弾性論(Chen)

Size: px
Start display at page:

Download "弾性論(Chen)"

Transcription

1 Phase-field by T.Koyama

2 Phase-field da da a( ) a + { } a d + d δ (-) δ (-) eigen a a a ε ε δ δ (-) da ε (-4) a d ε ε + δε ( ) (-5) δε d (-6) V u ul δεl + l (-7) eigen el ε ε ε (-8) σ el C ε el C { ε ε } (-9) l l l l l σ el, j el σ ( ) (-) j (-9),(-)(-7) C u δ l Clεδ l j l j (-)

3 d d δ δ ex( i ) { δ ex( i ) ( π) ( π) (-) d d ui ui ex( i ) { ui ex( i ) ( π) ( π) (-) d d δε δε ex( i ) { δε ex( ) ( π) i ( π) (-4) (-) C u δ l Clεδ l j l j C u ic εδ δ l j l l l j C u ic εδ δ l j l l l j (-5) G ( ) C (-6) i l j l σ C ε δ (-7) l l u ig C εδ δ ig σ δ (-8) i l l j i j (-7) δε i { ui j + uj ( ) i} (-9) (-8) δεl i { u l + ul } iu l iig σδ G C εδ δ i j l i l j mn mn (-) δε l el el Est Clε εl { C { l ε + δε ε εl + δεl εl d (-) ε ε ε ε δε ε (-) el +

4 { E }{ st } Cl ε + δε ε εl + δεl εl Clε εl + Clε δεl Clε εl Clδε εl Clδε δεl Clδε εl + + d Clε εl Clε δεl + Clε εl l l l l l l C ε ε d+ C ε δε C d ε ε + Clδε εld + Clδε δεl d Clδε εl d Clεi j εld Clε δεl + Clε εl l l l l l l C ε ε d+ C ε d C d δε ε ε + Clεl δε d + Clδε δεl d Cl δε ( ) εl d Clεl ε Clε δεl + l l C ε ε Clε εl d + Cl ε εl d Cl δε εl d + Clδε δεl V C lε εl + Clδδ lε { δ Cl δε εl (-) V V d Clε εl + Clδδlε { δ} niσω j σlnl δ n ( π ) n / δε d ε d δ ε δ d C δε ε d C δε δε d (-4) l l l l Phase-field eigen { } + ε ε δ δ ε (-) el + ε ε ε ε δε ε δ { δ } ε s ( ) (-)

5 σ el C ε el C { ε ε } (-) l l l l l d d δ δ ex( i ) { δ ex( i ) ( π) ( π) d d s { δs ex( i ) { δs ex( i ) ( π) ( π) d d δε δε ex( i ) { δε ex( i ) ( π) ( π) d d ui ui ex( i ) { u ex i i ( π) ( π) (-4) σ [ { ] el u εl Cl εl εl Cl Cl j j j l j u l δ l l lεδ l l εl j l j j j C C C + C C u ic εδ { δ} i C ε { s } l j l l l j s j l l s i lεδ l j{ δ } i j lεl { } u ig C ig C s i σ j{ δ } i j σ { } ig ig s (-5) δε i { u + u } iu l l l l i σ j l{ δ } i j l σ { } G + G s (-6) 4

6 el el el el Est Clε εl σ ε el el { { σ ε ε σ ε + δε ε el el el σ ε d + σ δε d σ ε d el el el el σ, ε d+ σ s ui s nj s ds σ j ui s d σ ε s el el d d σ ε σ ε el el Clεl ε d Clεl ε d Cl{ εl + δεl εl } ε d { Cl εl + δεl εl ε l l l l l l l l C ε ε d+ C δε ε d C d C d ε ε ε ε + Clε εl d Clε δεl d Clε εld Clε εl + Cl l d ε ε Clε δεl d (-7) Clε εld Clε εl + l l C ε ε Clε εld Clε εδl{ δ + εl d s s + εδ{ δ ε εδl{ δ εl ( + + ) d Clεεld Clεεδl{ δ d Clε εl d + { d { s d s s d + + εδδ l δ εδ l δ ε ε εl q, q (-8) (-7) 5

7 Clε δεl Cl { l d εδ δ + ε δε C d + C s d lεδ { δ } δεl l ε δεl d C + C lεδ { δ } δεl lε ( π ) d C C s + lεδ { δ } lε { δ } δεl π { δ δεl( ) π s s d σl{ δ + σl { δ Gi ( ) σjl{ δ + Gi ( ) jl σ { ( π ) s s d { { { { Gi jl σl δ + σl δ σ δ + σ ( π ) d G + G s } sq jσ i σl l{ δ } { δ } { { jσ i σl l δ ( π ) sq q, d + + jσ Gi σll{ δ} { } { { jσgi σl l δ δ ( π) ( π) d + sq jσgi σll{ δ { δ jσ Gi σl ( π ) q, s d s s l{ δ { q d ( π ) d ( π ) d + jσ Gi σll{ δ} { } ( π ) (-9) d d d δ δε d { δ} ex( i ) δε ex( i ) ( π) ( π) d d { δ δεl ex{ i( + ) } d ' ' ' ' ( π) ( π) l l d d' { δ} δεl ( ' ) δ( + ' ) ' ( π ) ( π ) d { δ} δεl π d (-) d { } ( ) (-) π ε δεl d ε δ δεl (-8)(-9)(-7) 6

