ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx

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1 i B p89 4. ψ x, tψx, t = ψ R x, t iψ I x, t ψ R x, t + iψ I x, t = ψ R x, t + ψ I x, t p 5.8 π π π F e ix + F e ix + F 3 e 3ix F e ix + F e ix + F 3 e 3ix dx πψ x πψx p39 7. AX = X A [ a b c d x y ] = x y a c b d p p x > b = 4 c a

2 ii p ϕ x, t = C ϕ xe i ħ E t +C ϕ xe i ħ E t ψ x,t ψ x,t p79 やは時間変化しないことに注意 振動 粒子はだいたい このあたりにいる 粒子はだいたい このあたりにいる p35 D.3 Aψ Cϕdx = aψ ψ C Aϕ dx bϕ p356 D.56 x = L 6L 9π C C e i ħ E E t + C C e i ħ E E t p6 5 x, p fx 3. fx 3. p59 p59 h

3 iii p346 D.7 cx cx fxδcx c dcx = cx cx f t c δt c p347 D.8 f t cx c δt c = c f c > = c f c < c f cx p3 h E h E 5 t = h E p4 7. X X = Y x = k y x + k y 5 h E h E h E

4 iv p8 mm a m m λa m =.6 m λ a m = a m mm m 4 λ m m a m = m m 3 a m 4 a m a m a m a m 4 a m m a m m λ a m = a m = mm = m λm 4 λ a m 4 mm m m 3 m λm 4 λ λ λ a mm m m 3m 4 3 m λm 4 λ 3 λ λ a mm m m 3m 4 3 m m.7 a m m a m a m a m m λ = m a m mm m p43 ψ n = πn! ξ n e ξ = πn! e ξ n e ξ ξ ξ.73 p73 L ± l, m = ħ ll + mm ± l, m ±.8 = l ml±m+

5 v p74.63 dx = sin θdθ d dθ = sin θ d dx p76 L e imϕ Θ m l θ = iħe iϕ θ i cot θ e imϕ Θ m l θ ϕ = iħe im ϕ d dθ + m cot θ Θ m l θ Θ m l.89 mħ m ħ L iħe iϕ d dθ + m cot θ p8 r > R p3 3.4 R l ρ = e ρ ρ l dl+ L n+l ρ dρ l+ = e ρ ρ l L l+ n+l ρ p36 D.8 [ [L z, L ± ] = iħ ϕ, ±iħe±iϕ [ = ±iħe ±iϕ iħ ϕ, e±iϕ =±ħ θ ± i cot θ ] ] ϕ θ ± i cot θ ϕ = ±ħ ±iħe ±iϕ θ ± i cot θ ϕ } {{ } L±

6 vi p9 6- [Â, ˆBĈ] = ˆB[Â, Ĉ] + [Â, ˆB]Ĉ p ψ ˆp ψ = ψ iħ ψdx x p54. p36 a = d v d = e r dθ r r r sin θ + e θ r d θ + dr + e ϕ r sin θ d ϕ dθ r sin θ cos θ + sin θ dr dϕ p36 D.8 p364 dϕ dϕ + r cos θ dθ dϕ l, l L x l, l = 4 l, l L + + L = 4 l, l L + L [L +,L ]+L L + = l, l [L +, L ] + L L + 4 ħl z = l, l ħlz l, l = ħ l L + +L l, l l, l l, l D.9

7 vii p9 5 g = ε E B p43.3 L π ke E L µe π = n h πke µ E πl = n h =nh p9 3 [Â, ˆB n ] = n ˆB n [Â, ˆB] Bn x = n B x Bn p4 6.9 d b a ψ x, tψx, tdx = [Jx, t] b a = Jb, t + Ja, t p α p dx p x x p = πħ dxe i ħ xp p p66 8- ψ

8 viii p69 ψ ψ dx ik ψψ ψψ dx + k ψψ dx > = ψ ψ ik ψ ψ ψ ψ + k ψ ψ >= 8.4 p7 [p p, x x ] = [p, x] = iħ 8.8 p ϕx x = L p83 N e αx e ikx = N απ dpe p k 4α e ipx 9. P x N x p 6 ψ O ψ E p ψ x = P ψx P ± p. dψ dx a, c b ax + bx + c > =

9 ix p3 k +κ = mv k +κ = mv ħ ħ kd κd p5 9 p p5 p m = h 8md p 4 9-4,9-5 p95 ke r 9 p3 p4 d dξ + ξ ψx = λ + ψx.5 a = a a = aa a a.59

10 x p4 ψ ξ = π 4 e ξ.6 ψ ξ = π 4 ξe ξ.63 ψ ξ = ψ 3 ξ = π 4 ξ e ξ.64 π 4 3 ξ 3 3ξe ξ.65 ψ 4 ξ = π 4 6 4ξ4 ξ + 3e ξ ψ 5 ξ = π 4 5 4ξ5 ξ 3 + 5ξe ξ p6 ħ µ r r r r ψ + L + V r ψ = Eψ.47 µr p p68 e mπi = m 6 p69 L ± = ±ħe ±iϕ θ ± i cot θ ϕ.69 5 e imϕ Lz L L L z 6 m -7 p79 p363

11 xi p75 dθ l lθ dθ = l cot θθ l lθ dθ l lθ = l cot θdθ Θ l l θ log Θ l lθ = l logsin θ + C Θ θ C.88 Θ l lθ = A sin l θ A e C p76 L e imϕ Θ m l θ = ħe iϕ θ i cot θ e imϕ Θ m l θ ϕ = ħe im ϕ d dθ + m cot θ Θ m l θ Θ m l.89 p75.9 d dθ = sin θ d dcos θ p79 l l! x m d l+m x l = N m dx l+m l x m d l m x l dx l m. p p74

