[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)

Size: px
Start display at page:

Download "[ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization)"

Transcription

1 . D : E = κ < : E = k > : : β A Green (Dated: June 6, 5) Schrödinger ħh m + g δ (D ) (x ) Ψ(x ) = E Ψ(x ) D D = D = (dimensional transmutation) g <. m D Schrödinger : ħh m + g δ (D ) (x ) Ψ(x ) = E Ψ(x ). (.) E g ħh m ħh m : ħh = m =. (.) Schrödinger (.): + g δ (D ) (x ) Ψ(x ) = E Ψ(x ). (.3) ħh = m =, D δ (D ) (x ) /

2 [ ] = L [δ (D ) (x )] = L D [g ] = L D [E ] = L Table : ħh = m = D D, V (x ) = g δ (D ) (x ) E g D E (Table )D = Schrödinger (.3)D = (regularization) (renormalization)... D Schrödinger (.3) Green D D... : E = κ < E < E = κ (κ > ) Schrödinger : + V (x ) Ψ(x ) = κ Ψ(x ). (.4) V (x ) = g δ (D ) (x ) (.4) : Ψ(x ) = d D y G B (x,y ;κ)v (y )Ψ(y ). (.5) G B (x,y ;κ) Green : κ + x GB (x,y ;κ) = δ (D ) (x y ). (.6) (A): G B (x,y ;κ) = d D p e i p (x y ) (π) D κ p = π κ π x y D K D κ x y. (.7) K ν (z ) ν (.5) Schrödinger (.4) (.5) κ + x (.6) ( κ + )Ψ(x ) = V (x )Ψ(x ) Schrödinger (.4)(.5) Ψ /

3 (.5) Schrödinger V (y ) = g δ (D ) (y ) (.5) y : Ψ(x ) = g G B (x,;κ)ψ(). (.8) Ψ() E = κ < Green (.7) (.8) x = : Ψ() = g G B (,;κ)ψ(). (.9) : = g G B (,;κ). (.) κ κ g E = κ (.) (.) (self-consistency condition)... : E = k > E = k > (k > ) Schrödinger : + V (x ) Ψ(x ) = k Ψ(x ). (.) V (x ) = g δ (D ) (x ) V (x ) x Schrödinger (.) Ψ + k ( k ) : Ψ + k (x ) = ei k x + d D y G + (x,y ;k)v (y )Ψ + k (y ). (.) G + (x,y ;k) Green k + x G + (x,y ;k) = δ (D ) (x y ) (.3) : G + (x,y ;k) = d D p e i p (x y ) (π) D k p + i ε = i 4 k π x y D H () D k x y. (.4) ε H ν () (z ) ν (.) Lippmann- Schwinger. (.) Schrödinger (.4) (.) k + x (.3) (k + )Ψ + k (x ) = V (x )Ψ+ k (x ) Schrödinger (.)(.) Ψ + k Schrödinger (.) Lippmann-Schwinger Schrödinger Schrödinger (.4) ε 3/

4 (.) V (y ) = g δ (D ) (y ) y : x = : Ψ + k () : (.5): Ψ + k (x ) = ei k x + g G + (x,;k)ψ + k (). (.5) Ψ + k () = + g G + (,;k)ψ + k (). (.6) Ψ + k () = g G + (,;k). (.7) Ψ + k (x ) = ei k x + g G + (x,;k) g G + (,;k). (.8) Green Schrödinger H ν () : (z ) z G + (x,y ;k) k D 3 i[k x y (D e 3)π/4] 4π π x y D : πz ei (z (ν+)π/4) Green as x y. (.9) Ψ + k (x ) ei k x [k x (D 3)π/4] ei + f (k) x D as x. (.) f (k) :.. f (k) = k D 3 g Ψ + k 4π π () = k D 3 g 4π π g G + (,;k). (.) Schrödinger (.)(.) G B (,;κ) G + (,;k) : d D p G B (,;κ) = = for D, (.a) (π) κ + p d G + D p (,;k) = = for D. (.b) (π) k p + i ε g D = 4/

5 ... : (.a)(.b) p : d p (π) d p (π). (.3) p <Λ Λ (.) : d p (.) = g G B (,;κ) = g (π) κ + p Λ p <Λ Λ πp d p = g (π) κ + p = g d p 4π κ + p = g κ 4π log + Λ. (.4) κ Λ (.4) Λ = Λ : κ + Λ Λ log = log + κ Λ = log + log + κ κ κ Λ κ Λ κ = log + κ +. (.5) κ Λ Λ Taylor log(+ x ) = x x + Λ (.4) logλ (.5) Λ (.):... : g G B (,;κ) = g 4π log Λ Λ + O (Λ ). (.6) (.a)(.b) D ε D = ε : d p d D (π) p (π), (.7a) D κ g g µ D. (.7b) µ D g (D D ) µ (.) : d (.) = g µ D G B (,;κ) = g µ D D p (π) D κ + p g µ D = (4π) D / Γ (D /) d p κ + p = g (µ/κ) D (4π) D / Γ (D /) p D d x x D / + x = g (µ/κ) D Γ ( D /) (4π) D /. (.8) 5/

