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

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1 I ( de Broglie (de Broglie p λ k h Planck ( Js p = h λ = k ( h π : Dirac k B Boltzmann ( J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = kg GaAs( a T = 300 K 3 fg f = 4, g = 5 f = g = 0 f = g = 9 f.g 0 kg = 45 kg 00 m/s ( = 4 b ( (L C R L C R (*, 3? (* Newton? a 3 b II (

2 4 7 ( y A C y A ( t 0 y( x,t 0 x B t 0 t W x W 0 : ( x y x A y ω y A (t = a sin(ωt + ϕ ( y A a t t 0 W 0 x t W t t 0 W 0 A C v x = v(t t 0 A C B (t t 0 C x t y(x, t t 0 y A (t 0 y(x, t = y A (t 0 ( x = v(t t 0 t 0 = t x v ( y(x, t = y A t x v (3 (4 t x y(x, t [ ( y(x, t = a sin ω t x ] = a sin(ωt kx (5 v ( k k = ω v = πf v = πf fλ = π λ π ( (5 p.7 (. (6

3 I ( y(x, t = A cos(kx ωt + ϕ ( ( v p (3b kx ωt + ϕ = ( ( v p = dx dt = ω k =? (3 y = A cos(kx ωt, y = A cos [(k + δkx (ω + δωt] (4 k δk, ω δω y = y + y [ ( δk = A cos x δω ] δk t ( [ cos k (x ω k t + ] (δk x δω t ( (5 p.: δk ( x δω δk t = δω v g lim δk 0 δk = dω dk (7 (6

4 ( W = 4.0 ev nm ( fg f = 4, g = 5 f.g 0 7 m m =450 nm f(x g(x ( u(x, t f(x + vt + g(x vt (8 u(x, t t = v u(x, t x (9 clue cos A + cos B = cos (A + B cos (A B (0

5 Schrödinger ( Ψ ( exp [ i(ωt kx] ( de Broglie Einstein( p = k, E = ω (a E = p m ( (b = Ψ exp [ i(ωt kx] ˆpΨ i t Ψ = ( i x Ψ = kψ = i ( iωψ = ωψ = E = p m ωψ = k m ψ i t ψ = ˆp m ψ = m x ψ ( H = p i q i L T + V = p + V (r ( m Ψ(r, t exp( iωt E = ω i Ψ(r, t = }{{ t Ĥ Ψ(r, t } ( E Schrödinger (3a ( ω Ĥ ( E = m + V (r ( ( (3b ( m Newton dp x dt = mdv x = 0 ( =0 [N] (4 dt 3

6 I (008 5 Schrödinger ( Ψ ( exp [ i(ωt kx] ( de Broglie Einstein( p = k, E = ω (a E = p m ( (b = Ψ exp [ i(ωt kx] ˆpΨ i t Ψ = ( i x Ψ = kψ = i ( iωψ = ωψ = E = p m ωψ = k m ψ i t ψ = ˆp m ψ = m x ψ ( H = p i q i L T + V = p + V (r ( m Ψ(r, t exp( iωt E = ω i Ψ(r, t = }{{ t Ĥ Ψ(r, t } ( E Schrödinger (3a ( ω Ĥ ( E = m + V (r ( ( (3b ( m Newton dp x dt = mdv x = 0 ( =0 [N] (4 dt

7 ( ( t Ψ(r, t i = ĤΨ(r, t (5 t ( Ĥ Ψ(r, t = ψ(r T (t (r (t Ψ(r, t = t t ĤΨ(r, t = (ψ(rt (t = ψ(rdt (t dt (6 Ĥ (ψ(rt (t = T (tĥψ(r Ĥ (7 (5 Ψ(r, t = ψ(r T (t dt (t i ψ(r = T (t dt Ĥψ(r dt (t i dt Ĥψ(r = = = T (t ψ(r = = E ( Ĥψ(r = Eψ(r (4a = dt (t i dt = ET (t (4b Ĥψ(r = Eψ(r (4a (

8 : Ψ(r, t = A exp[ i(ωt k r] ( r 0 P 0 n P r ( k = (k x, k y, k z 0 k n (3 k a λ π/ k = π/k r 0 = λn (,( r 0 n k, r (4 brace A exp[ i(ωt k r] (8 ( r ( = ωt k r = ( = C 0 (9 C 0 = ωt r k (5 (4 (4 π (3 (6 (4 (5 (7 k = (k x, 0, 0 a 4 0 : 3