8 Est Clε εld Clε εl + Cl l d Cl l d ε ε ε δε C d C d C s d lε εl lε εδl{ δ lε εl s + εδδ l{ δ d + ε δl{ δ} ε d + ε εl sq d jσgi σll{ δ { δ ( π ) jσ i σl l{ δ } { } G s sq jσ i σl l{ δ } { sq G s, q d ( π ) } lε εl ε lεl lεδδl ( δ ) sq lδε εl ( δ ) l ε εl q, q d ( π ) V V C V C s + C V + VC s + C s s d jσgi σll{ δ { δ ( π ) q, q, j σ G s i σl l{ δ { d ( π ) s d q jσ Gi σl l{ δ { sq ( π ) (-) ε (-) ε a ε ε a (-) 7

9 Est (-) (-) E st ε l εl lεl lεl VC V C s (-4) G E Vσ ε (-5) a st G (-6) (-5)(-) G VC V C s V a lεl lεl σ a l Cl l ε σ + ε (-7) C C S l l l hase-field hase-field (-8) [] el el Est Clε εl d Cl{ ε δε ε }{ εl δεl εl } + + (4-) [] 8

10 E C d C d C Cl{ ε ε { εl + δεl εl d el el el el st lε εl lεl ε lεl ε d (4-) (4-)δε hase-field (4-) δε hase-field δε hase-field (4-)δε hase-field (4-) (4-)(4-) E st C { ( l εl + δεl εl )} (4-) (4-) E δε { + + { st l C l εl δεl εl Cl ε ε C C + C l l{ εl εl } lδεl l{ ε ε } C ε ε C δε l{ l l l l δε { ( δεl l{ ε ε } ε ε + C { δε C C C { } l l{ εl εl } lδεl l{ ε ε } ε ε Cl{ εl εl } Clδεl Cl{ ε ( ε + δε ε C l { l l l } δε l ) ε } { ε ε } ε ε )} (4-4) δε { ε ε } l δε { ε ε } l δε ( ) δε { ε ε } l ε ε δε δε δε { ε ε } l l l l δε δε ε ε { ε ε } l l δε l { ε ε } δε l { ε ε } (4-5) 9

11 Est E st el Clε l C { { l εl + δεl εl Clε δ εl + δεl εl } E s s s st Est el C lε l ε C ε + δε ε l{ l l l } s s Clε { εl + δεl εl } (4-6) s εδ s ε s (4-7) d δε δε ex( i ) ( π ) G + G s i i σl j l{ δ } i j l σl { } ex( ) ( π ) G + s i d i j l σl { δ } σl { } ex( ) ( π ) s d Gi ( n) njnl σl { δ } + σ l { } ex( i ) ( π ) d (4-8) eigen eigen eigen eigen eigen (-)eigen σ

12 s σ Clεl Cl εδl{ δ} + εl (5-) lεδ l{ δ lεl σ{ δ σ ( C + C s + s ) σ { σ } { σ σ ex( i ) s σ{ δ} + σ ex( i ) d (5-) σ { δ}ex( i ) d+ σ s ex( i ) d { } { } σ δ + σ (4-8) d δε δε ex( i ) ( π ) G n n + s i d G n n n i i ( n) j l σl{ δ } σl { } ex( ) ( π ) i j l{ σ l ex π d (5-) (4-8) eigen (5-)

転位の応力場について

転位の応力場について y.koyama f ( F ( F( f( xp( i d f( F( xp( i ( π xp( i d δ ( F( δ ( f( δ ( xp( i f ( δ ( F( f ( xp( i d xp( i d F( f( xp( i d ' F( xp( i xp( i ' ' d ' F( xp{ i( } d ' ' ' d ' ' F( ' xp{ i( } d ' ' ' F( '

More information

³ÎΨÏÀ

³ÎΨÏÀ 2017 12 12 Makoto Nakashima 2017 12 12 1 / 22 2.1. C, D π- C, D. A 1, A 2 C A 1 A 2 C A 3, A 4 D A 1 A 2 D Makoto Nakashima 2017 12 12 2 / 22 . (,, L p - ). Makoto Nakashima 2017 12 12 3 / 22 . (,, L p

