) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4

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Download ") ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4"

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1 1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h = J sec) hc=197.3 MeV fm = kev pm= ev nm = ev Å 1pm=1 1 m 1nm=1 9 m 1Å=1 1 m 1eV = J 3. p x = i h x x T x = p x m ˆT x = h m 8) x 9) = 1) [ h m x + V x) 1 ] φ = Eφ 11)

2 ) ] [ h m x + y + + V x) φ = Eφ 1) z E = i h t 13) x << 1) N n n= = N N + 1) 14) N n n= = N N + 1)N + 1) 6 15) N n 3 n= = 1 4 N N + 1) 16) N n 4 n= = N 3 N + 1)N + 1)3N + 3N 1) 17) fa + x) = fa) + f a)x + 1! f a)x +... fa) + f a)x 18) 1 1 x = 1 + x + x + x x 19) e x = 1 + x + 1! x + 1 3! x = lim 1 + x n 1 + x ) n n) sin x = x + 1 3! x x 1) cos x = 1 + 1! x ) 1 + x = x 1 8 x ) ln1 ± x) = ±x + x ± x = n 6. :m :v :p :T :E :c ±1) n xn n 4)

3 E = m = T = p = ) 1MeV J) mc ) + pc) = mc + T = m c 5) m 6) 1 v c p = m v = E c v 7) E = mc 8) m g) mc MeV) mc mc ) pc << mc ) E = ) [ pc mc ) + pc) = mc 1 + mc ) ] pc = mc + p mc mc m 9) pc >> mc ) m = ) 7. E = pc 3) sin A sin B = 1 [cosa B) cosa + B)] 31) cos A cos B = 1 [cosa B) + cosa + B)] 3) sin A cos B = 1 [sina B) + sina + B)] 33) 8. e iθ = cos θ + i sin θ 34) e iθ = cos θ i sin θ 35) e inθ = cos θ + i sin θ) n = cos nθ + i sin nθ 36) cos θ = eiθ + e iθ 37) sin θ = eiθ e iθ i 38) 3

4 9. cosh x = ex + e x 39) sinh x = ex e x 4) cosh x sinh x = 1 41) F k) = fx) = 1 π 1 π fx)e ikx dx 4) F k)e ikx dk 43) I n α) I α) = I 1 α) = J n α) J α) = J 1 α) = I α) I 1 α) ax + bx + c = a π x n n 1)!! e αx dr = α n+1 n 44) π e αx 1 dx = α 45) π x e αx 3 dx = 46) 4 α x n+1 e αx dr = n! α n+1 47) xe αx dx = 1 α e ax +bx+c) dx = e b 4a c 48) x 3 e αx dx = 1 α 49) x + b ) b a 4a + c 5) e a x+ a) b π dx = a e b 4a c 51) I α) = I α) = e αx dx e αy dy 5) π = e αx +y ) dxdy = dr rdθe αr 53) π π 1 = re αr dr z αr = e α z dz 54) π = 55) α I 1 α) = π α I 3 α) = 4 α 56) 4

5 1. { 1 i = j δ ij = i j 57) 13. δx) 1 δx)dx = 1 58) fx)δx)dx = f) 59) fx)δx a)dx = fa) 6) 14. xδx) = 61) x a)δx a) = xδx a) = aδx a) 6) δx a)δx b) = δa b) 63) δfx)) = 1 f x i ) δx x i) 64) x i fx i )=) δax) = 1 δx) a 65) δx a ) = 1 [δx a) + δx + a)] a 66) 1 δx a)x b)) = [δx a) + δx b)] 67) a b δx) = 1 e ikx dk 68) π δ r) = δx)δy)δz) 69) fx)δ x a) = f a) 7) 1 r + k ) eikr r + k cos kr ) r µ ) e µr r = 4πδr) 71) = 4πδr) 7) = 4πδr) 73) = 4πδr) 74) 75) 15. r dθ =rdθ 5

6 16. r dω = rdθ r sin θdφ = r sin θdθdφ = r dcos θ)dφ 76) 17. d f dg ) = f d g dx dx dx + df dx d dg f g) = f dx dx + g df dx f dg dx dx = [f g] dg dx = f d g dx + d dx [ f d g dx dx = f dg ] [ g df ] + dx dx 77) g df dx 78) dx g df ) g d f 79) dx dx g d f dx 8) dx x f g ) 78) 8) f dg dx dx = f d g dx dx = 18. g df dx 81) dx g d f dx 8) dx V S n a) V fd 3 r = b) f g + f g)d 3 r = V V f g g f)d 3 r = S S f d S 83) S f g nds 84) f g g f) nds 85) 19. x 3 π4 dx = e x ). I = 1 gu) e βu u ) + 1 du = u 6 gu)du + π 6 g u ) β + 7π4 g u ) ) 36 β 4