6 3 d D p f (p) = d p p D f (p ) (f p = p ) 4 x = p /κ B (p,q ) = πd / Γ (D /) x p Γ (p)γ (q ) d x = ( + x ) p +q Γ (p + q ) (.9) Γ () = (.8) D = D =.99 ε (.8): (.8) = g (µ/κ)ε Γ (ε/) (4π) ε/ D = ε (.3) = g ε/ 4π µ Γ (ε/). (.3) 4π κ ε (4πµ /κ ) ε/ Γ (ε/) ε = : ε/ 4π µ = e ε log 4π µ κ = + ε 4π κ log µ + O (ε ), (.3a) κ Γ (ε/) = ε γ + O (ε). (.3b) γ.577 Euler (.3)(.) : g µ D G B (,;κ) = g µ γ + log(4π) + log + O (ε). (.33) 4π ε κ..3. Λ ε g Λ ε (.6)(.33) g ( E = κ ) (.) : (): (): g = 4π g = 4π Λ µ log + log + O (Λ ), (.34a) µ κ µ γ + log(4π) + log + O (ε). (.34b) ε κ (.34a) µ : (): (): := lim Λ := lim ε g + Λ 4π log, (.35a) µ g + 4π ε γ + log(4π). (.35b) := lim ε g + 4π ε minimal subtraction ( MS) (.35b) modified minimal subtraction ( MS ( )) MS 6/

7 g Λ ε g (.34a)(.34b) Λ ε : = 4π log µ. (.36) κ E B = κ : 4π E B = µ exp. (.37) (.35a)(.35b)(.) D = : κ f (k) = 8πk g G + (,;k). (.38) G + (x,y ;k) = G B (x,y ; i k) κ = k G + (,;k) : Λ µ log + log + O (Λ ) (), G + 4π µ k (,;k) = (.39) µ γ + log(4π) + log + O (ε) (). 4π ε k (.35a)(.35b) f (k) = 8πk + 4π log µ (.4) k (.36) = µ 4π log( E B ) (.4) π f (k) = k log EB k (.4) E B E B g E B.3. β µ E B (.) µ E B µ E B µ : µ d d µ E B =. (.4) 7/

8 d d µ µ d d µ (.4) µ E B µ (µ) E B (µ, (µ)) µ d E B d µ = µ E B µ + µ d d µ (.4): E B µ µ + β(g R ) E B =. (.43) β( ) β : β( ) := µ d d µ. (.44) E B (.37) µ β : = µ E B µ + β( ) E B = µ e 4π/ + β( ) µ e 4π/ 4π gr = 4π gr β( ) E B. (.45) (.45) β (.46) (.47) d g R ( < t < ) : ḡ (t ) β( ) = π g R (.46) µ d d µ = π g R (.47) d g R = d µ π µ µ µet = µe t d µ π µ. (.48) µ ḡ (t ) µe t : ḡ (t ) = π t. (.49) ḡ (t ) (running coupling constant) ḡ (t ) e t k e t k ke t : f (ke t ; ) = 8πke t + 4π log µ k e t = e t 8πk t π + 4π log µ = e t 8πk ḡ (t ) + 4π log µ k k = e t f (k;ḡ (t )). (.5) 8/

9 ḡ (t ) π π t Figure : ḡ (t ) =. t = g π t R () ( ) e t / (.5) k e t k ḡ (t ) ḡ (t ) t ḡ (t ) () t. : C. Thorn, Quark confinement in the infinite-momentum frame, Phys. Rev. D9 (979) [INSPIRE]. Roman Jackiw 99 : R. Jackiw, Delta function potentials in two-dimensional and three-dimensional quantum mechanics, in M. A. B. Bég Memorial Volume, A. Ali and P. Hoodbhoy, eds. World Scientific Publishing, 99 [INSPIRE]; scanned version available in the KEK library: (self-adjoint extension) ( ) Jackiw Schrödinger (.3) Schrödinger. 9/

10 S[ψ,ψ ] = d t d x i ψ t ψ ψ g ψ 4 (.) 4 Γ (4) Γ (4) Oren Bergman 99 O. Bergman, Nonrelativistic field theoretic scale anomaly, Phys. Rev. D46 (99) [INSPIRE].. Lippmann-Schwinger Bernard Lippmann Julian Schwinger B. A. Lippmann and J. Schwinger, Variational Principles for Scattering Processes. I, Phys. Rev. 79 (95) [INSPIRE] Steven Weinberg Lectures on Quantum Mechanics..3Sidney Coleman Erick Weinberg 973 Coleman-Weinberg S. Coleman and E. Weinberg, Radiative Corrections as the Origin of Spontaneous Symmetry Breaking, Phys. Rev. D7 (973) [INSPIRE] Kenneth G. Wilson K. G. Wilson and J. B. Kogut, The renormalization group and the ε expansion, Phys. Rep. (974) [INSPIRE] A. Green Green G B G B : ( κ + x )G B (x,y ;κ) = δ (D ) (x y ). (A.) Fourier : G B (x,y ;κ) = d D p (π) D e i p (x y ) κ + p. (A.) Schwinger : κ + p = d t e t (κ +p ). (A.3) /

