A A. ω ν = ω/π E = hω. E

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1 B 8.9.4, : : MIT I,II A.P. E.F.,, 993 I,,, 999, 7 I,II, 95 A A A A

2 A A. ω ν = ω/π E = hω. E h p p = hk. k ω k x e iωt+ikx.3 ψx, t E n p n n n a n ψx, t = n e iω nt+ik n x a n.4 a n. h i i h t ψx, t = n hω n e iω nt+ik n x a n = n E n e iω nt+ik n x a n.5 m p V x E mv / = p /m V x E = p + V x.6 m E n = p n + V x.7 m

3 .5 i h t ψx, t = p n m + V x e iω nt+ik n x a n = hkn m n n = h + V x e iωnt+iknx a n m i x n h = + V x e iω nt+ik n x a n m i x n h = + V x m i x + V x e iω nt+ik n x a n ψx, t.8 x, y, z 3 [ i h ] t ψx, t = h h h V x ψx, t.9 m i x i y i z. A. ψx, t. 3. Ω ω n Ωu n = ω n u n. 4. E p x ψx = a n u n x. n a n ψ u n 5. u n, u m = δ n,m. δ n,m n = m ψ, φ ψ, φ ψ, φ = d 3 xψx φx.3 3

4 Ω ω n a n ψ a n =.4 n Ω ψx Ω Ω = a n ω n = dxψ xωψx.5 n.3 A Hux = Eux H = h d + V x,.6 m dx {, x > a, V x =, x < a.7.6 E k me k = h h d ux m dx = Eux.8.9 d ux/dx = k ux k sin kx cos kx A B ux = A cos kx + B sin kx, a < x < a. k E ux = a =, ux = a =. A cos ka + B sin ka =, A cos ka + B sin ka =. 4

5 A cos ka =, B sin ka =.3 A, B B =, cos ka =,.4 A =, sin ka =.5 k n k = n + π a.6 ux = A cos ux = B sin n + πx, n =,, 4,,.7 a n + πx, n =, 3, 5,,.8 a E n = π h n + 8ma, n =,,, 3,,.9 A B n n E = π h 8ma.3 x a p = h/a h /ma x x.8 5

6 V x = V x, H x = Hx.3 ux Hxux = Eux, Hxu x = H xu x = Eu x.3 E u x = cux.33 x x ux = cu x.34 c =, c = ±.35 c = c = E ux.9 n... { V, x > a, V x =, x < a. x = du x = =. dx ux = =.3 6

7 V V d ux = κ ux,.4 dx mv E κ = h C D.5 ux = Ce κx + De κx, x > a.6 D ψx, t <.7 D =, ux = Ce κx, x > a.8 x < a. k me k = h..9 ux = A cos kx..3 ux = B sin kx,

8 x = a ux x = a a ϵ a + ϵ du du x = a + ϵ x = a ϵ = dx dx a+ϵ a ϵ dx d ux dx = h m a+ϵ a ϵ dx V x E ux. ϵ ϵ x = a A C.3 A cos ka = Ce κa.3 ka sin ka = κce κa.4 k tan ka = κ.5 x = a. B C.6 B sin ka = Ce κa.6 kb cos ka = κce κa.7 k cot ka = κ k κ ξ = ka, η = κa.9 ξ + η = mv a h. η = ξ tan ξ,. η = ξ cot ξ. V a < V a π h.3 8m 8

9 n π h 8m < V a n + π h 8m.4 n + V E.4 V E E E n < x < n nπ a < k n+π a { A cos kx, x < a, ux = A cos kae κ x a, x > a.5 A = a + κ.5.6 ux = { B sin kx, x < a, x x B sin kae κ x a, x > a.8 B = a + κ.7.8 V E E V.5 κ A B.3.4 V ±.6 9

10 3 3. ω c V x = mω c x 3. Hux = Eux, 3. H = p m + mω c x = h d + mω c m dx x 3.3 mωc Q = x, 3.4 h P = m hωc p = h d m hωc i dx = i d dq 3.5 H = hω c P + Q 3.6 [Q, P ] = i 3.7 a = Q + ip, a = Q ip 3.8?? Q, P a, a [a, a ] = 3.9 H = hω c a a + aa = hω c N +, N = a a 3. N a a N ν u ν [N, a] = a, [N, a ] = a 3. Nu ν = νu ν 3.