More information

橡00扉.PDF

橡00扉.PDF SQ2.1 SQ2.2 ( ) 19971998 1981-97

More information

20169 3 4 5003 n=3,000 61.8% 38.2% n=3,000 20 7.3% 30 21.3% 40 34.8% 50 36.6% n=3,000 3.0% 2.0% 1.5% 12.1% 14.0% 41.4% 25.9% n=3,000 37.7% % 24.8% 28.8% 1.9% 3.1% 0.2% n=3,000 500 64.0% 500 1,000 31.3%

More information

社葬事前手続き

社葬事前手続き 2 ... 4... 4... 5 1... 5 2... 5 3... 5 4... 5 5... 5 6... 5 7... 5 8... 6 9... 6 10... 6... 6 1... 6 2... 6 3... 7 4... 7... 8 1 2.... 8 2 2.... 9 3 4.. 3 4. 1 2 3 4 5 6 7 5 8 9 10 I 1 6 2 EL 3 4 24 7

More information

syuryoku

syuryoku 248 24622 24 P.5 EX P.212 2 P271 5. P.534 P.690 P.690 P.690 P.690 P.691 P.691 P.691 P.702 P.702 P.702 P.702 1S 30% 3 1S 3% 1S 30% 3 1S 3% P.702 P.702 P.702 P.702 45 60 P.702 P.702 P.704 H17.12.22 H22.4.1

More information

土壌環境行政の最新動向(環境省 水・大気環境局土壌環境課)

土壌環境行政の最新動向(環境省 水・大気環境局土壌環境課) 201022 1 18801970 19101970 19201960 1970-2 1975 1980 1986 1991 1994 3 1999 20022009 4 5 () () () () ( ( ) () 6 7 Ex Ex Ex 8 25 9 10 11 16619 123 12 13 14 5 18() 15 187 1811 16 17 3,000 2241 18 19 ( 50

More information

2.

2. 2. 10 2. 2. 1995/12006/111995/42006/12 2. 10 1995120061119954200612 02505 025 05 025 02505 0303 02505 250100 250 200 100200 5010050 100200 100 100 50100 100200 50100 10 75100100 0250512 02505 1 025051205

More information

laplace.dvi

laplace.dvi Λ 2.1 2004.2.20 1 Λ 1 2 Ay = u 2 2 A 2 u " # a 11 a 12 A = ; u = a 21 a 22 " # u 1 u 2 y Ay = u (1) A (1) y = A 1 u y A 2 x i i i =1; 2 Ax 1 = 1 x 1 ; Ax 2 = 2 x 2 (2) x 1 x 2 =0 (3) (3) (2) x 1 x 2 x

More information

73

73 73 74 ( u w + bw) d = Ɣ t tw dɣ u = N u + N u + N 3 u 3 + N 4 u 4 + [K ] {u = {F 75 u δu L σ (L) σ dx σ + dσ x δu b δu + d(δu) ALW W = L b δu dv + Aσ (L)δu(L) δu = (= ) W = A L b δu dx + Aσ (L)δu(L) Aσ

More information

all.dvi

all.dvi 5,, Euclid.,..,... Euclid,.,.,, e i (i =,, ). 6 x a x e e e x.:,,. a,,. a a = a e + a e + a e = {e, e, e } a (.) = a i e i = a i e i (.) i= {a,a,a } T ( T ),.,,,,. (.),.,...,,. a 0 0 a = a 0 + a + a 0

More information

19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional

19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional 19 σ = P/A o σ B Maximum tensile strength σ 0. 0.% 0.% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional limit ε p = 0.% ε e = σ 0. /E plastic strain ε = ε e

More information

( )

( ) 7..-8..8.......................................................................... 4.................................... 3...................................... 3..3.................................. 4.3....................................

More information

FdData社会地理

FdData社会地理 [ [ 1(3 ) [ 2(3 ) A C [ [ [ 3(2 ) (1) X Y Z (2) X Y Z 3,000m [ 4(3 ) [ [ [ 5(2 ) ( ) 1 [ [ 6( ) (1) A (2) (1) B [ 7(3 ) (1) A (2) A (3) A 2 [ 8(2 ) [ 9(3 ) 2 [ 10(2 ) A H [ [ 11( ) A H 3 3 [ 12(2 ) [ (

More information

消防力適正配置調査報告

消防力適正配置調査報告 8 5 5 20 11 22 4 25 1.1 1 1.2 1 1.3 2 2.1 6 2.2 6 2.3 8 2.4 8 2.5 9 3.1 10 3.2 10 3.3 13 4.1 15 4.2 17 4.3 19 4.4 21 4.5 23 (1) - 1 - (2) (1) ()1 ( ) 8 1 1 143 116 (2) 1-2 - 26 24 19 24 6 21 24 4 19 24