11 Schwinger t Schwinger (A.3)(A.): G B (x,y ;κ) = = d D p (π) D d t e t κ = (4π) D / = κ 4π π x y d t e t (κ +p )+i p (x y ) x y 4t D j = d t t D / e t κ D x y 4t d s e s D / d p j π e t p j i x j y j t κ x y (s + s ). (A.4) s = κt / x y Green G B (x,y ;κ) = π κ π x y D K D κ x y (A.5) 3 : κ e κ x y G B (x,y ;κ) = π K κ x y for D = ; for D = ; (A.6) K / (z ) = K / (z ) = π z e z 4π x y e κ x y for D = 3. /

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)

φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1) φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x

More information

Chebyshev Schrödinger Heisenberg H = 1 2m p2 + V (x), m = 1, h = 1 1/36 1 V (x) = { 0 (0 < x < L) (otherwise) ψ n (x) = 2 L sin (n + 1)π x L, n = 0, 1, 2,... Feynman K (a, b; T ) = e i EnT/ h ψ n (a)ψ

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

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n

2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n . X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n

More information

Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t)

Radiation from moving charges#1 Liénard-Wiechert potential Yuji Chinone 1 Maxwell Maxwell MKS E (x, t) + B (x, t) t = 0 (1) B (x, t) = 0 (2) B (x, t) Radiation from moving harges# Liénard-Wiehert potential Yuji Chinone Maxwell Maxwell MKS E x, t + B x, t = B x, t = B x, t E x, t = µ j x, t 3 E x, t = ε ρ x, t 4 ε µ ε µ = E B ρ j A x, t φ x, t A x, t

More information

1 (Contents) (4) Why Has the Superstring Theory Collapsed? Noboru NAKANISHI 2 2. A Periodic Potential Problem

1 (Contents) (4) Why Has the Superstring Theory Collapsed? Noboru NAKANISHI 2 2. A Periodic Potential Problem 7 1 2017 3 2017 3 18 1 (Contents) 1. 2 2. (4) 13 3. 1 20 4. 25 5. 28 6. 29 1. Why Has the Superstring Theory Collapsed? Noboru NAKANISHI 2 2. A Periodic Potential Problem in Quantum Mechanics (4) Kenji

More information

main.dvi

main.dvi SGC - 48 208X Y Z Z 2006 1930 β Z 2006! 1 2 3 Z 1930 SGC -12, 2001 5 6 http://www.saiensu.co.jp/support.htm http://www.shinshu-u.ac.jp/ haru/ xy.z :-P 3 4 2006 3 ii 1 1 1.1... 1 1.2 1930... 1 1.3 1930...

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

微分積分 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, 初版 1 刷発行時のものです.

微分積分 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, 初版 1 刷発行時のものです. 微分積分 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. ttp://www.morikita.co.jp/books/mid/00571 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 ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)

More information

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.

4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5. A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c

More information

4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = cm 896 Zeeman Zeeman Zeeman Lorentz

4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n 2 n = n +,n +2, n = Lyman n =2 Balmer n =3 Paschen R Rydberg R = cm 896 Zeeman Zeeman Zeeman Lorentz 2 Rutherford 2. Rutherford N. Bohr Rutherford 859 Kirchhoff Bunsen 86 Maxwell Maxwell 885 Balmer λ Balmer λ = 364.56 n 2 n 2 4 Lyman, Paschen 3 nm, n =3, 4, 5, 4 2 Rutherford 89 Rydberg λ = R ( n 2 ) n

More information

1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji

1 (Contents) (1) Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji 8 4 2018 6 2018 6 7 1 (Contents) 1. 2 2. (1) 22 3. 31 1. Beginning of the Universe, Dark Energy and Dark Matter Noboru NAKANISHI 2 2. Problem of Heat Exchanger (1) Kenji SETO 22 3. Editorial Comments Tadashi

More information

Anderson ( ) Anderson / 14

Anderson ( ) Anderson / 14 Anderson 2008 12 ( ) Anderson 2008 12 1 / 14 Anderson ( ) Anderson 2008 12 2 / 14 Anderson P.W.Anderson 1958 ( ) Anderson 2008 12 3 / 14 Anderson tight binding Anderson tight binding Z d u (x) = V i u

More information

E 1/2 3/ () +3/2 +3/ () +1/2 +1/ / E [1] B (3.2) F E 4.1 y x E = (E x,, ) j y 4.1 E int = (, E y, ) j y = (Hall ef