9 I ( ( ( t Ψ(r, t i = ĤΨ(r, t ( t ( Ĥ Ψ(r, t = ψ(r T (t (r (t Ψ(r, t t = t (ψ(rt (t = ψ(rdt (t dt ( ĤΨ(r, t = Ĥ (ψ(rt (t = T (tĥψ(r Ĥ (3 ( Ψ(r, t = ψ(r T (t = dt (t i ψ(r = T (t dt Ĥψ(r i dt (t dt T (t = Ĥψ(r = ψ(r = = E ( Ĥψ(r = Eψ(r = (4a dt (t i dt = ET (t (4b Ĥψ(r = Eψ(r (4a ( (4b T (t T (t = exp ( i E ( t = exp( iωt ω = E (5 Ψ(r, t = ψ(r exp ( i E t (6 (

10 Born Ψ(x, t Ψ(x, t t x Ψ(x, t dx t x x + dx dx : dx Ψ(x, t = = Ψ(r, t Ψ(r, t dr = Ψ(r, t Ψ(r, tdr = (7 E Ψ(r, t Ψ(r, t = ψ(r exp ( i E t (8 ψ(r ( Schrödinger ρ(r, t = Ψ(r, t = ψ(r t (9 ψ(r dr = (0 ψ(r  < A > < A >= dr ψ (râψ(r ( x < x >= dr ψ (rxψ(r ( p x ˆp x = i < p x >= dr ψ (r i x ψ(r (3 x

11 Schrödinger Schrödinger ( a ( r ( = x, y, z, r = x + y + z ( q [C] r q [C] F q = e ( (3 (,( V (r b (4 (3 H m 0 p (5 (: (4 p ˆp = i x x ˆx = x (5 Ĥ ψ(x Schrödinger ( a, p.30 5 ( (004. b r 3

12 I ( Born Ψ(x, t Ψ(x, t t x Ψ(x, t dx t x x + dx dx : dx Ψ(x, t = = Ψ(r, t Ψ(r, t dr = Ψ(r, t Ψ(r, tdr = ( E Ψ(r, t Ψ(r, t = ψ(r exp ( i E t ( ψ(r ( Schrödinger ρ(r, t = Ψ(r, t = ψ(r t (3 ψ(r dr = (4 Â Âφ(r = a φ(r (5 φ(r A a A ψ(r A a Schrödinger Ĥψ(r = Eψ(r E Ĥ ψ(r ( ψ (r drψ (râψ(r = a ( p.37

13  ψ (Âψ(r (râϕ(rdr = ϕ(rdr (6 [ ] = ψ Âϕ = Âψ ϕ (Dirac (7  ( ( = ψ, ϕ ψ (rϕ(rdr = ψ ϕ = 0 (8 ψ(r  < A > < A >= dr ψ (râψ(r (9 x < x >= dr ψ (rxψ(r (0 p x ˆp x = i < p x >= dr ψ (r i x ψ(r ( x,, (005 p

14 0 x L ( ψ n (x = L sin ( nπ L x ( ( Hamiltonian Ĥ = : d (n =,, 3, d m dx Schrödinger m dx ψ n(x = E n ψ n (x (0 x L ( E n ( n subscript n (3 ( ψ n (x ˆp x ( (4 ( ψ n (x L 0 ψ m(xψ n (xdx = ψ m ψ n = δ mn (3 a a Fourier 3

15 I (008 5 A Au = λu ( λ( u λ A u  Âψ(r = a ψ(r [  ψ = a ψ (Dirac ] ( a  ψ(r  A u = λ u (  ψ(r = a ψ(r (  ψ = a ψ (3 [ ] ψ(x = A exp(ikx ˆpψ(x = i (* ˆp ( = i x [A exp(ikx] = k A exp(ikx = k ψ(x (4 x ( (* ˆp k  ψ (Âψ(r (râϕ(rdr = ϕ(rdr (5 [ ] = ψ Âϕ = Âψ ϕ (Dirac (6  ( (** ( pp ( =