11 a 3.9 au ν, au ν = u ν, a au ν = u ν, Nu ν = νu ν, u ν, 3.3 a u ν, a u ν = u ν, aa u ν = u ν, N + u ν = ν + u ν, u ν ν 3.5 Nau ν = an u ν = ν au ν, 3.6 Na u ν = a N + u ν = ν + a u ν 3.7 au ν ν > N ν νu ν, u ν, a u ν ν > N ν + ν + u ν, u ν ν a N ν ν a ν N u N ν =,,,, n, u ν u, u = a u, u = a u,, u n = n! a n u, E = hω c, E = + hω c, E = + E n = hω c,, n + hω c, 3.9 u a 3.3 au, au = u au = 3. E = hω c hω c hω c N a N a N N

12 3. u n x 3. au = Q + ip u = Q + d u = 3. dq u x e Q mωc 4 u x = e mωc Q 4 = e mω c h x 3. π h π h n n u n x = a n mωc 4 u = Q d n e Q 3.3 n! π h n! n dq ψq Q d ψq = e Q dq d dq e Q ψq 3.4 Q d n = dq = = Q d dq = e Q d Q dq e 3.5 Q d dq Q d dq Q d dq = e Q d dq n e Q n e Q n e Q d dq e d dq e d dq Q Q e Q e Q = d dq e Q n e Q 3.6 u n x = = mωc π h mωc π h mωc π h 4 e Q d n! n dq 4 n! n e Q H n Q 4 n! n mωc e h n e Q mωc x H n h x 3.7 H n Q Hermite H n Q = n e Q dn dq n e Q, n =,,,, 3.8 n H Q =, H Q = Q, H Q = 4Q, H 3 Q = 8Q 3 Q, 3.9 n H n Q n n H n Q = n H n Q 3.3

13 4 4. L = x p 4. L = h i x 4. L x, L y, L z L x = h i y z z y, L y = h i z x x z, L z = h i x y y x 4.3 [L x, L y ] = i hl z, [L y, L z ] = i hl x, [L z, L x ] = i hl y [L i, L j ] = i h ε ijk L k 4.5,, 3 x, y, z k= 4. L L J J [J x, J y ] = i hj z, [J y, J z ] = i hj x, [J z, J x ] = i hj y 4.6 L 4.3 J J = Jx + Jy + Jz, 4.7 [J, J x ] = [J, J y ] = [J, J z ] = 4.8 J J x J y J z z J z J z J z u jm J u jm = jj + h u jm, J z u jm = m hu jm 4.9 3

14 j J jj + h J J = J jj + m jj + 4. J x J y J + = J x + ij y, J = J x ij y 4. J + J [J z, J ± ] = ± hj ±, [J +, J ] = hj z 4. J x, J y J ± J J = J +J + J J + + Jz, 4.3 [J, J z ] =, [J, J ± ] = 4.4 z J z J J J + = J J z J z + h, J + J = J J z J z h 4.5 u jm u jm J J + u jm = j mj + m + h u jm, 4.6 J + J u jm = j + mj m + h u jm 4.7 J + J J ± = J 4.8 J + u jm = J + u jm, J + u jm = u jm, J J + u jm = j mj + m + h u jm, 4.9 J u jm = J u jm, J u jm = u jm, J + J u jm = j + mj m + h u jm 4. j j m j 4. J J ± u jm = jj + h J ± u jm, 4. J z J ± u jm = m ± hj ± u jm 4.3 J + u jm z + h j h z m h 4.9 j mj + m + J z j 4

15 J J z j h J J z j h j. J jj + h j j j =,,, 3,, 4.4. J z m h m z m 3. J J z jm m = j, j +,, j 4.5 j + j + J x, J y, J z j + j u jm J J z h j z h m = j, j,, j j z h u jm c jm J u jm = c jm u jm c jm c jm = [jj + mm ] h, 4.7 c jm c jm J u jm = j + mj + m hu jm 4.8 J + u jm u jm+ J + J J u jm c jm J + u jm J + u jm = j mj + + m hu jm+ 4.9 J z m h = j h J + J z J + u jj = 4.3 5