E 1/2 3/ () +3/2 +3/ () +1/2 +1/ / E [1] B (3.2) F E 4.1 y x E = (E x,, ) j y 4.1 E int = (, E y, ) j y = (Hall ef 4 213 5 8 4.1.1 () f A exp( E/k B ) f E = A [ k B exp E ] = f k B k B = f (2 E /3n). 1 k B /2 σ = e 2 τ(e)d(e) 2E 3nf 3m 2 E de = ne2 τ E m (4.1) E E τ E = τe E = / τ(e)e 3/2 f de E 3/2 f de (4.2) f (3.2)

More information

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3

u = u(t, x 1,..., x d ) : R R d C λ i = 1 := x 2 1 x 2 d d Euclid Laplace Schrödinger N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3 2 2 1 5 5 Schrödinger i u t + u = λ u 2 u. u = u(t, x 1,..., x d ) : R R d C λ i = 1 := 2 + + 2 x 2 1 x 2 d d Euclid Laplace Schrödinger 3 1 1.1 N := {1, 2, 3,... } Z := {..., 3, 2, 1,, 1, 2, 3,... } Q

More information

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices

q quark L left-handed lepton. λ Gell-Mann SU(3), a = 8 σ Pauli, i =, 2, 3 U() T a T i 2 Ỹ = 60 traceless tr Ỹ 2 = 2 notation. 2 off-diagonal matrices Grand Unification M.Dine, Supersymmetry And String Theory: Beyond the Standard Model 6 2009 2 24 by Standard Model Coupling constant θ-parameter 8 Charge quantization. hypercharge charge Gauge group. simple

More information

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x

Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)

More information

( )

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

More information

: , 2.0, 3.0, 2.0, (%) ( 2.

: , 2.0, 3.0, 2.0, (%) ( 2. 2017 1 2 1.1...................................... 2 1.2......................................... 4 1.3........................................... 10 1.4................................. 14 1.5..........................................

More information

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e

N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >

More information

第86回日本感染症学会総会学術集会後抄録(I)

第86回日本感染症学会総会学術集会後抄録(I) κ κ κ κ κ κ μ μ β β β γ α α β β γ α β α α α γ α β β γ μ β β μ μ α ββ β β β β β β β β β β β β β β β β β β γ β μ μ μ μμ μ μ μ μ β β μ μ μ μ μ μ μ μ μ μ μ μ μ μ β

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

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin

More information

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H

V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H 199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)

More information

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論 email: takahash@sci.u-hyogo.ac.jp May 14, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 262 Today s Lecture: Mode-mode Coupling Theory 100 / 262 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear

More information

arxiv: v1(astro-ph.co)

arxiv: v1(astro-ph.co) arxiv:1311.0281v1(astro-ph.co) R µν 1 2 Rg µν + Λg µν = 8πG c 4 T µν Λ f(r) R f(r) Galileon φ(t) Massive Gravity etc... Action S = d 4 x g (L GG + L m ) L GG = K(φ,X) G 3 (φ,x)φ + G 4 (φ,x)r + G 4X (φ)

More information

QMI13a.dvi

QMI13a.dvi I (2013 (MAEDA, Atsutaka) 25 10 15 [ I I [] ( ) 0. (a) (b) Plank Compton de Broglie Bohr 1. (a) Einstein- de Broglie (b) (c) 1 (d) 2. Schrödinger (a) Schrödinger (b) Schrödinger (c) (d) 3. (a) (b) (c)

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

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

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

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i

July 28, H H 0 H int = H H 0 H int = H int (x)d 3 x Schrödinger Picture Ψ(t) S =e iht Ψ H O S Heisenberg Picture Ψ H O H (t) =e iht O S e i July 8, 4. H H H int H H H int H int (x)d 3 x Schrödinger Picture Ψ(t) S e iht Ψ H O S Heisenberg Picture Ψ H O H (t) e iht O S e iht Interaction Picture Ψ(t) D e iht Ψ(t) S O D (t) e iht O S e ih t (Dirac

More information

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google

I A A441 : April 21, 2014 Version : Kawahira, Tomoki TA (Kondo, Hirotaka ) Google I4 - : April, 4 Version :. Kwhir, Tomoki TA (Kondo, Hirotk) Google http://www.mth.ngoy-u.c.jp/~kwhir/courses/4s-biseki.html pdf 4 4 4 4 8 e 5 5 9 etc. 5 6 6 6 9 n etc. 6 6 6 3 6 3 7 7 etc 7 4 7 7 8 5 59

More information

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1

QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 QCD 1 QCD GeV 2014 QCD 2015 QCD SU(3) QCD A µ g µν QCD 1 (vierbein) QCD QCD 1 1: QCD QCD Γ ρ µν A µ R σ µνρ F µν g µν A µ Lagrangian gr TrFµν F µν No. Yes. Yes. No. No! Yes! [1] Nash & Sen [2] Riemann

More information

DVIOUT-fujin

DVIOUT-fujin 2005 Limit Distribution of Quantum Walks and Weyl Equation 2006 3 2 1 2 2 4 2.1...................... 4 2.2......................... 5 2.3..................... 6 3 8 3.1........... 8 3.2..........................