16 ψ, ϕ ψ (rϕ(rdr = ψ ϕ = 0 (7  a n ψ n (r  ψ n (r = a n ψ n (r (8 ψ n ψ n = ( (9 a m ψ m (r (n m (** ψ m ψ n = 0 (0 (9,(0 (m = n ψ m ψ n = δ mn = 0 (m n ( Dirac φ(r, ψ(r drφ Dirac (rψ(r = φ ψ ( bra ket ( dr u m (ru n (r = δ mn Dirac = u m u n = δ mn Φ(r = m ϕ mu m (r  [Â] mn = dr u mâ(ru n(r A Dirac = Φ = m ϕ m u m Dirac = [Â] mn = u m Âu n = u m  u n = m  n A P i A i < A > < A >= P i A i Schrödinger Ĥψ(r = Eψ(r ψ A < A > < A > drψ Âψ = ψ  ψ ( A < A >

17 φ(r, ψ(r, ξ(r : Â(φ + ξ = Âφ + Âξ : Âφ = ϕ, ˆBφ = ξ = ( Â + ˆBφ = ϕ + ξ Âφ = ϕ, ˆBϕ = ξ ˆB (Âφ = ξ = ˆBÂφ = ξ φ Â ˆB ξ ] Â, ˆB [Â, ˆB Â ˆB [Â, ˆB] ˆB Â ] = 0 Â, ˆB, [Â, ˆB 0 Â, ˆB, ψ ( Â, ˆB Â ˆB ( [Â, ˆB] ψ = Â ˆBψ ˆBÂψ 3

18 0 x L ψ n (x = L exp ( i nπ L x ( (n = 0, ±, ±, ψ n (x + L = ψ n (x a ψ n (x L 0 ψ m(xψ n (xdx = ψ m ψ n = δ mn ( 3 ˆp x ( = i d dx b 4 c < p x > 5 Hamiltonian Ĥ = d d m dx E n a p b c ( d Hamiltonian ( (Hamiltonian =Schrödinger 4

19 I ( ( Ψ A (x, t, Ψ B (x, t k ω Ψ A (x, t = exp { i [(ω + ωt (k + kx]} Ψ B (x, t = exp { i [(ω ωt (k kx]} (a (b Ψ A (x, t + Ψ B (x, t = exp { i [(ω + ωt (k + kx]} + exp { i [(ω ωt (k kx]} = exp [ i (ωt kx] {exp [ i ( ω t k x] + exp [i ( ω t k x]} = cos( ω t kx exp [ i (ωt kx] ( ( : ( ( v g dω/dk (p.8 φ(x, t ( /4 ] exp [ π π (k k 0 exp [ i (ω(kt kx] dk (3 =a(k (.8 ω E(k = ω = p m = k m ( ( (.7, (.3 (TA

20 t = 0 (.5 φ(x, t = 0 = exp (π /4 ( x exp(ik 0 x φ(x, t = 0 = φ(x, t = 0 φ(x, t = 0 exp( ik 0 x = (π = (π exp ( x /4 exp ( x / exp ( x (π /4 exp(ik 0 x (4 : (Gauss FWHM= ln.67 (.8 φ(x, t = 0 = a(k = π exp ( x exp(ik (π /4 0 x a, b > 0 e a x +ibx π dx = a(k = φ(x, 0e ikx dx (5 b a e 4a (6 e x ( 4π 3 /4 +i(k0 kx dx = ( π /4 ] exp [ (k k 0 p.44 (* E(k = ω = k (.9 (.8 m φ(x, t = ( /4 ] exp [ (k k0 exp [ i (ω(kt kx] dk π π ( /4 ( ] = dk exp [ 4π 3 (k k 0 + i kx k 0 m (7 (8

21 (6 φ(x, t = ( π ( / [ ] /4 + i m t exp x i k 0 x + (i k0 /mt + (i t/m (9 φ(x, t = ( x k 0 [ exp t ] m (0 π (t (t ( (t = + m 4 t ( t 0 k 0 m t t (t k/m ( Hamiltonian Ĥ d Ĥ = + V (x ( m dx [Ĥ, x ] (3 φ(x [Ĥ, x ] φ(x HP? 3

