h = h/2π 3 V (x) E ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 関 数 値

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Download "34 2 2 h = h/2π 3 V (x) E 4 2 1 ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 1.2 1 関 数 値 0.8 0.6 0.4 0.2 0 15 10 5 0 5 10 "

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1 x 1 T x T 0 F = ma T ψ) 1 x ψ(x) h2 d 2 ψ(x) + V (x)ψ(x) = Eψ(x) (2.1) 2m dx 2 1

2 h = h/2π 3 V (x) E ψ = sin kxk = 2π/λ λ = h/p p = h/λ = kh/2π = k h 5 2 ψ = e ax2 ガウス 型 関 数 関 数 値 : (+ ) 2 Web(http://schrodinger.haun.org/) 3 ω = 2πν E = hν = hω ω 2π ω h h 4 F = ma 5 k k k

3 V (x) = 0 h2 d 2 sin(kx) 2m dx 2 = E sin(kx) h2 2m ( k2 ) sin(kx) = E sin(kx) k 2 h 2 2m sin(kx) = E sin(kx) k 2 h 2 2m = E (2.2) 2 sin(kx) x 7 k h = p p 2 /2m = (mv) 2 /2m = mv 2 /2 d dx (e ax2 ) = 2ae ax2 x d 2 dx 2 (e ax2 ) = 4a 2 e ax2 x 2 2ae ax2 (2.3) h2 2m (4a2 x 2 2a)e ax2 = Ee ax2 h2 2m (4a2 x 2 2a) = E (2.4) x E x sin(kx) 8

4 V = 1 2 Kx2 10 d 2 h2 2m d 2 x ψ Kx2 ψ = Eψ (2.5) ψ = sin kx k2 h 2 2m sin kx Kx2 sin kx = E sin kx k2 h 2 2m Kx2 = E (2.6) h2 2m (4a2 x 2 2a)e ax Kx2 e ax2 = Ee ax2 h2 2m (4a2 x 2 2a) Kx2 = E ( 1 2 K h2 4a 2 2m )x2 + h2 a m = E (2.7) 1 2 K h2 4a 2 2m = 0 (2.8) 9 10 K K

5 h 2 4a 2 2m = 1 2 K a 2 = mk 4 h 2 mk a = 2 h (2.9) E = h2 a m = 1 2 h K m = hω 2 = hν 2 (2.10) ( 1 2 K h2 4a 2 2m )x2 = 0 (2.11) Web

6 :

7 /4 2.3: 2.4: 15 16

8 x Ψ(x) x x + x Ψ(x) 2 x 2 s3 v 18 Ψ(x) / 2 1/2 2 1/2 1/ /

9 / c t tc 22

10 Z 0 d 2 h2 2m dz ψ = Eψ 2 (2.12)

11 ψ = sin kz (2.13) 26 k 2 h 2 2m sin kz = E sin kz (2.14) E = k2 h 2 2m (2.15) k 2 ψ 2 = sin 2 kz (2.16) 27 ψ = const 0 0 = Eψ (2.17) E = 0 E k = 2π λ

12 x = sin z y = cos z (2.18) 2.5: x y z z ψ 2 = x 2 + y 2 = sin 2 z + cos 2 z = 1 = const (2.19) 2 x y xy (a, b) 2 2 2

13 a b = b a a(b + c) = ab + bc 2 2.6: 2 2 (a, b) ab ab 2 (a, b) + (c, d) = (a + c, b + d) (2.20) a + b = b + a (a, b) + (c, d) = (c, d) + (a, b) (2.21) (0, 0) (a, b) ( a, b) (a, 0) (b, 0) = (ab, 0) ( a, 0) (b, 0) = ( ab, 0) (2.22) b b 2 (0, 1) (a, 0) = (0, a) (0, b) (a, 0) = (0, ba) (2.23)

14 (a, 0) (0, 1) = (0, 1) (a, 0) = (0, a) (a, 0) (0, b) = (0, b) (a, 0) = (0, ba) (2.24) 1 a a b 90 b : (0, 1) (0, 1) = ( 1, 0) ( 1, 0) (0, 1) = (0, 1) (0, 1) (0, 1) = (1, 0) (2.25) 2 (a, b) (c, d) = (ac bd, ad + bc) (2.26) (a, b) (1, 0) = (a, b) (2.27) (1, 0) (a, b) (c, d) = (ac bd, ad + bc) = (1, 0) (2.28)

15 (c, d) ac bd = 1 ad + bc = 0 c d c = a a 2 + b 2 d = b a 2 + b 2 (2.29) (2.30) (a, b) (a/(a 2 + b 2 ), b/(a 2 + b 2 )) (0, 1) (0, 1) = ( 1, 0) (0, 1) = i 2 i (a + bi) + (c + di) = (c + di) + (a + bi) = (a + b) + (c + d)i (a + bi)(c + di) = (c + di)(a + bi) = (ac bd) + (ad + bc)i (2.31) (0, 1) i 2 2 i

16 a, b) i 90 1 = i i a + bi 1 = a bi 180 ab ab 180 i a i = a 1 i a a b a b a b ai b 90 a a b 180 a ai b 270 a 2 3 A =a + a i) B = (b + b i) A = α cos θ + αi sin θ = α(cos θ + i sin θ) B = β cos θ + bi sin θ = β(cos θ + sin θ ) α = a 2 + a 2 β = b 2 + b 2 (2.32) α(cos θ + i sin θ)β(cos θ + i sin θ ) = αβ((cos θ cos θ sin θ sin θ ) + i(cos θ sin θ + sin θ cos θ )) (2.33) = ab(cos(θ + θ ) + i sin(θ + θ )) e ±iθ = cos θ ± i sin θ (2.34)