More information

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising

,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising ,, Andrej Gendiar (Density Matrix Renormalization Group, DMRG) 1 10 S.R. White [1, 2] 2 DMRG ( ) [3, 2] DMRG Baxter [4, 5] 2 Ising 2 1 Ising 1 1 Ising Model 1 Ising 1 Ising Model N Ising (σ i = ±1) (Free

More information

II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re

II ( ) (7/31) II (  [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier

More information

基礎数学I

基礎数学I I & II ii ii........... 22................. 25 12............... 28.................. 28.................... 31............. 32.................. 34 3 1 9.................... 1....................... 1............

More information

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint (

9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) Ĥ0 ψ n (r) ω n Schrödinger Ĥ 0 ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ0 + Ĥint ( 9 1. (Ti:Al 2 O 3 ) (DCM) (Cr:Al 2 O 3 ) (Cr:BeAl 2 O 4 ) 2. 2.1 Ĥ ψ n (r) ω n Schrödinger Ĥ ψ n (r) = ω n ψ n (r), (1) ω i ψ (r, t) = [Ĥ + Ĥint (t)] ψ (r, t), (2) Ĥ int (t) = eˆxe cos ωt ˆdE cos ωt, (3)

More information

Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W

Black-Scholes [1] Nelson [2] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [2][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-W 003 7 14 Black-Scholes [1] Nelson [] Schrödinger 1 Black Scholes [1] Black-Scholes Nelson [][3][4] Schrödinger Nelson Parisi Wu [5] Nelson Parisi-Wu Nelson e-mail: takatoshi-tasaki@nifty.com kabutaro@mocha.freemail.ne.jp

More information

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT

I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345

More information

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes )

( ) ) ) ) 5) 1 J = σe 2 6) ) 9) 1955 Statistical-Mechanical Theory of Irreversible Processes ) ( 3 7 4 ) 2 2 ) 8 2 954 2) 955 3) 5) J = σe 2 6) 955 7) 9) 955 Statistical-Mechanical Theory of Irreversible Processes 957 ) 3 4 2 A B H (t) = Ae iωt B(t) = B(ω)e iωt B(ω) = [ Φ R (ω) Φ R () ] iω Φ R (t)

More information

ssp2_fixed.dvi

ssp2_fixed.dvi 13 12 30 2 1 3 1.1... 3 1.2... 4 1.3 Bravais... 4 1.4 Miller... 4 2 X 5 2.1 Bragg... 5 2.2... 5 2.3... 7 3 Brillouin 13 3.1... 13 3.2 Brillouin... 13 3.3 Brillouin... 14 3.4 Bloch... 16 3.5 Bloch... 17

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

Xray.dvi

Xray.dvi 1 X 1 X 1 1.1.............................. 1 1.2.................................. 3 1.3........................ 3 2 4 2.1.................................. 6 2.2 n ( )............. 6 3 7 3.1 ( ).....................

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

1. 1.1....................... 1.2............................ 1.3.................... 1.4.................. 2. 2.1.................... 2.2..................... 2.3.................... 3. 3.1.....................

More information

ohpmain.dvi

ohpmain.dvi fujisawa@ism.ac.jp 1 Contents 1. 2. 3. 4. γ- 2 1. 3 10 5.6, 5.7, 5.4, 5.5, 5.8, 5.5, 5.3, 5.6, 5.4, 5.2. 5.5 5.6 +5.7 +5.4 +5.5 +5.8 +5.5 +5.3 +5.6 +5.4 +5.2 =5.5. 10 outlier 5 5.6, 5.7, 5.4, 5.5, 5.8,

More information

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ

Aharonov-Bohm(AB) S 0 1/ 2 1/ 2 S t = 1/ 2 1/2 1/2 1/, (12.1) 2 1/2 1/2 *1 AB ( ) 0 e iθ AB S AB = e iθ, AB 0 θ 2π ϕ = e ϕ (ϕ ) ϕ 1 13 6 8 3.6.3 - Aharonov-BohmAB) S 1/ 1/ S t = 1/ 1/ 1/ 1/, 1.1) 1/ 1/ *1 AB ) e iθ AB S AB = e iθ, AB θ π ϕ = e ϕ ϕ ) ϕ 1.) S S ) e iθ S w = e iθ 1.3) θ θ AB??) S t = 4 sin θ 1 + e iθ AB e iθ AB + e

More information

i E B Maxwell Maxwell Newton Newton Schrödinger Newton Maxwell Kepler Maxwell Maxwell B H B ii Newton i 1 1.1.......................... 1 1.2 Coulomb.......................... 2 1.3.........................

