)aQ̂,μν (t) =00−δ Ĥ,μν0−δω(t) − iγ(t)μν

U9X9dVʇʉʪƑU9X9yV ʇಐӇɸʪʇƐaĤaR ,μν (t) ʊ԰ʝʫʪঋഒ Jk (t), Fk (t) ʍʞɫεʉʂʅɣʪɲʇɫʮɪ
ʪƑঢʇடํʍɮʩɲʞࣰٛ U9X9RV ʱѳɶʅƐಝগஞˡ˱˽˚˝ʴ̅ʱ֑ʠʪʇ

Ĥu (t) = {ω +Fk (t)}ξ † ξ − {ω +k

+ 2iAK

Fk∗ (t)}ξ˜† ξ˜kFk (t){n(t) − Nk }ξ † ξ˜† + ĤR

U9X93V

k

ʇʉʪƑ

9X9

ࠏϾС‫޶ޟ‬൥ʇʍಐӇৈছʆ QM@b?2HH ʍߢԨΧਮ१ʱ෗ߏɶʅ֑ʠɾ U9X93V ʎƐࠏϾС‫޶ޟ‬൥ʆ֑ʠɾ࡬ต෋୩С‫޶ޟ‬ʊ੆ɸʪอ‫ۄ‬ˡ˱˽˚˝ʴ̅ UjX8NV ʇடɷ‫ح‬ʱʇʪɲʇɫʮɪʪƑɼɲʆಝഥ‫ ۽‬h6. ʇࠏϾС‫޶ޟ‬൥ʍɼʫɽʫʍൣ൥ʆഐ๽ສ A ʍՎ੊ડʱ֑ʠƐຜࠖʱಐӇɸʪƑಝഥ‫ ۽‬h6. ʆഐ๽ສ A ʍՎ੊ડʎƐ Ah6. = H ⟪1| AH (t) |ρ⟫H = H ⟪1| AI (t) h 2tT −i dsHI (s) |ρ⟫H

U9X9NVʇʉʪƑɼɲʆƐগஞʍ y ߣʱ֑ʠʪʇƐ

Ah6.,0 = H ⟪1| AI (t) |ρ⟫Htt

= H ⟪1| ei −∞ dsĤU (s) AS e−i −∞ dsĤU (s) |ρ⟫Ht

= H ⟪1| AS e−i −∞ dsHu (s) |ρ⟫H

U9X8yVʇʉʪƑφൣʆࠏϾС‫޶ޟ‬൥ʱ๑ɣʅՎ੊ડʱ‫ޟك‬ɸʪʇƐ

ASP = S ⟪1| AS |ρa (t)⟫St= H ⟪1| AS e−i −∞ dsHe,S (s) |ρa ⟫H

U9X8RVʇʉʪƑ࡞ʂʅƐAh6.,0 = ASP ʇʉʪƑɸʉʮʀƐࠏϾС‫޶ޟ‬൥ʆ֑ʝʪഐ๽ສʍՎ੊ડʎಝഥ‫ ۽‬h6. ʆɮʩɲʲɿ‫گ‬ʍ y ߣʍՎ੊ડʊஉɶɮʉʪɲʇɫߪɴʫɾƑRN

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 9 ࣍ ಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʊʧʪ `2b2`pBQ` KQ/2H ʍ҈উ

9X8ɮʩɲʞࣰٛʍূʒൣʊʧʪܿεৈছʍٗѢʎƐʺ˟˽ˀƪʽʸ̅ˑƪ‫܈‬ʇɶʅδ Ĥ = δ ĤR + iδ ĤI
δ ĤR = δω(t)ξ¯μ ξ μU9X8kVU9X8jVμ ¯μ μδ ĤI = −γ(t) ξ ξ

