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

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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) Vx) + V 1 x /) x ξh n ξ) = 1 H n+1ξ) + nh n 1 ξ) iii) x E Vx) Vx) E. Vx) m V cos x + 1) π x π Vx) = x < π, x > π V x t = x V

2 199 L b a l v σ C R R V 1. T Qt) l L, T l/v. C Q c t) Qt) Q c t) jt) Vt) 3. Qt) Vt) Vt) ˆVω) ) Vt) ˆQω) ω ˆVω) ˆVω) ω = Qt) = ˆQω)e iωt dω, Vt) = ˆVω)e iωt dω 4. Vt) 1MΩ R = 1kΩ, C = Farad, l = 3m, L =.3m, v = m/sec l N = e = Coulomb R 5Ω a b V L l beam mettalic pipe capacitance C resistance R V potential difference earth

3 S = L x L y L x, L y x y m n k B π T 1.. ρe) dne)/de ρe) µ µ n T E F T 3. k B T E F x d f dx = π dx 3, f x) = 1 e x E F E E E F + E ρe) = E E F α α > 1 k B T E

4 X X X X R 6.cm S 1 A G X 1.54Å 1. X A) X X X B) C) X D) E) X D) F) G) G) S 3.3cm 5. S = 5.9cm 1 sin 13.5 = =.471rad sin 15.8 =.73 sin.6 =.35 sin 5.4 =.49 sin 8. =.473 sin 3.3 =.55 sin 3.7 = =.55rad 41. =.719rad 5.8 =.886rad 56.4 =.983rad 6.6 = 1.58rad 65.4 = 1.141rad

5

6 S V δ r 1 r ) V δ r) r 1, r p 1 m + p m V δ r 1 r ) ψ r 1, r ) = εψ r 1, r ) 1) m p 1, p 1. ψ r 1, r ) r 1 r ψ r 1, r ) ψ r 1, r ). ψ r 1, r ) g) ψ r 1, r ) = e i r 1 r ) g) ) 3. g) ε 1 )g) V g q) S = εg ) 3) q ε) = k /m δ r) = 1 S q e i q r ) q g q) k S q g q) = C C 3) g) V C g) = ε) ε ε V S 1 ε) ε = 1 4) 5. 4) ε) ε ε ε 4) ε mv /π 1 E = ε ] E = ε exp [ 4π mv

7 M = kg R 1 = 1 6 m R = 1 4 m U G. L = 3 1 m n ν 1 U E ν 3. 1/6 ν e ν e ν e + p n + e + M D = kg n ν σ = 1 46 m 1/ E ν = 1MeV 4. t E min E max G = m 3 kg 1 s N A = eV = J

8 φ C N C C ) ψn C C N) 3. α- β- 4. 1

9 i) x Vx) V 1 x + x4 4 + ) V 1 x ) 1) Schrödinger eq. d ) E m dx ψ + + V mω x ψ = d { me + dx ψ + V ) mω ) x } ψ = V = mω ) x = αx d dx = 1 d α dx, x = αx) α 4 = mω ) d dx ψ + λ X )ψ = ), λ = me + V ) α = E + V ) ω 3) 4) u = e ± X u + 1 X )u = X ± 4) 4) e ± X e X 4) ψx) = e X ϕx) 4) d dϕ ϕ X + λ 1)ϕ = 5) dx dx λ 1) = nn =, 1, ) X e X e X λ = n + 1n =, 1, ) 6) 3) E = E n = n + 1 ) ω V 7) 7) 6) 5) n Hermite N ψ n X) = Ne X Hn X) ψx) dx = 1 1 = N e X H n X) αdx = αn π n n! N = α π n n!) 1 π 1 = mω n n! π 1 [ ψ n x) = mω n n! exp mω ] ) mω x H n x 8)

10 x E n V n + 1 ) ω V ii) ) Vx) = V cos x 1 + x iii) E n = n Vx) n = = = = = = = V 4 N α 5 N e X H nx) e X H nx)x 4 dx e X XH n X)) X dx V 4 x4 N e α) x Hn V ) 4 α4 X 4 αdx e X H nx)x 4 dx e X [ 1 H n+1 + nh n 1 ] X dx x α) V 4 x4 ) dx e X [ 1 4 XH n+1) + nxh n+1 )XH n 1 ) + n XH n 1 ) ] dx e 1 { 1 X 4 H n+ + n + 1)H n { 1 H n + n 1)H n } { 1 + n } H n+ + n + 1)H n } + n { 1 H n + n 1)H n } dx H n [ ] [ { = e X 4 4 H n+ + 4 n + 1) + nn + 1) } ] n Hn + n n 1) Hn dx 4 [ ] 1 n + = e X 16 H 1) n+ + Hn + n n 1) Hn dx 4 = 1 π n+ n + 1) n + )! + π n n! + n n 1) π n n )! 16 4 = { } n + 1)n + ) π n n + 1) nn 1) n! = π n n! 6n + 6n E n = n Vx) n = V 4 N α 5 π n n! 6n + 6n α 5 π n n! 6n + 6n + 3 = V 4 = V 3 α π n n! 4 ) n + n + 1) mω = 3m n + n + 1) V = mω ) Vx) = V cos x eex = V 1 + ee x 1 V x + 1 ) 4 x4 = V 1 1 x ee ) + e E + 1 V 4 x4 V