More information

Microsoft Word - 01Ł\”ƒ.doc

Microsoft Word - 01Ł\”ƒ.doc 226821,416* 13,226 22 62,640 46,289 13,226 28.6 * 8,030 4,788 408 13,226 2,249 2,868 55 5,173 2,153 716 93 2,962 3,628 1,204 260 5,092 173 10 361 25.5% 40 220 112 50.9% 4,922 804 16.3% 3040 141 54 38.3%

More information

入門講座 

入門講座  第 8 章弾性歪エネルギー評価法 () () 8- Khhtun の弾性歪エネルギ- 評価ここでも簡単のため A-B 元系における不規則相の整合相分離を考え この相分解組織の弾性歪エネルギーを評価する 手順は ステップ ) まず位置 の関数として与えられる濃度場 () を用いて egen 歪場 ε () を定義する ステップ ) 次に全歪場 ε () を均一全歪 ε とそこからの変動量 δε ()

More information

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)

(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1) 1. 1.1...,. 1.1.1 V, V x, y, x y x + y x + y V,, V x α, αx αx V,, (i) (viii) : x, y, z V, α, β C, (i) x + y = y + x. (ii) (x + y) + z = x + (y + z). 1 (iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 1 19 3 19.1................... 3 19.............................. 4 19.3............................... 6 19.4.............................. 8 19.5.............................

More information

Netcommunity SYSTEM X7000 IPコードレス電話機 取扱説明書

Netcommunity SYSTEM X7000 IPコードレス電話機 取扱説明書 4 5 6 7 8 9 . 4 DS 0 4 5 4 4 4 5 5 6 7 8 9 0 4 5 6 7 8 9 4 5 6 4 0 4 4 4 4 5 6 7 8 9 40 4 4 4 4 44 45 4 6 7 5 46 47 4 5 6 48 49 50 5 4 5 4 5 6 5 5 6 4 54 4 5 6 7 55 5 6 4 56 4 5 6 57 4 5 6 7 58 4

More information

.A. D.S

.A. D.S 1999-1- .A. D.S 1996 2001 1999-2- -3- 1 p.16 17 18 19 2-4- 1-5- 1~2 1~2 2 5 1 34 2 10 3 2.6 2.85 3.05 2.9 2.9 3.16 4 7 9 9 17 9 25 10 3 10 8 10 17 10 18 10 22 11 29-6- 1 p.1-7- p.5-8- p.9 10 12 13-9- 2

More information

Phase field法を用いた材料組織形成過程の計算機シミュレ-ション

Phase field法を用いた材料組織形成過程の計算機シミュレ-ション Cahn-Hilliard Cahn-Hilliard 4 Cahn-Hilliard 5 x F (x F v F µ x (- µ v (- v M F M µ (- M F J v (- J v M µ (- Fik (-4 (- (-5 J D D µ M (-4 (-5 µ µ + RT ln a γ a γ (-5 (-6 D D * (ln γ * (ln γ D D +, D D +

More information

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1

1 I 1.1 ± e = = - = C C MKSA [m], [Kg] [s] [A] 1C 1A 1 MKSA 1C 1C +q q +q q 1 1 I 1.1 ± e = = - =1.602 10 19 C C MKA [m], [Kg] [s] [A] 1C 1A 1 MKA 1C 1C +q q +q q 1 1.1 r 1,2 q 1, q 2 r 12 2 q 1, q 2 2 F 12 = k q 1q 2 r 12 2 (1.1) k 2 k 2 ( r 1 r 2 ) ( r 2 r 1 ) q 1 q 2 (q 1 q 2

More information

Gmech08.dvi

Gmech08.dvi 63 6 6.1 6.1.1 v = v 0 =v 0x,v 0y, 0) t =0 x 0,y 0, 0) t x x 0 + v 0x t v x v 0x = y = y 0 + v 0y t, v = v y = v 0y 6.1) z 0 0 v z yv z zv y zv x xv z xv y yv x = 0 0 x 0 v 0y y 0 v 0x 6.) 6.) 6.1) 6.)

More information

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2

No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2 No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j

More information

QMI_10.dvi

QMI_10.dvi ... black body radiation black body black body radiation Gustav Kirchhoff 859 895 W. Wien O.R. Lummer cavity radiation ν ν +dν f T (ν) f T (ν)dν = 8πν2 c 3 kt dν (Rayleigh Jeans) (.) f T (ν) spectral energy

More information

1

1 GL (a) (b) Ph l P N P h l l Ph Ph Ph Ph l l l l P Ph l P N h l P l .9 αl B βlt D E. 5.5 L r..8 e g s e,e l l W l s l g W W s g l l W W e s g e s g r e l ( s ) l ( l s ) r e l ( s ) l ( l s ) e R e r