22 I ( Newton < x >= Ψ x Ψ = dxψ (x, txψ(x, t ( (a < p x >= Ψ ˆp x Ψ = dxψ (x, t d Ψ(x, t ( (b i dx d < x > = dt m < p x > d < p x > =< F (x > dt (a (b [ ] ˆx = x ˆx t = 0 d < x > dt = d dt Ψ ˆx Ψ Ψ = t ˆx Ψ Ψ = t ˆx Ψ + + Ψ ˆx t Ψ + Ψ ˆx Ψ t Ψ ˆx Ψ t (3 Schrödinger Ψ(x, t t = i ĤΨ(x, t (4 d < x > = i ĤΨ dt ˆx Ψ i Ψ ˆx ĤΨ ( = i Ψ Ĥˆx Ψ i Ψ ˆxĤ Ψ = i Ψ Ĥˆx ˆxĤ Ψ Ĥ = i < [Ĥ, x ] > (.6 (5 [Ĥ, x ] = i m ˆpx (6 (5 9 d < x > dt = m < px > (7

23 ˆp x ˆpx = 0 t d < p x > dt = d dt Ψ ˆp x Ψ Ψ = t ˆpx Ψ + Ψ Ψ ˆpx t = i ĤΨ ˆp x Ψ i Ψ ˆp x ĤΨ = i < [Ĥ, ˆpx ] > (.3 (8 ] [Ĥ, ˆpx ϕ(x = Ĥˆp xϕ(x ˆp x Ĥϕ(x d < p x > dt = [ = m = i ] d m dx + V (x i dϕ dx i d 3 ϕ i dx + V (x dϕ 3 i dx + m d [ d ϕ dx m d 3 ϕ i dx 3 i ] dx + V (xϕ [ dv (x dx dϕ ϕ + V (x dx dv (x ϕ(x (9 dx = i ] [Ĥ, < ˆpx >= < dv (x dx >=< F (x > (.3 (0 ] ( Schrödinger m Ψ(x, t = Ψ(x, t ( i t (5 8 [ Ψ(x, t = A exp i (kx E ] t ( k ( E A Ψ(x, t ( < p > (? ( Ehrenfest? (3 < v > < p > m [clue:( ] (4 ( E (, m, k [clue: ( (.] (3

24 x p x x p x E t ( ( = h π [Js] [ ] A B  ˆB < A > < B > A B A B ( A =< A > < A > = ϕ ( < A > ϕ = ϕ ( < A > ϕ ( B =< B > < B > = ϕ ( ˆB < B > ϕ = ϕ ( ˆB < B > ϕ (4a (4b  ˆB 3 ( A = ( < A >ϕ ( < A >ϕ = ( < A >ϕ (5a ( B = ( ˆB < B >ϕ ( ˆB < B >ϕ = ( ˆB < B >ϕ (5b Schwarz ( ( A ( B = ( < A >ϕ ( ˆB < B >ϕ ( < A >ϕ ( ˆB < B >ϕ = ϕ ( < A >( ˆB < B >ϕ (6   < A >  ˆB < B > ˆB  ˆB = (  ˆB ˆB  + (  ˆB + ˆB  [ ] Â, ˆB [ ] Â, ˆB = [Â, ˆB] + { } Â, ˆB (7 (8 ( η ξ ] ] η [Â, ˆB ξ = [Â, ˆB η ξ (9a { } { } η Â, ˆB ξ = Â, ˆB η ξ (9b { } ( [Â, ˆB] Â, ˆB (7 <  ˆB > = ] [Â, < ˆB > + } < { Â, ˆB > (0 <  ˆB > = ] 4 [Â, < ˆB > + } 4 < { Â, ˆB > ] 4 [Â, < ˆB > ( ( 3

25 (6,(7,( ( A ( B ] 4 [Â, < ˆB > (3 Â = x, ˆB = ˆpx [x, ˆp x ] = i ( x ( p x 4 [x, ˆp x] = 4 i = 4 (4 ( x( p x (5 [Ê, t] = i ( E( t (6 Schwartz ψ ϕ ψ ϕ [ ] ψ 0 0 ψ 0 (A ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ϕ ψ = ψ ϕ ϕ ψ ϕ ϕ ψ ψ ϕ ϕ ψ + ψ ϕ ϕ ψ ( = ψ ϕ ϕ ψ ϕ ϕ ψ ϕ ψ ψ ϕ ψ ψ + ψ ϕ ϕ ψ ψ ψ ψ ψ (7 t ϕ ψ ψ ψ t = ψ ϕ ψ (A = ψ ( ϕ ϕ t ϕ ψ t ψ ϕ + t t ψ ψ = ψ ϕ tψ ϕ tψ = ψ ϕ tψ 0 (8 Ĉ ψ Ĉϕ = Ĉψ ϕ C < C > ϕ Ĉϕ < C > = ϕ Ĉϕ = Ĉϕ ϕ = ϕ Ĉϕ =< C > ( Ĉ ˆD ψ ˆDϕ = ˆDψ ϕ ˆD < D > = ϕ ˆDϕ = ˆDϕ ϕ = ϕ ˆDϕ = < D > 4