16 m h = j h J J u j j = 4.3 j j + u jj J u jm = j + m! j!j m! u j j J + u jm = j m! j!j + m! j m J u jj 4.3 h j+m J+ u j j 4.33 h 4.3 j j u j m, u j m u jm J k = J k + J k, k = x, y, z 4.34 z u j j u j j 4.35 J z u j j u j j = j + j hu j j u j j 4.36 z j + j J u j j u j j = j + j j + j + h u j j u j j 4.37 z u j+j j +j = u j j u j j 4.38 J = J + J j + j z J z J z j + j h = j + j hu j +j,j +j = J u j +j,j +j = J u J u j j u j j + u j j J u j j j j u j j = j hu j j u j j + j hu j j u j j 4.39 z J z J z z J z j + j h u j,j u j,j, u j,j u j,j 4.4 6

17 J j + j j + j z j + j h j + j j + j j + j u j +j,j +j = j u j j + j j u j j + j u j j u j j 4.4 J z. j j j = j j, j j +,, j + j 4.4. j j + j Hx, p ; x, p = Hx, p ; x, p 5. α β u α,β x ; x = u α x u β x, 5. u β,α x ; x = u β x u α x 5.3 u S u A u S = u α,β + u β,α, u A = u α,β u β,α 5.4 c S c A u = c S u S + c A u A, c S + c A = 5.5 7

18 c S c A u t ψ i h ψx, t = Hψx, t 5.6 t ψx ; x ; t = c S ψ S x ; x ; t + c A ψ A x ; x ; t 5.7 t x, x P x ; x ; t = ψx ; x ; t + ψx ; x ; t 5.8 [ P x ; x ; t = c S ψ S x ; x ; t + c A ψ A x ; x ; t ] 5.9 c S, c A c S, c A Bose 5. 8

19 ψx, x ψx, x = φ α x φ β x 5. P P ψx, x = ψx, x 5. ψ S x, x = ψx, x + ψx, x = + P ψx, x 5. ψ A x, x = ψx, x ψx, x = P ψx, x 5.3 ψx, x ± ψx, x = ψx, x + ψx, x ± Re [ψx, x ψ x, x ] 5.4 ψ A x, x = ψ A x, x =, T E n n P n P n = e En/kBT Z 5.6 k B Z Z = n=,, e nϵτ /kbt 5.7 τ ϵ τ n nϵ τ n 5.6 P n = e nϵ τ /k B T 5.8 Z ϵ n Z n =,,, 5.9 Z = + e ϵ/kbt + e ϵ/kbt + = 9 e ϵ/kbt 5.

20 ϵ τ e ϵ/k BT e ϵ/k BT = e ϵ/k BT 5. ϵ τ n =, 5. Z Z = + e ϵ/k BT 5.3 ϵ τ e ϵ/kbt + e ϵ/kbt = e ϵ/kbt ϵ k B T T ν = ω/π hν = hω = k B T 5.5 h = h/π 6 6. V x, t 3 V x V x, t = V r 6. r, θ, φ x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ 6. x, y, z r, θ, φ [ h m r r + r r r sin θ sin θ + θ θ r sin θ ] φ u + V ru = Eu 6.3

21 r, θ, φ e r = sin θ cos φ, sin θ sin φ, cos θ, e θ = cos θ cos φ, cos θ sin φ, sin θ, e φ = sin φ, cos φ, 6.4 e i e j = δ ij, e i e j = k ε ijk e k, i, j, k = r, θ, φ 6.5 ε ijk 3 ε 3 = +,, 3 r, θ, φ x x = re r 6.6 r, θ, φ e r θ e θ θ e φ θ = e θ, = e r, =, e r φ = sin θe φ, e θ φ = cos θe φ, e φ φ = sin θe r cos θe θ dx fr, θ, φ dx = dre r + rdθe θ + r sin θdφe φ 6.8 df = dr f f + dθ r θ + dφ f φ, f = dre r + rdθe θ + r sin θdφe φ e r r + e f θ r θ + e f φ, r sin θ φ = dx f 6.9 = e r r + e θ r θ + e φ r sin θ φ e r, e θ, e φ = e r e r, φ r + e θ r θ + e φ r sin θ r + e θ r θ + e φ r sin θ = e r e r + e θ e θ r r r θ + e θ r + e φ e φ r sin θ r sin θ φ + sin θe φ r + cos θe φ, r θ [ = + r r r + r + cot θ θ θ + ] sin θ φ φ