More information

B

B B YES NO 5 7 6 1 4 3 2 BB BB BB AA AA BB 510J B B A 510J B A A A A A A 510J B A 510J B A A A A A 510J M = σ Z Z = M σ AAA π T T = a ZP ZP = a AAA π B M + M 2 +T 2 M T Me = = 1 + 1 + 2 2 M σ Te = M 2 +T

More information

(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x

(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x Compton Scattering Beaming exp [i k x ωt] k λ k π/λ ω πν k ω/c k x ωt ω k α c, k k x ωt η αβ k α x β diag + ++ x β ct, x O O x O O v k α k α β, γ k γ k βk, k γ k + βk k γ k k, k γ k + βk 3 k k 4 k 3 k

More information

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb

18 2 F 12 r 2 r 1 (3) Coulomb km Coulomb M = kg F G = ( ) ( ) ( ) 2 = [N]. Coulomb r 1 r 2 r 1 r 2 2 Coulomb Gauss Coulomb 2.1 Coulomb 1 2 r 1 r 2 1 2 F 12 2 1 F 21 F 12 = F 21 = 1 4πε 0 1 2 r 1 r 2 2 r 1 r 2 r 1 r 2 (2.1) Coulomb ε 0 = 107 4πc 2 =8.854 187 817 10 12 C 2 N 1 m 2 (2.2)

More information

n ξ n,i, i = 1,, n S n ξ n,i n 0 R 1,.. σ 1 σ i .10.14.15 0 1 0 1 1 3.14 3.18 3.19 3.14 3.14,. ii 1 1 1.1..................................... 1 1............................... 3 1.3.........................

More information

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C602E646F63>

<4D F736F F D B B83578B6594BB2D834A836F815B82D082C88C602E646F63> スピントロニクスの基礎 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/077461 このサンプルページの内容は, 初版 1 刷発行時のものです. i 1 2 ii 3 5 4 AMR (anisotropic magnetoresistance effect) GMR (giant magnetoresistance

More information

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P

9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L) du (L) = f (9.3) dx (9.) P 9 (Finite Element Method; FEM) 9. 9. P(0) P(x) u(x) (a) P(L) f P(0) P(x) (b) 9. P(L) 9. 05 L x P(x) P(0) P(x) u(x) u(x) (0 < = x < = L) P(x) E(x) A(x) P(L) f ( d EA du ) = 0 (9.) dx dx u(0) = 0 (9.2) E(L)A(L)

More information

006 11 8 0 3 1 5 1.1..................... 5 1......................... 6 1.3.................... 6 1.4.................. 8 1.5................... 8 1.6................... 10 1.6.1......................

More information

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,.

,. Black-Scholes u t t, x c u 0 t, x x u t t, x c u t, x x u t t, x + σ x u t, x + rx ut, x rux, t 0 x x,,.,. Step 3, 7,,, Step 6., Step 4,. Step 5,,. 9 α ν β Ξ ξ Γ γ o δ Π π ε ρ ζ Σ σ η τ Θ θ Υ υ ι Φ φ κ χ Λ λ Ψ ψ µ Ω ω Def, Prop, Th, Lem, Note, Remark, Ex,, Proof, R, N, Q, C [a, b {x R : a x b} : a, b {x R : a < x < b} : [a, b {x R : a x < b} : a,

More information

( ) ) AGD 2) 7) 1

( ) ) AGD 2) 7) 1 ( 9 5 6 ) ) AGD ) 7) S. ψ (r, t) ψ(r, t) (r, t) Ĥ ψ(r, t) = e iĥt/ħ ψ(r, )e iĥt/ħ ˆn(r, t) = ψ (r, t)ψ(r, t) () : ψ(r, t)ψ (r, t) ψ (r, t)ψ(r, t) = δ(r r ) () ψ(r, t)ψ(r, t) ψ(r, t)ψ(r, t) = (3) ψ (r,

More information

Mathematical Logic I 12 Contents I Zorn

Mathematical Logic I 12 Contents I Zorn Mathematical Logic I 12 Contents I 2 1 3 1.1............................. 3 1.2.......................... 5 1.3 Zorn.................. 5 2 6 2.1.............................. 6 2.2..............................

More information

all.dvi

all.dvi 29 4 Green-Lagrange,,.,,,,,,.,,,,,,,,,, E, σ, ε σ = Eε,,.. 4.1? l, l 1 (l 1 l) ε ε = l 1 l l (4.1) F l l 1 F 30 4 Green-Lagrange Δz Δδ γ = Δδ (4.2) Δz π/2 φ γ = π 2 φ (4.3) γ tan γ γ,sin γ γ ( π ) γ tan

More information

n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m

n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m 1 1 1 + 1 4 + + 1 n 2 + π2 6 x [10 n x] x = lim n 10 n n 10 k x 1.1. a 1, a 2,, a n, (a n ) n=1 {a n } n=1 1.2 ( ). {a n } n=1 Q ε > 0 N N m, n N a m a n < ε 1 1. ε = 10 1 N m, n N a m a n < ε = 10 1 N