26 I (008 6 (6 Schrödinger [ Ψ(x, t = A exp i (k x x E ] t ( A Ψ(x, t = Ψ(x, t Ψ(x, t = dx Ψ(x, t = ( = A dx ( s x [ s x (x, t Re = Ψ (x, t ˆp x Ψ(x, t m ( Ψ (x, t m i ] (Re x Ψ(x, t Ψ(x, t m i x Ψ (x, t (3 a s x (x, t = k x ρ(x, t (4 m ρ(x, t Ψ(x, t (clue: (3 Ψ(x, t = Ψ (x, tψ(x, t < v x > (? s x ρ < v x > 3 n [m 3 ] < v > [ms ] j [Am ] ( q [C] analogy( a x

27 ρ = Ψ(r, t = Ψ (r, tψ(r, t s(r, t Re [Ψ (r, t ˆpm ] Ψ(r, t = ( Ψ (r, t m i Ψ(r, t Ψ(r, t m i Ψ (r, t ρ t = s ( [ ] Ψ(r, t ρ(r, t Ψ(r, t ρ(r, t t = Ψ(r, t t = t [Ψ (r, tψ(r, t] = Ψ (r, t Ψ(r, t + Ψ Ψ(r, t (r, t t t Ψ(r, t Schrödinger (5 ĤΨ(r, t = i Ψ(r, t t (A ρ(r, t t Hamiltonian = [ Ψ ĤΨ i ΨĤΨ ] (7 Ĥ = ˆp + V (r = m m + V (r (8 (A = i [ Ψ Ψ Ψ Ψ ] (9 m (φ ψ = φ ψ + φ ψ ( (A = i m [Ψ Ψ Ψ Ψ ] [ ( = Ψ m i Ψ Ψ ] m i Ψ (6 (0 s(r, t Re [Ψ (r, t ˆpm ] Ψ(r, t = [ Ψ (r, t m i Ψ(r, t Ψ(r, t ] m i Ψ (r, t ( (7 ρ(r, t t = s(r, t ( 3 (r, t t = J(r, t ( : [Cm 3 ] J : [Am ] (3 3 g (f x x = f g x x + f g x 3 W.H. Heit I,II

28 ρ ( (, s ( J ( (4 (7 [] x, y, z e x,e y,e z ψ = ψ x e x + ψ y e y + ψ z e z φ φ ψ = φ ψ x e x + φ ψ y e y + φ ψ z e z divergence (φ ψ = ( φ ψ x x + x ( φ ψ + ( φ ψ y x z = φ ψ x x + φ ψ x + φ ψ y y + φ ψ y + φ z ( ( φ φ φ ψ = ex + ey + x y z ez x = φ ψ + φ ψ ψ z + φ ψ z ψ ψ ex + ey + y z ez ( ψ + φ x + ψ y + ψ z [Gauss ] A(r V S S n A(r n ds = Adr (5 :S :V ( 4 A A (divergence A ( A 0 n= 0 ( x + x, y, z x ( x, y, z z ( x, y, z A ( x, y+ y, z 0 n= 0 y ( x, y+ y, z ( x + x, y+ y, z : Gauss : Gauss A ( x x y y z z A (5 A(x, y, z n x z = A y (x, y, z x z ( y (6 A A(x, y + y, z n x z = A y(x, y + y, z x z ( y 4 Gauss Gauss 3

29 Taylor A y (x, y + y, z A y (x, y, z + A y(x, y, z y y A A y (x, y + y, z x z A y (x, y, z x z + A y(x, y, z x y z (7 y (6 (7 ( =, 3,3 Ay(x, y, z x y z (8 y ( = A x(x, y, z x y z x ( 3 3 = A z(x, y, z x y z z (5 = A(r nds = ( Ax x + A y y + A z x y z = A V z ( (9a (9b (0 (5 4