22 L = h i re r e r, = h i = h i e φ r + e θ r θ + e φ r sin θ, φ θ e θ sin θ φ sin φ cot θ cos φ θ φ, cos φ cot θ sin φ θ φ, z L z = h i L = h e φ θ e θ e φ sin θ φ θ e θ, sin θ φ [ = h e φ e φ θ e θ sin θ θ φ + e r sin θ φ e θ e φ φ θ e θ sin θ φ sin θe r + cos θe θ θ e cos θ φ sin θ [ ] = h, = h [ sin θ θ + cot θ θ + sin θ sin θ θ θ φ + sin θ φ ] φ φ ], φ H = h m r r + L + V r 6.5 r r mr 4. z m h Y m θ, φ Y m L Y m θ, φ = h + Y m θ, φ 6.6 L z Y m θ, φ = hmy m θ, φ 6.7 θ, φ Y m θ, φ ur, θ, φ =,m R ry m θ, φ [ h d m r r dr + R ] dr dr mr h + Y m θ, φ + V rr Y m θ, φ = E,m,m Y m R Y m θ, φ 6.9 h d m r r dr + h + dr dr mr R + V rr = ER 6. r 6.

23 6. z m h z 6.3 L z u jm θ, φ = h i C φ u jmθ, φ = m hu jm θ, φ 6. u jm = Ce imφ 6. φ φ + π 6.3 z h m z j j =,,, θ, φ z h m =,,, + Y m θ, φ Y Y Y ± a b c : =, Y m. a =, m =, b =, m =, c =, m = ±, m Y = 4π, 6.4 Y = 3 ± 3 cos θ, Y = 4π 8π sin θe±iφ, 6.5 3

24 Y 3 = Y = 5 6π 3 cos θ, Y ± Y ± = 5 = 8π sin θ cos θe±iφ, π sin θe ±iφ π 5 cos3 θ 3 cos θ, Y 3 ± = 64π sin θ5 cos θ e ±iφ, Y 3 ± = 3π sin θ cos θe ±iφ, Y 3 ±3 = 64π sin3 θe ±3iφ θ, φ π θ, φ + π 6.3 Y m π θ, φ + π = Y m θ, φ 6.3 z xy,, 3 xy z xy Y.5 Y ± - Y ± a b c : = Y m. a m =, b m = ±, c m = ± m h z z = Y m, m = +,, x, y, z 4

25 Y Y ± Y ± 3 - Y ± a b c d 3: = 3 Y m 3. a m =, b m = ±, c m = ±, d m = ±3. x, y, z 3 3 Y z = 4π r, 6.3 Y + Y 3 x = 4π r, 6.33 Y + Y 3 i y = π r 4. Y m θ, φ Y m θ, φ L Y m θ, φ = h + Y m θ, φ, L z Y m θ, φ = hmy m θ, φ, L ± Y m θ, φ = h m + ± my m± θ, φ z Y θ, φ z 6. φ L z Y θ, φ = h i φ Y θ, φ = hy θ, φ, 6.36 Y θ, φ exp iφ 6.37 L ± 6. [ L ± = L x ± il y = he ±iφ ± θ + i cot θ ] φ

26 θ h z z L Y 4.3 N = L Y θ, φ = he iφ [ θ + cot θ ] Y θ, φ 6.39 Y θ, φ = N sin θe iφ 6.4 Y θ, φ Y θ, φ Y, Y π dφ π sin θdθy θ, φ Y θ, φ 6.4 N Y θ, φ = z L + +! 4π sin θe iφ 6.4! L + Y θ, φ = he iφ [ θ + cot θ ] Y θ, φ 6.35 h Y θ, φ Y = hn sin θ cos θe i φ 6.43 θ, φ = N sin θ cos θe i φ +! = 4π! sin θ cos θe i φ 6.44 L + z Y θ Θθ L + [ e imφ Θθ ] = he im+φ [ θ m cot θ ] Θθ = he im+φ sin θ m d [ sin θ m Θθ ] dθ = he im+φ sin θ m+ d [sin θ m Θθ ] 6.45 d cos θ θ, φ L + m + Y m θ, φ 6.45 = = = Y m θ, φ = + 4π + 4π + 4π L+ + m + m h m! + m! m! + m! e imφ! e imφ! m+ Y θ, φ [ ] [ ] d d m cot θ cot θ sin θ dθ dθ [ ] d m cot θ sin θ + dθ d d cos θ sin θ m! +m +m d + m! e imφ sin θ m sin θ 6.46! d cos θ 6