7 1. q 1 q... D 1 q 1 ) + D q ) +...) φq 1, q,...) = 88) φq 1, q,...) φ 1 q 1 )φ q )... D i q i ) q i D 1 q 1 )φ 1 q 1 )) φ q )φ 3 q 3 )... + D q )φ q )) φ 1 q 1 )φ 3 q 3 ) +... = 89) φ D 1 q 1 )φ 1 q 1 ) φ 1 q 1 ) + D q )φ q ) φ q ) +... = 9) q 1 q... ɛ i i=1,,..) D 1 q 1 )φ 1 q 1 ) φ 1 q 1 ) D q )φ q ) φ q ) = ɛ 1 D 1 q 1 )φ 1 q 1 ) = ɛ 1 φ 1 q 1 ) 91) = ɛ D q )φ q ) = ɛ φ q ) 9)... 93) ɛ 1 + ɛ +... = 94). D i q i )φ i q i ) = ɛ i φ i q i ) 95) V = ) ) h m x + y + φ = Eφ 96) z φx, y, z) = Xx)Y y)zz) 97) h d Xx) m dx Y y)zz) + d Y y) dy ) Xx)Zx) + d Zz) Xx)Y y) = EXx)Y y)zz) dz 98) 7

8 Xx)Y y)zz) h 1 d Xx) + 1 d Y y) + 1 d ) Zz) = E 99) m Xx) dx Y y) dy Zz) dz x, y, z) E h 1 m Xx) h 1 m Y y) h 1 m Zz) 3. d Xx) dx = ɛ x 1) d Y y) dy = ɛ y 11) d Zz) dz = ɛ z 1) ɛ x + ɛ y + ɛ z = E 13) h 1 m Xx) d Xx) dx = ɛ x 14) ) mɛx Xx) = exp i x h 15) exp x e x y z ) ) ) mɛx mɛy mɛz φx, y, z) = A exp i x exp i y exp i z 16) h h h A L L L Xx) Xx + L) = xx) 17) ) ) mɛx mɛx exp i x + L) = exp i x 18) h h mɛx L = n x π 19) h ɛ x = 1 ) nx π h 11) m L 8

9 n x p x ɛ x = p x m 111) 11) p x = n xπ h L = n xh L 11) p x L =, ± h, ± h, ± 3h, ) φx, y, z) = A exp i p x = A exp ) h x exp i p y i πn ) x L x exp ) h y exp i p ) z h z i πn ) y L y exp A A 4. i πn z L z ) 114) 115) 113) ) x h p x = h) y z 1 h 3 N 3N h 3N 6N h h 3N p p + dp V 4πp + dp) 3 4πp3 = 4πp dp 116) 3 3 V 4πV p dpv 117) dw = 4πV p dp h 3 118) N N 3N )V p N N V N h 3N 9

10 1 g g = 119) g = 1) S g = S + 1) 11) W = g V p N V N 1) h 3N N! 5. dw = g 4πV p dp 13) h 3 ɛ a) ɛ = p m 14) p = mɛ 15) dp = m m p dɛ = dɛ 16) ɛ p dp = ɛm 3 dɛ 17) b) πv m)3/ dw = g ɛdɛ 18) h 3 ɛ = pc 19) p = ɛ c 13) dp = dɛ c 131) 1

11 dw = g 4πV h 3 c 3 ɛ dɛ 13) 6. fx)dx = πi i Resfz i )) 133) Res fz) n = 1 fz) = 7. a) n= A n z a) n 134) T E e E k B T 135) k B erg K 1 b) S = k B ln W E) = k B ln ΩE)dE 136) W E) E ± de Ω = ) c) S E = 1 T 137) 8. H H ) H ij H ji 138) AB) ij = AB) ji = A jnbni = ) ) B A = B A ) in nj ij n n 139) AB) = B A 14) ABC...) =...C B A 141) 11

12 9. H ) ij H = H 14) H ji = H ij 143) H ) ii = H ii = H ii 144) 3. U I 31. U = U 1 145) U U = UU = I 146) H : U = e ih 147) U = 1 + ih + 1! ih) +... = 1 + ih 1! H ) H U = 1 ih 1! H ) +... = 1 ih 1! H) +... = e ih 149) U 3. φ n x) n x φ n > n > φ n x) =< x φ n > 15) φ nx) =< φ n x > 151) φ n x) ψ m x) ψnx)φ n x)dx = < ψ m x >< x φ n > dx 15) 1