More information

QMII_10.dvi

QMII_10.dvi 65 1 1.1 1.1.1 1.1 H H () = E (), (1.1) H ν () = E ν () ν (). (1.) () () = δ, (1.3) μ () ν () = δ(μ ν). (1.4) E E ν () E () H 1.1: H α(t) = c (t) () + dνc ν (t) ν (), (1.5) H () () + dν ν () ν () = 1 (1.6)

More information

反D中間子と核子のエキゾチックな 束縛状態と散乱状態の解析

反D中間子と核子のエキゾチックな   束縛状態と散乱状態の解析 .... D 1 in collaboration with 1, 2, 1 RCNP 1, KEK 2 . Exotic hadron qqq q q Θ + Λ(1405) etc. uudd s? KN quasi-bound state? . D(B)-N bound state { { D D0 ( cu) B = D ( cd), B = + ( bu) B 0 ( bd) D(B)-N

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

eto-vol1.dvi

eto-vol1.dvi ( 1) 1 ( [1] ) [] ( ) (AC) [3] [4, 5, 6] 3 (i) AC (ii) (iii) 3 AC [3, 7] [4, 5, 6] 1.1 ( e; e>0) Ze r v [ 1(a)] v [ 1(a )] B = μ 0 4π Zer v r 3 = μ 0 4π 1 Ze l m r 3, μ 0 l = mr v ( l s ) s μ s = μ B s

More information

19 /

19 / 19 / 1 1.1............................... 1. Andreev............................... 3 1.3..................................... 3 1.4..................................... 4 / 5.1......................................

More information

kawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2

kawa (Spin-Orbit Tomography: Kawahara and Fujii 21,Kawahara and Fujii 211,Fujii & Kawahara submitted) 2 van Cittert-Zernike Appendix A V 2 Hanbury-Brown Twiss (ver. 1.) 24 2 1 1 1 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 3 3 Hanbury-Brown Twiss ( ) 4 3.1............................................

More information

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6

O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-1 O1-2 O1-3 O1-4 O1-5 O1-6 O1-7 O1-8 O1-9 O1-10 O1-11 O1-12 O1-13 O1-14 O1-15 O1-16 O1-17 O1-18 O1-19 O1-20 O1-21 O1-22 O1-23 O1-24 O1-25 O1-26 O1-27 O1-28 O1-29 O1-30 O1-31 O1-32 O1-33 O1-34 O1-35

More information

I A A441 : April 15, 2013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida )

I A A441 : April 15, 2013 Version : 1.1 I   Kawahira, Tomoki TA (Shigehiro, Yoshida ) I013 00-1 : April 15, 013 Version : 1.1 I Kawahira, Tomoki TA (Shigehiro, Yoshida) http://www.math.nagoya-u.ac.jp/~kawahira/courses/13s-tenbou.html pdf * 4 15 4 5 13 e πi = 1 5 0 5 7 3 4 6 3 6 10 6 17

More information

Untitled

Untitled II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j

More information

koji07-02.dvi

koji07-02.dvi 007 I II III 1,, 3, 4, 5, 6, 7 5 4 1 ε-n 1 ε-n ε-n ε-n. {a } =1 a ε N N a a N= a a

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

d > 2 α B(y) y (5.1) s 2 = c z = x d 1+α dx ln u 1 ] 2u ψ(u) c z y 1 d 2 + α c z y t y y t- s 2 2 s 2 > d > 2 T c y T c y = T t c = T c /T 1 (3.

d > 2 α B(y) y (5.1) s 2 = c z = x d 1+α dx ln u 1 ] 2u ψ(u) c z y 1 d 2 + α c z y t y y t- s 2 2 s 2 > d > 2 T c y T c y = T t c = T c /T 1 (3. 5 S 2 tot = S 2 T (y, t) + S 2 (y) = const. Z 2 (4.22) σ 2 /4 y = y z y t = T/T 1 2 (3.9) (3.15) s 2 = A(y, t) B(y) (5.1) A(y, t) = x d 1+α dx ln u 1 ] 2u ψ(u), u = x(y + x 2 )/t s 2 T A 3T d S 2 tot S

More information

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha

Euler, Yang-Mills Clebsch variable Helicity ( Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity (i) Yang-Mills 3 A T (T A) Poisson Ha Euler, Yang-ills Clebsch variable Helicity Tosiaki Kori ) School of Sciences and Technology, Waseda Uiversity i) Yang-ills 3 A T T A) Poisson Hamilton ii) Clebsch parametrization iii) Y- Y-iv) Euler,v)

More information

·«¤ê¤³¤ß·²¤È¥ß¥ì¥Ë¥¢¥àÌäÂê

·«¤ê¤³¤ß·²¤È¥ß¥ì¥Ë¥¢¥àÌäÂê .. 1 10-11 Nov., 2016 1 email:keiichi.r.ito@gmail.com, ito@kurims.kyoto-u.ac.jp ( ) 10-11 Nov., 2016 1 / 45 Clay Institute.1 Construction of 4D YM Field Theory (Jaffe, Witten) Jaffe, Balaban (1980).2 Solution

More information

nsg02-13/ky045059301600033210

nsg02-13/ky045059301600033210 φ φ φ φ κ κ α α μ μ α α μ χ et al Neurosci. Res. Trpv J Physiol μ μ α α α β in vivo β β β β β β β β in vitro β γ μ δ μδ δ δ α θ α θ α In Biomechanics at Micro- and Nanoscale Levels, Volume I W W v W

More information

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +

ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + 2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j

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

(Onsager )

(Onsager ) 05819311 (Onsager ) 1 2 2 Onsager 4 3 11 3.1................ 11 3.2............ 14 3.3................... 16 4 18 4.1........... 18 4.2............. 20 5 25 A 27 A.1................. 27 A.2..............