17 αe iθ βe iθ = αβe i(θ+θ ) (2.35) 2 i θ θ 1 αe iθ 1 α e iθ = 1 (2.36) D A ( SSH

18 x = cos θ, y = sin θ x y 2 ψ = cos θ + i sin θ d2 dz 2 e ikz = k 2 e ikz ( ) 20

19 h2 d 2 ψ 2m dz = h2 k 2 ψ = Eψ (2.37) 2 2m sin kz 2 ψ 2 = e ikz e ikz = e 0 (2.38) 2.4 Ψ(x) NΨ(x) NΨ(x) 2 dx = N 2 Ψ (x)ψ(x) dx 1 (2.39) N = 1 Ψ (x)ψ(x) dx (2.40) :

20 52 2 L L/2 34 N = 2 L (2.41) ψ ψ dτ = 1 τ 2 2 ψ ψ dx dy = 1 x = r cos ϕy = r sin ϕ r π ψ ψr dr dϕ = 1 r r 0 0 r

21 x ψ(x) 2 x ) 37 0

22 d 2 h2 ψ + V (x)ψ = Eψ 2m d 2 x ) ( h2 d 2 2m d 2 x + V (x) ψ = Eψ ψ ψ ()() = (() (2.42) H Hψ = Eψ (2.43)

23 k h ψ = e ikx ψ = ike ikx h i d dx (2.44) h i d dx ϵikx = h i ikeikx = k hψ = pψ (2.45) k h ψ Hψ dx * ψ Hψ dx = ψ Eψ dx = E ψ ψ dx = E

24 ψ = e ikx ψ = e ik x e ikx e ik x dx k = k 0 40 Hermitian operator H 2.7 ψ = cos kx h i d dx cos kx = h i k sin kx = p cos kx (2.46) 41 cos kx = eikx + e ikx 2 (2.47) 40 ψ e ikx e ikx (a+ib a-ib) 41

25 k h 2 k h +h h k h 1 1 sinkx ϕ 1, ϕ 2,,,, ϕ n ψ = c 1 ϕ 1 + c 2 ϕ 2 +,,,, c n ϕ n (2.48) ψ c 2 1 +C c 2 n = 1 ψ P ψ dx = (c 1 ϕ 1 + c 2 ϕ c n ϕ n)p (c 1 ϕ 1 + c 2 ϕ c n ϕ n ) dx = (c 1 ϕ 1 + c 2 ϕ c n ϕ n)(c 1 P 1 ϕ 1 + c 2 P 2 ϕ c n P n ϕ n ) dx = (c 2 1P 1 ϕ 1ϕ 1 + c 2 2P 2 ϕ 2ϕ c 2 np 2 ϕ nϕ 2 ) + (CrossT erm) dx = (2.49) c 2 1P 1 + c 2 2P c 2 np e ikx

26 58 2 e ikx /3 1/5 1/(2n-1) (2n-1) ko (2n+1) n= 4 f(x) = sin((2n 1)x) (2.50) (2n 1)π n= f(x) = n=1 8 1 π n sin ( ) nπ 2 2 sin (nx) 45 SNS 46 40

27 矩 形 波 基 本 波 0.5 矩 形 波 基 本 波 倍 波 5 倍 波 51 倍 波 倍 和 5 倍 和 51 倍 和 : n n n WG VB 1 WG igor igor igor igor

28 鋸 波 基 本 波 0.5 鋸 波 基 本 波 倍 波 3 倍 波 26 倍 波 倍 和 3 倍 和 26 倍 和 : k 0 F(x) F (x) = a n sin nk 0 x + b n cos nk 0 x (2.51) n=0 a n b n k : k 0 0 nk 0 0 y (

29 a n y ) F (x) = y=0 a(y) sin yx + b(y) cos yx dx (2.52) 49 k0=0 2 mathmatica e ax2 1 2a e k2 4a (2.53) (2.54) A1003 B1003 A958 B958 A906 B906 A854 B A1003 B1003 A958 B958 A906 B906 A854 B A802 B802 A750 B750 A698 B698 A646 B646 A594 B A802 B802 A750 B750 A698 B698 A646 B646 A594 B : a mathmatica

30 62 2 1/2 1/e a a a x = 0 0 x a k 51 k p p = k h k k 2.54 k k k 0 a = 0 k 0 h 51 k k

31 a a a x p x p h/2 (2.55) % )

32 64 2 ) g) 0.1nm) mm 400m 1cm 10g 800m m k h = p E = hν E t h/2 56

33

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E

7 9 7..................................... 9 7................................ 3 7.3...................................... 3 A A. ω ν = ω/π E = hω. E B 8.9.4, : : MIT I,II A.P. E.F.,, 993 I,,, 999, 7 I,II, 95 A A........................... A........................... 3.3 A.............................. 4.4....................................... 5 6..............................

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7 27 7.1........................................ 27 7.2.......................................... 28 1 ( a 3 = 3 = 3 a a > 0(a a a a < 0(a a a -1 1 6 26 11 5 1 ( 2 2 2 3 5 3.1...................................... 5 3.2....................................... 5 3.3....................................... 6 3.4....................................... 7

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