More information

xyz,, uvw,, Bernoulli-Euler u c c c v, w θ x c c c dv ( x) dw uxyz (,, ) = u( x) y z + ω( yz, ) φ dx dx c vxyz (,, ) = v( x) zθ x ( x) c wxyz (,, ) =

xyz,, uvw,, Bernoulli-Euler u c c c v, w θ x c c c dv ( x) dw uxyz (,, ) = u( x) y z + ω( yz, ) φ dx dx c vxyz (,, ) = v( x) zθ x ( x) c wxyz (,, ) = ,, uvw,, Bernoull-Euler u v, w θ dv ( ) dw u (,, ) u( ) ω(, ) φ d d v (,, ) v( ) θ ( ) w (,, ) w( ) θ ( ) (11.1) ω φ φ dθ / dφ v v θ u w u w 11.1 θ θ θ 11. vw, (11.1) u du d v d w ε d d d u v ω γ φ w u

More information

untitled

untitled - k k k = y. k = ky. y du dx = ε ux ( ) ux ( ) = ax+ b x u() = ; u( ) = AE u() = b= u () = a= ; a= d x du ε x = = = dx dx N = σ da = E ε da = EA ε A x A x x - σ x σ x = Eε x N = EAε x = EA = N = EA k =

More information

V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V

V 0 = + r pv (H) + qv (T ) = + r ps (H) + qs (T ) = S 0 X n+ (T ) = n S n+ (T ) + ( + r)(x n n S n ) = ( + r)x n + n (d r)s n = ( + r)v n + V n+(h) V I (..2) (0 < d < + r < u) X 0, X X = 0 S + ( + r)(x 0 0 S 0 ) () X 0 = 0, P (X 0) =, P (X > 0) > 0 0 H, T () X 0 = 0, X (H) = 0 us 0 ( + r) 0 S 0 = 0 S 0 (u r) X (T ) = 0 ds 0 ( + r) 0 S 0 = 0 S 0 (d r)

More information

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz

n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz 1 2 (a 1, a 2, a n ) (b 1, b 2, b n ) A (1.1) A = a 1 b 1 + a 2 b 2 + + a n b n (1.1) n A = a i b i (1.2) i=1 n i 1 n i=1 a i b i n i=1 A = a i b i (1.3) (1.3) (1.3) (1.1) (ummation convention) a 11 x

More information

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [

H 0 H = H 0 + V (t), V (t) = gµ B S α qb e e iωt i t Ψ(t) = [H 0 + V (t)]ψ(t) Φ(t) Ψ(t) = e ih0t Φ(t) H 0 e ih0t Φ(t) + ie ih0t t Φ(t) = [ 3 3. 3.. H H = H + V (t), V (t) = gµ B α B e e iωt i t Ψ(t) = [H + V (t)]ψ(t) Φ(t) Ψ(t) = e iht Φ(t) H e iht Φ(t) + ie iht t Φ(t) = [H + V (t)]e iht Φ(t) Φ(t) i t Φ(t) = V H(t)Φ(t), V H (t) = e iht V (t)e

More information

Gmech08.dvi

Gmech08.dvi 51 5 5.1 5.1.1 P r P z θ P P P z e r e, z ) r, θ, ) 5.1 z r e θ,, z r, θ, = r sin θ cos = r sin θ sin 5.1) e θ e z = r cos θ r, θ, 5.1: 0 r

More information

TOP URL 1

TOP URL   1 TOP URL http://amonphys.web.fc.com/ 3.............................. 3.............................. 4.3 4................... 5.4........................ 6.5........................ 8.6...........................7

More information

chap9.dvi

chap9.dvi 9 AR (i) (ii) MA (iii) (iv) (v) 9.1 2 1 AR 1 9.1.1 S S y j = (α i + β i j) D ij + η j, η j = ρ S η j S + ε j (j =1,,T) (1) i=1 {ε j } i.i.d(,σ 2 ) η j (j ) D ij j i S 1 S =1 D ij =1 S>1 S =4 (1) y j =

More information

all.dvi

all.dvi 72 9 Hooke,,,. Hooke. 9.1 Hooke 1 Hooke. 1, 1 Hooke. σ, ε, Young. σ ε (9.1), Young. τ γ G τ Gγ (9.2) X 1, X 2. Poisson, Poisson ν. ν ε 22 (9.) ε 11 F F X 2 X 1 9.1: Poisson 9.1. Hooke 7 Young Poisson G

More information

note1.dvi

note1.dvi (1) 1996 11 7 1 (1) 1. 1 dx dy d x τ xx x x, stress x + dx x τ xx x+dx dyd x x τ xx x dyd y τ xx x τ xx x+dx d dx y x dy 1. dx dy d x τ xy x τ x ρdxdyd x dx dy d ρdxdyd u x t = τ xx x+dx dyd τ xx x dyd