U9X89V_2 a11 (t) = 0

U9X88VʱূʒƐɮʩɲʞࣰٛ

11

AK a (t) = 0

U9X8eVʱѳɶɾٗѢʆɡʂɾƑɲʍɲʇʊʧʂʅƐࢀຑ޶ʍɮʩɲʝʫɾʺ˟˽ˀƪʎഉয়ॐʇʉʪƑɶɪɶƐ૾࣭ʍ࣪ʍສ޶໼ʆʎˡ˱˽˚˝ʴ̅ʍʺ˽˱ƪ˚१ʱҋɶʅɶʝɥɾʠƐಝগஞˡ˱˽˚˝ʴ̅ʍଜձʊഉয়ॐʍʺ˟˽ˀƪʱʇʪɲʇʎʉɣƑφൣʆಝഥ‫ ۽‬h6. ʊɩɰʪಝগஞˡ˱˽˚˝ʴ̅ʎ௺଺ʽʸ̅ˑƪ‫܈‬ʍ஡௬ʊʧʂʅՍʊಝʺ˽˱ƪ˚ʉ‫߲ح‬ʇʉʂʅɣʅƐಝʺ˽˱ƪ˚ʱৈଥʇɶɾմ໼ɫ
њఉʆɡʪƑ
ɲɲʆʎಝগஞˡ˱˽˚˝ʴ̅ʊ԰ʠʪࢀຑ޶ʺ˟˽ˀƪʱࠄॐʊ‫ڌ‬ʂɾ࣪܏ʊѕɫ՟ɲʪʍɪʱ‫ٵ‬ʅʞʪƑࠄॐʊ‫ڌ‬ʪʊʎʺ˟˽ˀƪʽʸ̅ˑƪ‫܈‬ʇɶʅδ Ĥ = δ ĤR = δω(t)ξ¯μ ξ μU9X8dVʇূʲʆƐQM@b?2HH ߭‫˽˟ʺڔ‬ˀƪʊ੆ɸʪɮʩɲʞࣰٛʇɶʅ

_2 a11 (t) = 0

U9X83VʇূʘʏʧɣƑɸʪʇɮʩɲʝʫɾಝগஞˡ˱˽˚˝ʴ̅ʇɶʅ

Ĥu (t) = {ω + _2



Fk (t)}ξ † ξ − {ω + _2k

+ 2iAK

Fk∗ (t)}ξ˜† ξ˜kFk (t){n(t) − Nk }ξ † ξ˜† + ĤRkɫமʨʫʪƑɸʪʇϞ഻ʍਫ਼φ‫܈‬ʇਫ਼௡‫܈‬ʆࢀຑ޶ʺ˟˽ˀƪʍൊ९ʇɶʅ‫ڇ‬ʫʪ

U9X8NVk Fk (t) ೼ɪʨʍՅ฿ɫࠄ೼ʊ‫ڌ‬ଜɴʫʅɣʪɲʇɫʮɪʪƑφൣʆࠏϾС‫޶ޟ‬൥ʆமʨʫɾอ‫ۄ‬ˡ˱˽˚˝ʴ̅ UjX8NV
ʊʎࠄ೼ɿɰʆʉɮ֥೼ɪʨʍՅ฿ʡ԰ʝʫʅɣʅɣʪʍʆƐɲʍɮʩɲʝʫɾಝগஞˡ˱˽˚˝ʴ̅ʱ๑ɣʅഐ๽ສʍՎ੊ડʱ֑ʠɾʇɶʅʡƐʈʍߣॐʊɩɣʅʡࠏϾС‫޶ޟ‬൥ʇφફɸʪɲʇʎʉɣƑࢀຑ޶ʺ˟˽ˀƪʊ֥ॐʍൊ९ʱљɧʪʍʎ૾࣭ʍ࣪ʍສ޶໼ʍ‫ٵ‬ൣʆʎε࣭ʉൣ൥ʆɡʪɫƐສ޶ҟൢ‫ػ‬ʱΑɥࠏϾС‫޶ޟ‬൥ʊɩɣʅʎ߭োʊ԰ʝʫʪ‫߲ح‬ʊʉʂʅɣʪɲʇɫʮɪʂɾƑʡʀʬʲɲʍٗѢʎ `2b2`pQB` KQ/2H ʊ‫ڌ‬ଜɴʫɾʡʍɿɫƐҟൢ‫ػ‬ʊಝഥ‫ ۽‬h6. ʱ๑ɣʪ‫ݣ‬ʍɮʩɲʞࣰٛʍূʒ
ൣʊɩɰʪφʃʍ޼धʊʉʪʆɡʬɥʇ޻ʮʫʪƑky