12 xii p8 q V P Q = 4πε R > r R + r Rr cos θ V P Q = q + r r 4πεR R cos }{{ θ 3 cos θ r 3 5 cos 3 θ 3 cos θ } R R P cos θ P cos θ P 3 cos θ.8 p86 ξ 3 ξ d ξ d Ql ξ + dξ dξ ξ d ξ 3 dq lξ ξql ξ dξ dξ d dξ Q lξ + dq l ξ + ξ dξ + ll + Ql ξ = ξ ξ ll + Q ξ l ξ = l + ξ Q l ξ =.4 p89-3 r r = x x + y y + z z p p p3 R l ρ = e ρ ρ l L l ρ 3.5 L n,lρ = ρ n + c n ρ n + c n ρ n + + c ρ + c ρ + c 3.36 pr = iħ r r = r r p59

13 xiii p3 3.6 C p34 s p ± s p µ ke r } {{ } E µ ke r } {{ } E ke s r ± s p ke r s + s p µ ke r E µ ke r } {{ } E ke s r ke s r + ke R d dρ χ ll + l χ ρ l + λ ρ χ l 4 χ l = ll + λ λ = n = 3 l =,, ρ ρ 3 l p339 d b a ψ x, tψx, tdx = b a ψ x, t ψx, t + ψ ψx, t x, t dx t t C.6 p343-8 π dθ sin θψ d sin θ ddθ sin θ dθ ϕ = p344 π dθψ d dθ sin θ ddθ ϕ ϕ ψ [ d dξ + α ] [ ] [ ] d d, ξ ξ dξ + ξ d =, ξ dξ dξ + ξ d + α, ξ [ ] ξ dξ d d [ ] C.7 ξ = dξ, ξ dξ + + αξ, d dξ

14 xiv p349 [Â, ˆBn ] = ] [Â, ˆB ˆBn + ˆB ] [Â, ˆB ˆBn + ˆB ] [Â, ˆBn D.7 [Â, ] ] ˆBn = [Â, ˆB ˆBn + ˆB [ Â, ˆB n ] D.8 p35 = iħ b a = iħ m [ ħ b a = iħ b m a m ψ x x ψ x + V x ] ψ ψx, t + ψ x, t [ ħ m ψ ψ ψ x x ψ ψ ψ x dx dx = iħ [ ψ m x ] ψ b ψ ψ x a x + V x ] ψ dx D.5 p a, b b a ψ = Ax A b a [ x = A dxx = A 3 3 ] b a = A b a A = 3 b a 3

15 xv p354 3 a a b, b b a b a ψ = Ax A b a = A dxx = A [ x 3 3 ] b a = A b a 3 D.43 A = x b a 3 x 4 = b a 3 b a [ dxx 4 4 x 5 = b a 3 5 ] b a = 3 b a D.44 p354 ψ = Ce k ħ x D.46 p355 sin A sin B = p356 cosa B cosa + B e ikx + R e ikx e ikx + Re ikx = + R + R e ikx + Re ikx = + + e ikx+iϕ + e ikx iϕ = + coskx + ϕ D.6 kx + ϕ = nπ kx + ϕ = n + π p357 k + κ = mv ħ κd = kd cot kd

16 xvi p36 L x ± il y = iħ sin ϕ cot θ cos ϕ θ = iħ sin ϕ ± i cos ϕ =±ie ±iϕ = iħe ±iϕ ±i θ cot θ ϕ = ±ħe ±iϕ θ ± i cot θ ϕ ϕ ± i θ cot θ cos ϕ ± i sin ϕ =e ±iϕ cos ϕ cot θ sin ϕ θ ϕ ϕ D.79 p363-7 p363-8 π dθ sin θψ sin θ d sin θ ddθ dθ ϕ = π sin θ ddθ ϕ dθψ d dθ = [ψ sin θ d ] π dθ ϕ sin =sin π= = [ dθ dψ δθ sin θ d ] π dθ ϕ + sin =sin π= π dθ dψ dθ sin θ d dθ ϕ π dθ d dθ sin θ dψ δθ D.85 ϕ p365 d ξ dξ + α ξ + d dξ D.93 P l+ l x l+ d l+ dx l+ x l l l m p67

17 xvii p5 隣近所の波と 位相がそろっている経路 位相がそろってここに到着する 隣近所の波と位相がそろわない経路の例 p83 p96 ψ ϕ Dirac

18 xviii p ψx, t = ψ x, tψx, tdx = p9 E t > h p44 π π π e inx π e in x dx = n n 7.3 p fx a n p48 ϕ ψ ψ ψ ϕ 3 ϕ 3 = ψ ϕ 7.34 ψ.. ϕ p5 FAQ p = iħ x E = iħ x t

19 Ax = ψ x, ta x ψx, tdx = ψ p, te i p x ħ dp A x πħ Ax = πħ = πħ dx = ψ p, ta ψ p, te i p x ħ dp [A ψp, te i px ħ dp dx iħ p 7.44 xix ] ψp, t e i px ħ dp e i ħ p p x dx ψ p, ta iħ ψp, tdpdp p =πħδp p iħ p ψp, tdp 7.47 p.3 8 Be κx Ae κx p3 5 p34 8 p5.9 e r, e θ, e ϕ z e θ, e ϕ e r, e θ, e ϕ 5 8 a + a = a + aa + a a a

20 xx p64.54 e ϕ p39 δs = = p335 tf t i tf t i m m dx dx + δx V x + δx dδx δx V x x tf t i m dx V x A.4 fx = c π π L n= dk F ke i nπ L x B.7 p335 B.9 L F k fx f n B.4 p334 c F k L c f n = c L L nπ i e x L fxdx F k = c e ikx fxdx B.9 p335 B. c =

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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