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

8 (2006 ) X ( ) 1. X X X 2. ( ) ( ) ( 1) X (a) (b) 1: (a) (b)

8 (2006 ) X ( ) 1. X X X 2. ( ) ( ) ( 1) X (a) (b) 1: (a) (b) 8 (2006 ) X ( ) 1. X X X 2. ( ) ( ) ( 1) X (a) (b) 1: (a) (b) X hkl 2θ ω 000 2: ω X 2θ X 3: X 2 X ω X 2θ X θ-2θ X X 2-1. ( ) ( 3) X 2θ ω 4 Si GaAs Si/Si GaAs/GaAs X 2θ : 2 2θ 000 ω 000 ω ω = θ 4: 2θ ω

More information

1 1.1 Excel Excel Excel log 1, log 2, log 3,, log 10 e = ln 10 log cm 1mm 1 10 =0.1mm = f(x) f(x) = n

1 1.1 Excel Excel Excel log 1, log 2, log 3,, log 10 e = ln 10 log cm 1mm 1 10 =0.1mm = f(x) f(x) = n 1 1.1 Excel Excel Excel log 1, log, log,, log e.7188188 ln log 1. 5cm 1mm 1 0.1mm 0.1 4 4 1 4.1 fx) fx) n0 f n) 0) x n n! n + 1 R n+1 x) fx) f0) + f 0) 1! x + f 0)! x + + f n) 0) x n + R n+1 x) n! 1 .

More information

( ) a C n ( R n ) R a R C n. a C n (or R n ) a 0 2. α C( R ) a C n αa = α a 3. a, b C n a + b a + b ( ) p 8..2 (p ) a = [a a n ] T C n p n a

( ) a C n ( R n ) R a R C n. a C n (or R n ) a 0 2. α C( R ) a C n αa = α a 3. a, b C n a + b a + b ( ) p 8..2 (p ) a = [a a n ] T C n p n a 9 8 m n mn N.J.Nigham, Accuracy and Stability of Numerical Algorithms 2nd ed., (SIAM) x x = x2 + y 2 = x + y = max( x, y ) x y x () (norm) (condition number) 8. R C a, b C a b 0 a, b a = a 0 0 0 n C n

More information

* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *

* 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) * * 1 1 (i) (ii) Brückner-Hartree-Fock (iii) (HF, BCS, HFB) (iv) (TDHF,TDHFB) (RPA) (QRPA) (v) (vi) *1 2004 1 1 ( ) ( ) 1.1 140 MeV 1.2 ( ) ( ) 1.3 2.6 10 8 s 7.6 10 17 s? Λ 2.5 10 10 s 6 10 24 s 1.4 ( m

More information

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ)

ʪ¼Á¤Î¥È¥Ý¥í¥¸¥«¥ë¸½¾Ý  (2016ǯ¥Î¡¼¥Ù¥ë¾Þ¤Ë´ØÏ¢¤·¤Æ) (2016 ) Dept. of Phys., Kyushu Univ. 2017 8 10 1 / 59 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER 2 / 59 ( ) ( ) (Dirac, t Hooft-Polyakov)

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

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence

Hanbury-Brown Twiss (ver. 2.0) van Cittert - Zernike mutual coherence Hanbury-Brown Twiss (ver. 2.) 25 4 4 1 2 2 2 2.1 van Cittert - Zernike..................................... 2 2.2 mutual coherence................................. 4 3 Hanbury-Brown Twiss ( ) 5 3.1............................................

More information

0406_total.pdf

0406_total.pdf 59 7 7.1 σ-ω σ-ω σ ω σ = σ(r), ω µ = δ µ,0 ω(r) (6-4) (iγ µ µ m U(r) γ 0 V (r))ψ(x) = 0 (7-1) U(r) = g σ σ(r), V (r) = g ω ω(r) σ(r) ω(r) (6-3) ( 2 + m 2 σ)σ(r) = g σ ψψ (7-2) ( 2 + m 2 ω)ω(r) = g ω ψγ

More information

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í

‚åŁÎ“·„´Šš‡ðŠp‡¢‡½‹âfi`fiI…A…‰…S…−…Y…•‡ÌMarkovŸA“½fiI›ð’Í Markov 2009 10 2 Markov 2009 10 2 1 / 25 1 (GA) 2 GA 3 4 Markov 2009 10 2 2 / 25 (GA) (GA) L ( 1) I := {0, 1} L f : I (0, ) M( 2) S := I M GA (GA) f (i) i I Markov 2009 10 2 3 / 25 (GA) ρ(i, j), i, j I