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 8 ࣍ ʝʇʠਫ਼8࣍ʝʇʠච໼ഞʆʎສ޶ҟൢ‫ػ‬ʊ੆ɸʪಝഥ‫ ۽‬h6. ʆʍɮʩɲʞࣰٛʱ૦ʘʪɾʠʊƐ`2b2`pQB` KQ/2H ʊɩɣʅࠏϾС‫޶ޟ‬൥ʇಝഥ‫ ۽‬h?2`KQ 6B2H/ .vMKB+b ʍಐӇʱ‫ۼ‬ʂɾƑਫ਼ k ࣍ʆʎ૨С‫߲ح޶ޟ‬ʱ஡௬ɶƐ௺଺छ‫؃‬ɫ૾࣭ʍ෋୩С‫޶ޟ‬ʇ੆жɸʪɲʇʣƐ௺଺ "Q;QHBm#Qp
ഷԋʊʧʂʅ஡௬ɴʫʪ ξ ૨С‫޶ޟ‬ʇ௺଺छ‫؃‬ʊʧʂʅ6Q+F ‫؃‬Ԩɫ‫ۥ‬२ʆɬʪɲʇʱӂ௳ɶɾƑਫ਼ j ࣍ʆʎࠏϾС‫޶ޟ‬൥ʱ๑ɣɾ `2b2`pQB` KQ/2H ʍ҈উʊʃɣʅࡲʘɾƑ‫଺੄׿‬ʊʎৌ੄‫ػ‬ʍ෋୩С‫޶ޟ‬ɪʨԖֻ‫ػ‬ʍ࣮൙ʱࣁɸʧɥʉࠏϾС‫޶ޟ‬൥ʱ஡௬ɶƐ࡬ต෋୩С‫޶ޟ‬ʍപɷɾߢԨౙୈൣପ߲
ʱமʨʫʪɲʇʱࡲʘɾƑ࡬ต෋୩С‫޶ޟ‬ʊ੆ɸʪߢԨౙୈС‫޶ޟ‬ʎอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʇ۵ɧʪɲ
ʇɫʆɬƐɼʍอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʎগஞୈҟʊʧʂʅ֑ʠʨʫʪɲʇʱজ෢ɶɾƑɼɶʅ‫଺੄׿‬ʊ`2b2`pQB` KQ/2H ʆʍ஡ࡰʱ‫ۼ‬ɣƐ࡬ต෋୩С‫޶ޟ‬ʊ੆ɸʪอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʍ೅߲ʱமɾƑਫ਼ 9 ࣍ʆʎಝഥ‫ ۽‬h6. ʱ๑ɣʅ `2b2`pQB` KQ/2H ʍ҈উʱ‫ۼ‬ɣƐਫ਼ j ࣍ʆ‫ۼ‬ʂɾࠏϾС‫޶ޟ‬൥ʇʍಐӇʱ‫ۼ‬ʂɾƑಝഥ‫ ۽‬h6. ʆʎ௺଺ʽʸ̅ˑƪ‫܈‬ʣʺ˟˽ˀƪʽʸ̅ˑƪ‫܈‬ʱ஡௬ɸʪɲʇʆƐξ ૨С‫޶ޟ‬ʇ௺଺छ‫؃‬ʊʧʂʅ‫ݴ‬ʨʫʪ6Q+F ‫؃‬Ԩɫ௰ίʍߢ‫ܗ‬ʊɩɣʅʡΜଜ଺ʊଜձɴʫʪɲʇʱ‫ٵ‬ɾƑɼʍࣣʆ୑౬Ԫॐʱ஡௬ɶƐ.vbQM ൣପ߲ʱ૾ɷʅଜձɴʫʪ߭‫˽˟ʺڔ‬ˀƪʱ‫ޟك‬ɶɾƑɼɶʅഥ‫۽‬h6. ʊɩɰʪ߭‫˽˟ʺڔ‬ˀƪʊ੆ɸʪ QM@b?2HH ɮʩɲʞʊൕʂɾಝഥ‫ ۽‬h6. ʆʍ QM@b?2HH ɮʩɲʞʱ஡௬ɶƐ‫଺੄׿‬ʉɮʩɲʞࣰٛʱѳɸɲʇʆಝগஞˡ˱˽˚˝ʴ̅ʱٔଜɶɾƑɼɲʆਫ਼ j ࣍ʆ֑ʠɾอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʇʍಐӇʱɶɾٗѢƐࠏϾС‫޶ޟ‬൥ʆ֑ʠʪอ‫ۄ‬ˡ˱˽˚˝ʴ̅ɫಝഥ‫ ۽‬h6.
ʍɮʩɲʲɿ‫گ‬ʍಝগஞˡ˱˽˚˝ʴ̅ʇடɷ‫ح‬ʱʇʪɲʇɫʮɪʩƐಝഥ‫ ۽‬h6. ʆʍগஞʍ y ߣʇࠏϾС‫޶ޟ‬൥ʆʍਂ‫ݴک‬๑ʍ k ߣʝʆʱ԰ʠɾอ‫ۄ‬ˡ˱˽˚˝ʴ̅ʊ԰ʠɾ‫߲ح‬ʆഐ๽ສʍՎ੊ડɫφફɸʪɲʇɫʮɪʂɾƑۡʊຜࠖʍഐ๽ສʍՎ੊ડɫφફɸʪɮʩɲʞࣰٛʊʃɣʅ۵‫ޅ‬ʱ‫ۼ‬ɣƐಝഥ‫ ۽‬h6. ʆଥΠɴʫʅɣʪɮʩɲʞࣰٛʍʑʇʃʆɡʪࢀຑ޶ʺ˟˽ˀƪʍ֥೼ʱಝগஞˡ˱˽˚˝ʴ̅ʍଜձʊ԰ʠʪൣ൥ɫࠏϾС‫޶ޟ‬൥ʊɩɣʅ߭োʊ੷२ɴʫʅɣʪɲʇɫʮɪʂɾƑɲʍɲʇɪʨƐಝഥ‫ ۽‬h6. ʆສ޶ҟൢ‫ػ‬ʱΑɥ࣪܏ʊʎಝগஞˡ˱˽˚˝ʴ̅ʊʺ˟˽ˀƪʍ֥೼ʍൊ९ʱљɧʪʧɥʉɮʩɲʞࣰٛʱѳɸɲʇɫ߭োʆɡʪɲʇɫߪܼɴʫʪʇٗ໼ɶɾƑච໼ഞʆʎࠏϾС‫޶ޟ‬൥ʇಝഥ‫ ۽‬h6. ʍಐӇɪʨຜࠖԨʊ຾ߛɫɡʪɲʇʱߪɶɾɫƐɡɮʝʆগ
ஞʍߣॐʍ౹Ωʱ‫ڌ‬ଜɶɾ҈উʆɡʩƐ‫ߣ܊‬ʆʎεʉʂɾٗѢʊʉʪɲʇɫ฽৳ɴʫʪƑಝഥ‫ ۽‬h6.
ʎ 62vMKM ऺ൥ʱ๑ɣɾ੄‫଺ػ‬ʉগஞ‫ޟك‬ɫњఉʆɡʪφൣƐࠏϾС‫޶ޟ‬൥ʊʎɼʍʧɥʉ‫ޟك‬ɫ‫ܪ‬௟ʉɾʠʊ‫ߣ܊‬ʍ૰খʍಐӇʎ௟ɶɣƑʝɾƐ‫҉ܩ‬ʎສ޶ҟൢ‫ػ‬ʊુ෾ɶɾ҈উʱ‫ۼ‬ʂɾɫಝഥ‫ ۽‬h6.