30 x p x x p x E t ( ( = h π [Js] [ ] A B  ˆB < A > < B > A B A B ( A =< A > < A > = ϕ ( < A > ϕ = ϕ ( < A > ϕ ( B =< B > < B > = ϕ ( ˆB < B > ϕ = ϕ ( ˆB < B > ϕ (a (b  ˆB 3 ( A = ( < A >ϕ ( < A >ϕ = ( < A >ϕ (a ( B = ( ˆB < B >ϕ ( ˆB < B >ϕ = ( ˆB < B >ϕ (b Schwarz ( ( A ( B = ( < A >ϕ ( ˆB < B >ϕ ( < A >ϕ ( ˆB < B >ϕ = ϕ ( < A >( ˆB < B >ϕ (3   < A >  ˆB < B > ˆB  ˆB = (  ˆB ˆB  + (  ˆB + ˆB  [ ] Â, ˆB [ ] Â, ˆB = [Â, ˆB] + { } Â, ˆB (4 (5 ( η ξ ] ] η [Â, ˆB ξ = [Â, ˆB η ξ (6a { } { } η Â, ˆB ξ = Â, ˆB η ξ (6b { } ( [Â, ˆB] Â, ˆB (4 <  ˆB > = ] [Â, < ˆB > + } < { Â, ˆB > (7 <  ˆB > = ] 4 [Â, < ˆB > + } 4 < { Â, ˆB > ] 4 [Â, < ˆB > (8 (9 5

31 (3,(4,(9 ( A ( B ] 4 [Â, < ˆB > (30 Â = x, ˆB = ˆpx [x, ˆp x ] = i ( x ( p x 4 [x, ˆp x] = 4 i = 4 (3 ( x( p x (3 [Ê, t] = i ( E( t (33 Schwartz ψ ϕ ψ ϕ [ ] ψ 0 0 ψ 0 (A ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ϕ ψ = ψ ϕ ϕ ψ ϕ ϕ ψ ψ ϕ ϕ ψ + ψ ϕ ϕ ψ ( = ψ ϕ ϕ ψ ϕ ϕ ψ ϕ ψ ψ ϕ ψ ψ + ψ ϕ ϕ ψ ψ ψ ψ ψ (34 t ϕ ψ ψ ψ t = ψ ϕ ψ (A = ψ ( ϕ ϕ t ϕ ψ t ψ ϕ + t t ψ ψ = ψ ϕ tψ ϕ tψ = ψ ϕ tψ 0 (35 Ĉ ψ Ĉϕ = Ĉψ ϕ C < C > ϕ Ĉϕ < C > = ϕ Ĉϕ = Ĉϕ ϕ = ϕ Ĉϕ =< C > ( Ĉ ˆD ψ ˆDϕ = ˆDψ ϕ ˆD < D > = ϕ ˆDϕ = ˆDϕ ϕ = ϕ ˆDϕ = < D > 6

32 I ( (6 9 (Newton m x(t F Newton t = 0 v 0 x 0 t φ(x = exp (π /4 ( x ( ( p.43 (.5 or 5 9 < x < 3 ( x x ( φ(x ( 4 3 < x > 5 3 < p x > 6 x p x ( x p x ( x < x > < x > ( p x < p x > < p x > (a (b 3 5 ( x ( p x a, b > 0 x n e a x (n!! π dx = n a 4n+ (3 (n!! = 3 (n e a x π +ibx b dx = a e 4a. (4 7 6 x p x : ( ( (Gauss FWHM= ln.67

33 x p x x p x E t ( ( = h π [Js] [ ] A B  ˆB < A > < B > A B A B ( A =< A > < A > = ϕ ( < A > ϕ = ϕ ( < A > ϕ ( B =< B > < B > = ϕ ( ˆB < B > ϕ = ϕ ( ˆB < B > ϕ (5a (5b  ˆB 3 ( A = ( < A >ϕ ( < A >ϕ = ( < A >ϕ (6a ( B = ( ˆB < B >ϕ ( ˆB < B >ϕ = ( ˆB < B >ϕ (6b Schwarz ( ( A ( B = ( < A >ϕ ( ˆB < B >ϕ ( < A >ϕ ( ˆB < B >ϕ = ϕ ( < A >( ˆB < B >ϕ (7   < A >  ˆB < B > ˆB  ˆB = (  ˆB ˆB  + (  ˆB + ˆB  [ ] Â, ˆB [ ] Â, ˆB = [Â, ˆB] + { } Â, ˆB (8 (9 ( η ξ ] ] η [Â, ˆB ξ = [Â, ˆB η ξ { } { } η Â, ˆB ξ = Â, ˆB η ξ { } ( [Â, ˆB] Â, ˆB (8 <  ˆB > = ] [Â, < ˆB > + } < { Â, ˆB > ( <  ˆB > = ] 4 [Â, < ˆB > + } 4 < { Â, ˆB > ] 4 [Â, < ˆB > (0a (0b ( (3