11 ) x ee V x ee V 1 e E E n = n + 1 ) ω V e E V V. ground state 1st excited state ground E ψ x) 1st excited E 1 ψ 1 x) ground state 1st excited state ψ ψ x), ψ 1 x) Schrödinger eq. i t ψ nx, t) = E n ψ n x, t) n =, 1) [ ψ x, t) = exp ie ] t ψ x), [ ψx, t) = C ψ x, t) + C 1 ψ 1 x, t) = exp [ ψ 1 x, t) = exp ie t ie 1t ] ψ 1 x) ] { C ψ x) + C 1 exp ψx, t + π E 1 E ) = ψ x, t), ψx, t + π E 1 E ) = ψx, t) [ ie1 E )t ] } ψ 1 x) ω = E 1 E x > x <

12 1 199 Qt) 1. RC l/v L/v σl σl L + l)/v l/v Qt) T t. Q c j d dt Qt) + Q ct)) = jt) 1) Vt) = R jt) = 1 C Q ct) ) +Q C ) 1) j, Q c dvt) + 1 dt RC Vt) = 1 C dqt) 3) C dt R j V 3. Qt) T Vt) = Ṽω)e iωt dω, Qt) = Qω)e iωt dω 4) 4) 3) iωṽω) + 1 RC Ṽω) = iω C Qω)... Ṽω) = iω Qω) 1 RC + iω C 5) Qt) t l/v σl Qω) Qω) = 1 π Qt)e iωt dt = 1 π Ṽω) l/v σl)e iωt dt = σl π e iωl/v 1 iω Ṽω) = 1 π Ṽω) = 1 π e iωl/v 1 σl 1 RC + iω C σl C sin ωl v ) 1 RC + ω RC π σl ωl C v ω ) 6) Ṽω)

13 Vω) ω 4. 6) Vt) Vt) 3) Vt) = Ṽω)e iωt dω = 1 σl e iωt l/v) e iωt dω π C 1 1 RC + iω RC + iωdω, t < ) σl = C e t RC, < t < l/v) σl C e t RC e l/v RC + 1), t > l/v) σl C = 1 en C l L = [V] l v = 1 6 [sec] 1 4 [sec] R = 1kΩ) RC = [sec] R = 5Ω) 16mV R=5Ω 16mV R= 1kΩ 1 µ s 1 µ s

14 Schrödinger ) m x + Ψ = EΨ 1) y Ψ = Xx)Yy) 1) ) X m X + Y = E Y X /X = k x, Y /Y = ky X = A x e ik xx, Y = A y e ik yy k x, k y Xx) = Xx + L x ) e ik xl x = 1 k x = πn x /L x n x =, ±1, ±, ) k y = πn y /L y n y =, ±1, ±, ) 1) Ψ = Xx)Yy) = Ae iπ nx Lx x+ ny Ly y ) A = A x A y ) A = 1/ S Ψ = 1 e iπ S nx Lx x+ ny Ly y ) E = m k x + ky) = π ) m nx L x ) + ny L y ), n x, n y =, ±1, ±, ). E n x -n y E = L x, L y melx /π, mel y /π NE) = π me L x π me L y π = ms π E ) NE F ) = ns ms π E F = ns = E F = π m n T N fermi f E) N = ρe) f E)dE = ) ρe) ρe) = dne) de = ms π ρe) 1 de e E µ k BT + 1

15 N = ms π de e E µ k BT + 1 = ms ) π k BT ln e µ k BT + 1 = ns µ T n T E F ) ) µ = k B T ln e π m n 1 k BT 1 = k B T ln e E F k BT 1 3) 3. UT) UT) = EρE) f E)dE = ms π EdE e E µ k BT + 1 k B T E F 3) µ E F β 1/k B T UT) ms π EdE e βe E F) + 1 CT) CT) = UT) 1 ms ) EdE T k B T β π e βe E F) + 1 = ms 1 π k B T x βe E F ) dx = βde CT) = ms 1 x β + E ) x F β ex dx π k B T βe F e x + 1) β = ms 1 1 x + βe F )xe x dx π k B T βe F β 3 e x + 1) 1/β E F βe F 1 ms 1 1 x + βe F )xe x dx π k B T β 3 e x + 1) EE E F )e βe E F) e βe E F) + 1 ) de x + βe F ) d dx f x) = d ) 1 e x dx e x = + 1 e x + 1) CT) ms π 1 k B T 1 d ms 1 β 3 x f x)dx = dx π k B T k BT) 3 ) π 3 = ms πk B T 3 4. ρe) = E E F α, α > 1for E F E E E F + E) UT) = EF + E E F E E E E F α de e βe µ) + 1 k B T E F, E CT) 1 k B T = 1 k B T EF + E E F E EF + E E F E ) E E E F α 1 de β e βe E F) + 1 E E E F α E E F)e βe E F) e βe E F) + 1 ) de