kR

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ਫ਼ 8 ࣍ ʝʇʠ߭੄ʎ‫ړ‬ງ‫ػ‬ʊʡ଼жњఉʉ๽໼ʆɡʪɲʇʱֽ૦ɶʅɩɮƑಝഥ‫ ۽‬h6. ʆʎ‫ړ‬ງ‫ػ‬ʇҟൢ‫ػ‬ʱφԥɶɾൣ൥ʆΑɥɲʇɫʆɬʪɾʠƐࠏϾС‫޶ޟ‬൥ɫอ‫ۄ‬ʆʉɣ‫ړ‬ງ‫ػ‬ʊɩɣʅʡ௺ԝ໾ʱ՝ࡲʆɬʪ๽
໼ʇɶʅՎ੊ɴʫƐໂց‫ػ޶ڀ‬ʊ੆ɸʪ҈উແʡਮ‫ݥ‬ɸʪ (NĜRk)ƑɶɪɶƐ๽໼଺ʉਜ෮ʆʎ਴ʍಝഥ
‫۽‬ʱ՝ࡲɸʪ๽໼ʇʍԪ‫ؤ‬ɫɸʘʅ෢ӂʊʉʂʅɣʪʇʎ‫ڊ‬ɧɹƐறʊ੆жɸʪʡʍɫ෗ɣʧɥʉ‫ړ‬ງ‫ػ‬ʊɩɣʅ๽໼ʇɶʅʍΦપʄɰʱʈʍʧɥʊ෢ӂʊɶʅɣɮɪʎ‫گܩ‬ʍѳ੠ʆɡʪƑkk

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

ࠗ߯චٰ֖ʱे‫ۼ‬ɸʪʊɡɾʩƐਵɮʍൣƧʍɩफ़໿ʊʉʩʝɶɾƑ‫઺ޗ‬ำซ‫࠺׃‬ʊʎ௺ऐʊɳ޼஡ʱɣ
ɾɿɬʝɶɾƑٰ֖ʍௐๆʊʃɣʅɿɰʆʉɮƐٰ֖ʊ‫ۈ‬ɪɥऐ‫ۥ‬ɧʉʈਵɮʍɲʇʱӌʏɺʅ૬ɬʝ
ɶɾƑ઺ਲิ҆߅ʎƐѕʡઢʨʉɪʂɾʊʡԪʮʨɹಝഥ‫ ۽‬h6. ʱʣʩɾɣʇ‫ڊ‬ɣ޳ʠɾ߈ʊਵɮʍ
޼஡ʱɶʅɮɿɴɣʝɶɾƑ‫۔ڀؘ‬໫߅ʊʎɳ߭तʍಝഥ‫ ۽‬h6. ʣສ޶ยਏൣପ߲ʍٰ֖ʍ‫ٽؼ‬ʊʡʇʄɣɾਵɮʍ࢕‫ڊ‬ʱ૬ɬʝɶɾƑʝɾƐ߈ʍɮɿʨʉɣկฆʣܹ‫ݟ‬ʉմ໼ʊʡ૎௷ʊʃɬɡʂʅɮɿ
ɴʩƐ๽҈ʱघʠʪɲʇɫʆɬʝɶɾƑ‫ࡿׄ܊‬φ߅ʎ‫ە‬ɣߏฐɪʨํƧʉ࣮൙ʱଥ֯ɶʅɮɿɴʩƐ෾ʍৈʍɲʇʆߏฐʍ‫ׇ‬ɮʉʩɫʀʉ߈ʊʇʂʅƐʇʅʡວɣޭُʇʉʩʝɶɾƑடՎʍᄹఛ‫ڸ‬ऩ߅Ɛ౫ਣ९࠻߅ʇʎˎ˱ʣٰ֖ʆʍմ໼ʱ૾ɶʅઢ߳ʱघʠʪɲʇɫʆɬʝɶɾƑʝɾƐЂπۗี߅Ɛඐ‫ۗڀ‬՞߅ƐፑҴບฤ߅ʇʎƐҟൢ‫ػ‬ʣສ޶ยਏൣପ߲ʊԪɸʪմ໼ʱ૾ɷʅਵɮʍઢ߳ʣߪܼʱ૬ɬʝɶɾƑɩफ़໿ʊʉʂɾ‫߾ٰ֖઺ޗ‬ʍқํʊҔʠʅԈࠗફɶʝɸƑ