34 (7,(8,(3 ( A ( B ] 4 [Â, < ˆB > (4 Â = x, ˆB = ˆpx [x, ˆp x ] = i ( x ( p x 4 [x, ˆp x] = 4 i = 4 (5 ( x( p x (6 [Ê, t] = i ( E( t (7 Schwartz ψ ϕ ψ ϕ [ ] ψ 0 0 ψ 0 (A ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ψ ϕ = ψ ϕ ϕ ψ ϕ ϕ ψ = ψ ϕ ϕ ψ ϕ ϕ ψ ψ ϕ ϕ ψ + ψ ϕ ϕ ψ ( = ψ ϕ ϕ ψ ϕ ϕ ψ ϕ ψ ψ ϕ ψ ψ + ψ ϕ ϕ ψ ψ ψ ψ ψ (8 t ϕ ψ ψ ψ t = ψ ϕ ψ (A = ψ ( ϕ ϕ t ϕ ψ t ψ ϕ + t t ψ ψ = ψ ϕ tψ ϕ tψ = ψ ϕ tψ 0 (9 Ĉ ψ Ĉϕ = Ĉψ ϕ C < C > ϕ Ĉϕ < C > = ϕ Ĉϕ = Ĉϕ ϕ = ϕ Ĉϕ =< C > ( Ĉ ˆD ψ ˆDϕ = ˆDψ ϕ ˆD < D > = ϕ ˆDϕ = ˆDϕ ϕ = ϕ ˆDϕ = < D > 3

35 I ( V = 0 Hamiltonian H = p x m Schrödinger d p x ˆp x = i x d = Ĥ = m dx ( ϕ(x = Eϕ(x ( m dx ϕ(x = A e ikx ( ( k = ± me ( (3 ϕ(x = A e ikx (k (4 k E = k m E k Schrödinger Φ(x, t (4 Φ(x, t = ϕ(x exp ( i E t = A exp[i(kx ωt] (ω = E (6 (5 ρ(x = ϕ ϕ(x = A ϕ ˆp x ϕ(x = k ( 3 Hamiltonian Ĥ = m = m ( x + y + z (7 Schrödinger ϕ(x, y, z ( m x + y + z ϕ(x, y, z = Eϕ(x, y, z (8 (5 [ ] X(x, Y (y, Z(z x y z ϕ(x, y, z ϕ(x, y, z = X(xY (yz(z (8 ϕ(x, y, z ( 0 X m X = E + Y m Y + Z m Z (9 ( TA

36 x y,z E x y z E y E z (8 m X (x = E xx(x m Y (y = E yy (y m Z (z = E z Z(z E = E x + E y + E z (0a (0b (0c (0d mex mey mez k x = ± k y = ± k z = ± ( 3 ϕ(x, y, z = A e i(kxx+kyy+kzz = A e ik r ( E = E x + E y + E z = ( kx + k y + k z = k m m (3 = ( nm nm x, y, z L x, L y, L z ϕ(x, y, z = ϕ(x + L x, y, z = ϕ(x, y + L y, z = ϕ(x, y, z + L z (4 Schrödinger ( k k x = n xπ L x, k y = n yπ L y, k z = n zπ L z, (n x, n y, n z = 0, ±, ±, (5 = ρ(rdr = A L x L y L z A V (V (6 ϕ(r = V e ik r (7 (3 (5 L x ( x+ Lx φ( x φ = L x :

37 + x < 0 V (x = 0 0 x L + x > L (8 V( x potential 0 L x : (QW:Quantum Well Schrödinger 0 < x < L Ĥψ(x = d ψ(x m dx = Eψ(x ( (9 x < 0, x > L ( ψ(x = 0 (0 (ψ(x x = 0, L ψ(x = 0 ( ψ + me ψ = 0. ψ = e ikx k +me/ = 0. k = ± me. ψ(x = e +ikx, e ikx k ( C,C ψ(x = C e +ikx + C e ikx. ( ψ(0 = 0 C e 0 + C e 0 = C + C = 0 C = C. ψ(x = C e +ikx C e ikx = ic sin kx. ( ψ(l = 0 i sin(kl = 0 kl = nπ (n =,, 3,. ( nπ ψ n (x = C sin L x = C sin(k n x (n =,, 3,. (3 (3 k n = nπ L = men (4 3