More information

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2

1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2 2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6

More information

50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq

50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq 49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r

More information

1'1153 1 ド Ui:;~

1'1153 1 ド Ui:;~ 1'1153 1 ド Ui:;~ ~f ~f ~f 17~ 3.~1

More information

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B

A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B 9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A

More information

第3章

第3章 5 5.. Maxwell Maxwell-Ampere E H D P J D roth = J+ = J+ E+ P ( ε P = σe+ εe + (5. ( NL P= ε χe+ P NL, J = σe (5. Faraday rot = µ H E (5. (5. (5. ( E ( roth rot rot = µ NL µσ E µε µ P E (5.4 = ( = grad

More information

QMI_09.dvi

QMI_09.dvi 25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 3.1.2 σ τ 2 2 ux, t) = ux, t) 3.1) 2 x2 ux, t) σ τ 2 u/ 2 m p E E = p2 3.2) E ν ω E = hν = hω. 3.3) k p k = p h. 3.4) 26 3 hω = E = p2 = h2 k 2 ψkx ωt) ψ 3.5) h

More information

QMI_10.dvi

QMI_10.dvi 25 3 19 Erwin Schrödinger 1925 3.1 3.1.1 σ τ x u u x t ux, t) u 3.1 t x P ux, t) Q θ P Q Δx x + Δx Q P ux + Δx, t) Q θ P u+δu x u x σ τ P x) Q x+δx) P Q x 3.1: θ P θ Q P Q equation of motion P τ Q τ σδx

More information

CVMに基づくNi-Al合金の

CVMに基づくNi-Al合金の CV N-A (-' by T.Koyama ennard-jones fcc α, β, γ, δ β α γ δ = or α, β. γ, δ α β γ ( αβγ w = = k k k ( αβγ w = ( αβγ ( αβγ w = w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( αβγ w = ( βγδ w = = k k k ( αγδ

More information

i

i 009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3

More information

Gmech08.dvi

Gmech08.dvi 145 13 13.1 13.1.1 0 m mg S 13.1 F 13.1 F /m S F F 13.1 F mg S F F mg 13.1: m d2 r 2 = F + F = 0 (13.1) 146 13 F = F (13.2) S S S S S P r S P r r = r 0 + r (13.3) r 0 S S m d2 r 2 = F (13.4) (13.3) d 2

More information

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i

v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i 1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [

More information

Part () () Γ Part ,

Part () () Γ Part , Contents a 6 6 6 6 6 6 6 7 7. 8.. 8.. 8.3. 8 Part. 9. 9.. 9.. 3. 3.. 3.. 3 4. 5 4.. 5 4.. 9 4.3. 3 Part. 6 5. () 6 5.. () 7 5.. 9 5.3. Γ 3 6. 3 6.. 3 6.. 3 6.3. 33 Part 3. 34 7. 34 7.. 34 7.. 34 8. 35

More information

nsg04-28/ky208684356100043077

nsg04-28/ky208684356100043077 δ!!! μ μ μ γ UBE3A Ube3a Ube3a δ !!!! α α α α α α α α α α μ μ α β α β β !!!!!!!! μ! Suncus murinus μ Ω! π μ Ω in vivo! μ μ μ!!! ! in situ! in vivo δ δ !!!!!!!!!! ! in vivo Orexin-Arch Orexin-Arch !!

More information

マイクロメカニクスの基礎と応用

マイクロメカニクスの基礎と応用 by.koyama ( ij ijk k ( (,, ijk jik ijk ijk ijk kij ( * * * * * * ( ( * k uk u + x x k u i (4 Estr ijkij k (5 (5 (5 * * * * 0 0 0 0 0 0 0 0 0 * * 0 0 0 (6 (7 Estr ijkijk ( + + + + + + + + + + + + + + +

More information

68 A mm 1/10 A. (a) (b) A.: (a) A.3 A.4 1 1

68 A mm 1/10 A. (a) (b) A.: (a) A.3 A.4 1 1 67 A Section A.1 0 1 0 1 Balmer 7 9 1 0.1 0.01 1 9 3 10:09 6 A.1: A.1 1 10 9 68 A 10 9 10 9 1 10 9 10 1 mm 1/10 A. (a) (b) A.: (a) A.3 A.4 1 1 A.1. 69 5 1 10 15 3 40 0 0 ¾ ¾ É f Á ½ j 30 A.3: A.4: 1/10

More information

x y x-y σ x + τ xy + X σ y B = + τ xy + Y B = S x = σ x l + τ xy m S y = σ y m + τ xy l σ x σ y τ xy X B Y B S x S y l m δu δv [ ( σx δu + τ )

x y x-y σ x + τ xy + X σ y B = + τ xy + Y B = S x = σ x l + τ xy m S y = σ y m + τ xy l σ x σ y τ xy X B Y B S x S y l m δu δv [ ( σx δu + τ ) 1 8 6 No-tension 1. 1 1.1................................ 1 1............................................ 5.1 - [B].................................. 5................................. 6.3..........................................