More information

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)

m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120) 2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ

More information

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc

1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc 013 6 30 BCS 1 1.1........................ 1................................ 3 1.3............................ 3 1.4............................... 5 1.5.................................... 5 6 3 7 4 8

More information

211 kotaro@math.titech.ac.jp 1 R *1 n n R n *2 R n = {(x 1,..., x n ) x 1,..., x n R}. R R 2 R 3 R n R n R n D D R n *3 ) (x 1,..., x n ) f(x 1,..., x n ) f D *4 n 2 n = 1 ( ) 1 f D R n f : D R 1.1. (x,

More information

Chern-Simons Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q

Chern-Simons   Jones 3 Chern-Simons 1 - Chern-Simons - Jones J(K; q) [1] Jones q 1 J (K + ; q) qj (K ; q) = (q 1/2 q Chern-Simons E-mail: fuji@th.phys.nagoya-u.ac.jp Jones 3 Chern-Simons - Chern-Simons - Jones J(K; q) []Jones q J (K + ; q) qj (K ; q) = (q /2 q /2 )J (K 0 ; q), () J( ; q) =. (2) K Figure : K +, K, K 0

More information

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ

p = mv p x > h/4π λ = h p m v Ψ 2 Ψ II p = mv p x > h/4π λ = h p m v Ψ 2 Ψ Ψ Ψ 2 0 x P'(x) m d 2 x = mω 2 x = kx = F(x) dt 2 x = cos(ωt + φ) mω 2 = k ω = m k v = dx = -ωsin(ωt + φ) dt = d 2 x dt 2 0 y v θ P(x,y) θ = ωt + φ ν = ω [Hz] 2π

More information

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e

( ) Note (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e ( ) Note 3 19 12 13 8 8.1 (e ) (µ ) (τ ) ( (ν e,e ) e- (ν µ, µ ) µ- (ν τ,τ ) τ- ) ( ) ( ) (SU(2) ) (W +,Z 0,W ) * 1) 3 * 2) [ ] [ ] [ ] ν e ν µ ν τ e µ τ, e R, µ R, τ R (1a) L ( ) ) * 3) W Z 1/2 ( - )

More information

( ) s n (n = 0, 1,...) n n = δ nn n n = I n=0 ψ = n C n n (1) C n = n ψ α = e 1 2 α 2 n=0 α, β α n n! n (2) β α = e 1 2 α 2 1

( ) s n (n = 0, 1,...) n n = δ nn n n = I n=0 ψ = n C n n (1) C n = n ψ α = e 1 2 α 2 n=0 α, β α n n! n (2) β α = e 1 2 α 2 1 (3.5 3.8) 03032s 2006.7.0 n (n = 0,,...) n n = δ nn n n = I n=0 ψ = n C n n () C n = n ψ α = e 2 α 2 n=0 α, β α n n (2) β α = e 2 α 2 2 β 2 n=0 =0 = e 2 α 2 β n α 2 β 2 n=0 = e 2 α 2 2 β 2 +β α β n α!

More information

I

I I 6 4 10 1 1 1.1............... 1 1................ 1 1.3.................... 1.4............... 1.4.1.............. 1.4................. 1.4.3........... 3 1.4.4.. 3 1.5.......... 3 1.5.1..............

More information

( ) Loewner SLE 13 February

( ) Loewner SLE 13 February ( ) Loewner SLE 3 February 00 G. F. Lawler, Conformally Invariant Processes in the Plane, (American Mathematical Society, 005)., Summer School 009 (009 8 7-9 ) . d- (BES d ) d B t = (Bt, B t,, Bd t ) (d

More information

確率論と統計学の資料

確率論と統計学の資料 5 June 015 ii........................ 1 1 1.1...................... 1 1........................... 3 1.3... 4 6.1........................... 6................... 7 ii ii.3.................. 8.4..........................

More information

1 Tokyo Daily Rainfall (mm) Days (mm)

1 Tokyo Daily Rainfall (mm) Days (mm) ( ) r-taka@maritime.kobe-u.ac.jp 1 Tokyo Daily Rainfall (mm) 0 100 200 300 0 10000 20000 30000 40000 50000 Days (mm) 1876 1 1 2013 12 31 Tokyo, 1876 Daily Rainfall (mm) 0 50 100 150 0 100 200 300 Tokyo,

More information

1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1)

1 9 v.0.1 c (2016/10/07) Minoru Suzuki T µ 1 (7.108) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) 1 9 v..1 c (216/1/7) Minoru Suzuki 1 1 9.1 9.1.1 T µ 1 (7.18) f(e ) = 1 e β(e µ) 1 E 1 f(e ) (Bose-Einstein distribution function) *1 (8.1) (9.1) E E µ = E f(e ) E µ (9.1) µ (9.2) µ 1 e β(e µ) 1 f(e )

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