《修士論文》

物性研究・電子版 Vol. 4, No. 2, 042601 （2015年5月号）

k9‫ޖ‬۵ഞ‫ٯ‬(R) aX LFDBK- S`Q;X h?2Q`X S?vbX ky- N93 URN83VX(k) _X wrMxB;- CX *?2KX S?vbX jj- Rjj3 URNeyVX(j) hX `BKBibm M/ >X lK2xr- S`Q;X h?2Q`X S?vbX dd- jk URN3dVX(9) >X SX "`2m2` M/ 6X S2i`m++BQM2- h?2 h?2Q`v Q7 PT2M ZmMimK avbi2Kb UPt7Q`/ lMBp2`bBiv
S`2bb- kyydVX(8) CX a+?rBM;2`- CX Ji?X S?vbX k- 9yd URNeRVX(e) GX oX E2H/vb?- aQpX S?vbX C1hS ky- Rjyd URNe8VX(d) GX E/MQz M/ :X "vK- ZmMimK aiiBbiB+H J2+?MB+b, :`22MǶb 6mM+iBQM J2i?Q/b BM
1[mBHB#`BmK M/ LQM2[mBHB#`BmK S`Q#H2Kb UqXX "2MDKBM- RNekVX(3) >X lK2xr- /pM+2/ 6B2H/ h?2Q`v, JB+`Q- J+`Q- M/ h?2`KH S?vbB+b UAS- RNNjVX(N) uX LFKm`- hX amM;- JX JBM2- JX PFmKm`- M/ uX uKMF- MMX S?vbX ULXuXV jk89ke UkyRyVX(Ry) uX LFKm` M/ uX uKMF- MMX S?vbX ULXuXV jke- Rydy UkyRRVX(RR) uX LFKm`- uX Emr?`- M/ uX uKMF- LQM2[mBHB#`BmK i?2`KQ }2H/ /vMKB+b 7Q`
i?2`KH `2HtiBQM T`Q+2bb Q7 +QM}M2/ +QH/ iQKB+ ;b- BM CSa *QM72`2M+2 S`Q+22/BM;b oQHX RkyR9X(Rk) uX Emr?`- uX LFKm`- M/ uX uKMF- LmK2`B+H MHvbBb Q7 [mMimK i`MbTQ`i
2[miBQM 7Q` #Qb2 ;b BM QM2 /BK2MbBQMH QTiB+H HiiB+2- BM CSa *QM72`2M+2 S`Q+22/BM;b oQHX RkyR9X(Rj) uX LFKm` M/ uX uKMF- MMX S?vbX ULXuXV jjR- 8R UkyRjVX(R9) JX a+?Kmix- wX S?vbX " jy- Nd URNd3VX(R8) >X *?m M/ >X lK2xr- AMiX CX JQ/X S?vbX  N- Rdyj URNN9VX(Re) >X lK2xr- h?2`KQ 6B2H/ .vMKB+b M/ *QM/2Mb2/ aii2b U1Hb2pB2` a+B2M+2 Gi/- RN3kVX(Rd) hX `BKBibm- JX :mB/- M/ >X lK2xr- 1m`QT?vbX G2iiX j URN3dVX(R3) hX `BKBibm- >X lK2xr- uX uKMF- M/ LX STbiKiBQm- S?vbB+  R93- kd URN33VX(RN) >X lK2xr M/ uX uKMF- /pX S?vbX jd- 8jR URN33VX