More information

2

2 1 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234 12123456789012345678901234

More information

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =,

.5 z = a + b + c n.6 = a sin t y = b cos t dy d a e e b e + e c e e e + e 3 s36 3 a + y = a, b > b 3 s363.7 y = + 3 y = + 3 s364.8 cos a 3 s365.9 y =, [ ] IC. r, θ r, θ π, y y = 3 3 = r cos θ r sin θ D D = {, y ; y }, y D r, θ ep y yddy D D 9 s96. d y dt + 3dy + y = cos t dt t = y = e π + e π +. t = π y =.9 s6.3 d y d + dy d + y = y =, dy d = 3 a, b

More information

chap10.dvi

chap10.dvi . q {y j } I( ( L y j =Δy j = u j = C l ε j l = C(L ε j, {ε j } i.i.d.(,i q ( l= y O p ( {u j } q {C l } A l C l

More information

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 第 2 版 1 刷発行時のものです.

医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます.   このサンプルページの内容は, 第 2 版 1 刷発行時のものです. 医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987

More information

,,..,. 1

,,..,. 1 016 9 3 6 0 016 1 0 1 10 1 1 17 1..,,..,. 1 1 c = h = G = ε 0 = 1. 1.1 L L T V 1.1. T, V. d dt L q i L q i = 0 1.. q i t L q i, q i, t L ϕ, ϕ, x µ x µ 1.3. ϕ x µ, L. S, L, L S = Ld 4 x 1.4 = Ld 3 xdt 1.5

More information

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0

80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0 79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t

More information

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ

SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )

More information

all.dvi

all.dvi 38 5 Cauchy.,,,,., σ.,, 3,,. 5.1 Cauchy (a) (b) (a) (b) 5.1: 5.1. Cauchy 39 F Q Newton F F F Q F Q 5.2: n n ds df n ( 5.1). df n n df(n) df n, t n. t n = df n (5.1) ds 40 5 Cauchy t l n mds df n 5.3: t

More information

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =

1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ = 1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A

More information

4/15 No.

4/15 No. 4/15 No. 1 4/15 No. 4/15 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = m ψ(r,t)+v(r)ψ(r,t) ψ(r,t) = ϕ(r)e iωt ψ(r,t) Wave function steady state m ϕ(r)+v(r)ϕ(r) = εϕ(r) Eigenvalue problem

More information

橡博論表紙.PDF

橡博論表紙.PDF Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction 2003 3 Study on Retaining Wall Design For Circular Deep Shaft Undergoing Lateral Pressure During Construction

More information

2005 2006.2.22-1 - 1 Fig. 1 2005 2006.2.22-2 - Element-Free Galerkin Method (EFGM) Meshless Local Petrov-Galerkin Method (MLPGM) 2005 2006.2.22-3 - 2 MLS u h (x) 1 p T (x) = [1, x, y]. (1) φ(x) 0.5 φ(x)

More information

untitled

untitled SPring-8 RFgun JASRI/SPring-8 6..7 Contents.. 3.. 5. 6. 7. 8. . 3 cavity γ E A = er 3 πε γ vb r B = v E c r c A B A ( ) F = e E + v B A A A A B dp e( v B+ E) = = m d dt dt ( γ v) dv e ( ) dt v B E v E

More information

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0

(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0 1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45

More information

朕醩佑宖醸æ−žã†®ã†�ã‡†ã†®æ··å’‹æŁ´æŁ°è¨‹çfl»ã…¢ã…⁄ã…«

朕醩佑宖醸æ−žã†®ã†�ã‡†ã†®æ··å’‹æŁ´æŁ°è¨‹çfl»ã…¢ã…⁄ã…« 1 / 34 Li-Yao,, Li-Yao The Life-Cycle Effects of House Price Changes (Li-Yao ),,,, ( ) 1 ω ( ) ω 1 γ Ct Ht t T βt U(C t, H t, N t) = N t N t t T βt N t 1 γ H t : t C t : t β : ω : γ : W. Li, R. Yao, The

More information

02-量子力学の復習

02-量子力学の復習 4/17 No. 1 4/17 No. 2 4/17 No. 3 Particle of mass m moving in a potential V(r) V(r) m i ψ t = 2 2m 2 ψ(r,t)+v(r)ψ(r,t) ψ(r,t) Wave function ψ(r,t) = ϕ(r)e iωt steady state 2 2m 2 ϕ(r)+v(r)ϕ(r) = εϕ(r)

More information