38 E n = kn ( m = nπ (n =,, 3, k n m L (5 (4 = L ψ(x dx = L 0 0 C sin (k n xdx = L (6 ψ(x = L sin(k nx. (7 (? 3: (n =,, 3 4

39 I ( ( V 0 x < L V (x = 0 L x L V 0 x > L ( V( x potential V 0 L 0 L x : (finite potential QW Schrödinger d m dx ψ(x + V 0ψ(x = Eψ(x d m dx ψ(x = Eψ(x L x L E (a,(b : x < L, L < x (a (i ψ(x dψ(x dx (ii ( =0 ψ(± = 0 (b ( C,C ψ(x = C e +ikx + C e ikx (3a = A cos(kx + B sin(kx (3b me k = (3c (

40 ( V 0 E > 0 (a d m ψ(x = dx (V 0 Eψ(x (4 (4 ψ(x = Ce κx + De κx m(v0 E κ = (κ (6 (5 (ii (x ± ψ(x = Ce κx x > L ψ(x = De κx x < L (7a (7b [ ( ] (i (3b (7a,(7b V (x V ( x = V (x Schrödinger ψ(x ( ψ( x = ψ(x (ψ (8a ψ( x = ψ(x (ψ (8b ( cos x x sin x x (a ψ(x (3b (7a,(7b Ce κx x > L ψ(x = A cos(kx x L Ce κx x < L (9 x = L 3 C exp ( ( κl = A cos kl (ψ κc exp ( ( κl = ka sin kl (0 ( dψ dx ( kl κ = k tan ( ψ(x = Ce κx (C κ = m(v 0 E/ (> 0 e κx e κx. 3 x = L

41 k κ (3c (6 E ( E (transcendental equation (bisection method Newton 4 ( α kl β κl ( (3c (6 E α + β = mv 0L (3 α,β r 0 = mv 0 L / 4 k κ ( ( (3 k κ ( (b ψ(x (3b (7a,(7b Ce κx x > L ψ(x = B sin(kx x L Ce κx x < L x = L 5 C exp ( ( κl = B sin kl (ψ κc exp ( ( κl = kb cos kl (5 ( dψ dx (4 ( kl κ = k cot (6 (a (6 (3 k κ β = α tanα β = α cot α 4 0 π/ π 3π/ α π : ( (3 (6 (3 4 ISBN (006 5 x = L 3

42 (3 6 = ψ(x dx = L/ C e κx dx } {{ } (A L/ + L/ a cos kxdx } {{ } (B + L/ C e κx dx } {{ } (C (7 (A = C [ e κx ] L/ κ = C (B = A L/ L/ (0 κ e κl = (C ((A x x (8 [ x + ] L/ sin kx = (L A + k k sin kl (9 ( + cos kx dx = A κc exp( κl = k A sin kl (0 L/ (7 (9 = A [ L + k ( + k κ ] sin kl ( A = ( ( L + k + k κ sin kl C (0 ( k κl C = sin kl exp A (3 κ B D ψ E ψ E 3: ( ( ( 6 A, B, C ( 4

43 (* x x V ( x = V (x [ ] d m dx + V (x ψ( x = Eψ( x (4 ψ(x ψ( x,ψ( x ψ(x ( (** ψ( x = Cψ(x (5 x x ψ(x = Cψ( x = C ψ(x C = ± C = C = (** ψ (x ψ (x E d m dx ψ (x + V 0 ψ (x = Eψ (x d ψ(x + V0ψ(x = Eψ(x m dx (6b (6a (6a ψ (x (6b ψ (x /m d ψ (x ψ dx (x + d ψ (x ψ dx (x = 0 (6c dw (x dx d dx Wronskian W (x ψ (x dψ (x dx ψ (x = 0, dψ (x dx W (x = dψ (x dx ψ (x + dψ (x dx ψ (x = C ( (7 ψ (± = 0, ψ (± = 0 C = 0 (6d (6e ψ (x ψ (x = ψ (x ψ (x (8 log ψ (x = log ψ (x + Const (9 ψ (x = Cψ (x (30 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.

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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