27 z Y θ, φ Y θ, φ z m h Y m θ, φ = Y m θ, φ = + 4π L + + m m h + m! m!! eimφ sin θ m m Y θ, φ m d sin θ 6.47 d cos θ L L [ e imφ Θθ ] = he im φ sin θ m+ d d cos θ [sin θm Θθ] 6.48 sin θ z m > 6.46 m 6.47 P m cos θ m = P cos θ Y m θ, φ = ϵ, m P m + m! 4π + m! P m cos θe imφ 6.49 { m, m >, ϵ, m =, m m cos θ = d sin θ m sin θ! d cos θ 6.5 L / h + L z / h m Y m θ, φ 6.49 γ η = e iγ Y θ, φ [ H = h m r r r r Hur, θ, φ = Eur, θ, φ r sin θ + ] sin θ θ θ sin θ φ + V r 6.53 u r = im rur, θ, φ = 6.54 r 7

28 6.49 Y m θ, φ r = 6.54 im rr r = 6.55 r 6. r h m r d r d = d dr dr dr + r d dr = d r 6.56 r dr d + h r + dr mr + V r E R r = 6.57 z m h z z r = 6.55 h m d + h + dr mr χ r = rr r 6.58 im χ r = 6.59 r + V r E χ r = 6.6 χ r V eff V eff = V r + + h mr 6.6 r > r = χ r χ r = r γ n= a n r n 6.6 γ V r = r η n= 6.59 V n r n, η > 6.63 a [γγ + ] = 6.64 γ = γ =

29 z m h u u, u = = drr R r dr χ r sin θdθdφ Y m θ, φ sin θdθdφ Y m θ, φ 6.67 r R r 6.55 r = /r r = 7 7. MKSA α ε α e 4πε hc Ze e V r = Zα hc r r 6.49 Y m θ, φ z m h 7. u m r, θ, φ = R ry m θ, φ 7.3 [ h d d + m r dr r h + dr mr Zα hc ] E R r = 7.4 r ρ = κr, κ = m E h m, λ = Zαc E 7.5 ρ R d ρ ρ dr [ λ + dρ dρ ρ ] + 4 ρ R = 7.6 ρ /4 d R dρ 4 R ρ 7.7 R ρ e ± ρ, ρ 7.8 R ρ e ρ 9

30 R ρ R e ρ R v R ρ = ρ e ρ v ρ v ρ d dρ v + [ + ρ] d dρ v + λ v = v v ρ = k= a k ρ k ρ a k+ = k k + + λ k + k + + a k 7. a k+ a k k 7.3 k v e +ρ 7.4 R e ρ v e ρ e + ρ e ρ ρ n λ = n + + n 7.5 n n λ 7.5 n E n = mz αc n a k = k + n kk + + a k + nk + n k = kk + + k k + a k = = k + nk + n + n kk k + + k + + a 7.7 3

31 7. dρe ρ ρ p+ [L p qρ] = [q + p!]3 p + q q! N. Bohr a B a B h mαc 7.5 κ = Z na B, 7.9 ρ = κr = Zr na B R n r 6.49 Y m θ, φ u nm r, θ, φ Z R n r = a B 3 n u nm = R n ry m θ, φ, 7. n! Zr Zr [n +!] 3 L + n e Zr na B 7. na B na B R n r a B n /r u r = nn m, r u nn m = Z n 7.3 a B Z = n = R /r m n =,,, 3, 4, 5, = s, p, d, f, g, h, 7.4 = p n =,, 3, 4, 5, 6, K, L, M, N, O, P,. K s n =, =. L s n =, = p n =, = Z R r = Z R r = a B R r = a B 3 Z a B 3 e Zr a B 7.5 Zr e Zr a B 7.6 a B 3 Zr 6a B e Zr a B 7.7 3

32 3. M 3s n = 3, = R 3 r = Z a B Zr Zr + a B 3a B e Zr 3a B 7.8 3p n = 3, = 3d n = 3, = R 3 r = Z a B Zr a B Zr e Zr 3a B 7.9 3a B R 3 r = Z a B Zr a B e Zr 3a B 7.3 r r u = r R r n n n z m h =,,, n, m =, +,, 7.3 n = m= n = + = n 7.3 = n E n E n h z m z z r 3 x, y, z r, θ, φ SU3 r O4 3

33 / Z Z +Z n =,, n =,,, n n, z = h, + h,, h z + / n, + n, /r n =,,, n E n = mz αc /n n n n,,, 8 3 3, 3, , 4, 4, , 5, 5, , 3 5, 6, 6, 3 86 : n,,, 8, 36, 54, 86, / n n, +, 8,, 8, 5, 8, 6, 33

34

24.15章.微分方程式

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