16 x βe E F ) = 1 k B T β E β E = 1 k B T β α 3 = 1 k B T β α 3 T α+1 ) x β + E x F β α x β ex dx e x + 1) β x + βe F ) x α xe x e x + 1) dx x α x e x dx e x + 1) 4

17 A B C D E F G X. 1.X.X S = 3.3cm θ = 3.3cm 6cm θ = 15.8 =.55[rad] S = 3.3cm n = 1 d =.8[Å] = a 5. S = 5.9cm θ = 5.9cm 6cm θ = 8. =.983[rad] d sin θ = λ d = 1.63[Å] h, k, l) 1 d = 1 a h + k + l ) h + k + l = 3 h, k, l) = 1, 1, 1)

18 ψ r 1, r ) = ψ r, r 1 ) ψ r 1, r ) = ψ r, r 1 ) ψ r 1, r ) r 1 = r ψ = δ r 1 r )ψ r 1, r ) = Schrödinger ψ r 1, r ) ψ r 1, r ){ V δ r 1 r )}ψ r 1, r )d r 1 d r = V ψ r, r) d r <. p 1 + p )ψ r 1, r ) = 1) R r R 1 r 1 + r ), r r 1 r ψ r 1, r ) ψ R, r ) 1) 1) p 1 + p )ψ r 1, r ) = i r 1 + r ) ψ r 1, r ) = i R ψ R, r ) = ψ R, r ) r ψ r 1, r ) = e i r 1 r ) g) ψ r 1, r ) = ψ r, r 1 ) e i r 1 r ) g) = e i r 1 r ) g ) g) = g ) g)

19 p 1 + p = 1 { p1 + p ) p 1 p ) } p 1 + p = + ) = i r 1 r i R p 1 p = ) = i r 1 r i r Schrödinger ψ R, r ) R p 1 m + p m V δ r ) ψ r ) = [ m d ] V δ r ) ψ r ) = εψ r ) d r ψ r ) = e i r g), δ r ) = 1 S q e i q r k m ei r g) V S e i + q) r g) = ε e i r g) q e i r k m g ) V g q) S = εg ) ) q 4. 1 S g q) = C q ) V C g) = ε) ε 3) 1 S g q) = C q = g q) = g q) q 1 S g) = C 4) 3) 4) V S 1 ε) ε = 1 5. V S 1 k m ε = 1

20 199 5 S V k 4π S πk k m εdk = mv 4π ε de E ε = 1 ε > ε < mv 4π log1 ε ε ) = 1 ε ε = < 1 exp 4π mv mv π 1 1 ) E = ε exp 4π mv

21 r r ) 3 Mr) = M GMr) R r r r + dr dur) dur) = GMr) dmr) = G r ) 3 r r M 3r dr M = 3GM r 4 dr R R 3 R 6 UR) = R dur) = 3GM 5R... U = UR 1 ) UR ) = 3GM 5 1 R 1 R 1 ) 1/) ρφdv 1/ φ φ GM/r r > R) GM/Rr /R 3/) r < R) UR) = 1 R GM r R R 3 ) 3 dmr) = 3 ) GM 4 R = 3GM 5R.. 4πL n ν = Ū E ν n ν = U 4πL Ē ν = 3GM 1 πl Ē 1 ) ν R R 1 3. N = n ν 6 σn p N p M D = kg N p = M D 1/ N A N A M = kg, L = 3 1 m, G = m 3 kg 1 s, Ē ν = 1MeV, R 1 = 1 6 m, R = 1 4 m n ν = 1 16 m M D = kg, N A = N p = n ν 6 σn p σn p 1[ m ] 1 σn p 1 46 [ m ] = [ m ] N = 3 1

22 E max, E min 1, t = t t 1 = L L = L 1 1 ) v v 1 c β β 1 E 1 = E = mc 1 β 1 mc 1 β 1) ) 3) β 1, β > 1) 3) ) c t L = mc 1 E 1 1 mc E 1 ) 1 ) mc E 1 c t L mc E ) ) mc E 1 = mc ) 1 1 E E1 mc >.. ) c t. mc / 1 L E 1 E1 1/ [ ] 1) β 3 1

23 α β α β. backbone C N, N C,C C, N C C=O 3. homopolypeptide hydrophilicity charge chemical energy in situ folding enzyme 4. ribonuclease urea urea C.B.Anfinsen) urea urea in vitro folding enzyme

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

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30 3 ............................................2 2...........................................2....................................2.2...................................2.3..............................

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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2 II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh

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