13 33. {φ n x)}:n = 1,,..) φ nx)φ m x)dx = δ nm 153) fx) fx) = m C m φ m x) 154) C n φ nx) φ nx)fx)dx = m C m φ nx)φ m x)dx = m C m δ nm = C n 155) fx) = m C m φ m x) = m [ ] φ my)fy)dy φ m x) 156) [ ] fx) = φ mx)φ m y) fy)dy 157) m φ mx)φ m y) = δx y) 158) m [ ] fx) = φ mx)φ m y) fy)dy = δx y)fy)dy = fx) 159) m {φ n x)}:n = 1,,..) φ mx)φ m y) = δx y) 16) m 1 φ m x) m > φ m < m { m >} < m n > = δ nm 161) m >< m = 1 16) m < ψ m A φ n >=< φ n A ψ m > 163) 13

14 A A = A) < ψ m A φ n >=< φ n A ψ m > =< ψ m A φ n > 164) ψ m > φ n > A < A >=< φ n A φ n >=< φ n A φ n > 165) A { n >} A n > = a n n > a n ) 166) < n m > = δ nm 167) n >< n = 1 168) n Ψ > = C n n > n 169) C n = < n Ψ > 17) 6 < x x >= δx x ) 171) fx ) = δx x )fx)dx 17) dx x >< x = 1 173) f > < x < x f > < x dx x >< x f >= dx < x x >< x f >= δx x )fx)dx = fx ) 174) < ψ m x >< x φ n > dx =< ψ m φ n > 175) φ e ikx k { 1 < x k >= π e ikr 1 L e ikr L 176) 14

15 { 1 < k x >= π e ikr 1 L e ikr L 177) φk) =< k φ >= < k x >< x φ > dx = 1 π e ikx φx)dx 178) φx) =< x φ >= < x k >< k φ > dk = 1 π e ikx φk)dk 179) 4 43) 179) dk k >< k = 1 18) 35. φx) φx) =< x φ > 181) < k x > x { x >} dx < k x >< x φ >=< k φ > 18) < ψ n x > dx < ψ n x >< x φ >=< ψ n φ > 183) {ψ n } A A UAU 36. n n A v i i = 1...n) Av i = α i v i 184) α v i α i Av i = α i Iv i 185) A α i I)v i = 186) 15

16 A α i I = 187) U UAU Uv i = α i Uv i 188) Uv i UAU α i 37. A U B UAU = B, 189) B B B =... 19) B nn A B A b i = Uv i =,,..., 1,,...) T v i A ) Bb i = B ii b i = UAU Uv i = UAv i = α i Uv i = α i b i 191) v i = U b i ) A B = UAU ) = UAU = B 19) B B ii = B ii 193) 38. H E i i = 1..z) { i >}i=1..z) H i >= E i i > 194) < i j > = δ ij 195) i >< i = 1 196) i 16

17 H ij < i H j > 197) E 1... E H ij = E j < i j >= E j δ ij = E z 198) H { φ n >, n = 1..z} < φ n φ m > = δ nm 199) φ n >< φ n = 1 ) n A A = {A nm : A nm =< φ n H φ m >} 1) { i >} { φ n >} U im < i φ m > ) U mi = U im =< φ m i > 3) UU ) ij = m U U) nm = i U im U jm = m U inu im = i < i φ m >< φ m j >=< i j >= δ ij 4) < φ n i >< i φ m >=< φ n φ m >= δ nm 5) U = U 1 U U B = UAU B ij = U in A nm U mj = < i φ n >< φ n H φ m >< φ m j >=< i H j >= E j δ ij 6) nm nm E 1... B = UAU E = E z 7) i > = n φ n >< φ n i >= n < i φ n >) φ n >= n φ n > U ni 8) φ n > = i i >< i φ n >= i i > U in 9) 17

18 { i >} { φ n >} U { i >} < φ n H φ m > φ n > 39. {φ n }n = 1,,..)  A = {A nm : A nm =< φ n  φ m >} 1) {φ n }n = 1,,..) ˆB B = {B nm : B nm =< φ n ˆB φ m >} 11)  ˆB Ĉ C = {C nm : C nm =< φ n Ĉ φ m >=< φ n  ˆB φ m >} 1) A ni B im = < φ n  φ i >< φ i ˆB φ m >=< φ n  ˆB φ m >=< φ n Ĉ φ m >= C nm 13) i i i φ i >< φ i = x x N {x 1, x,..., x N } x x ) < x >= 1 N x i 14) N i=1 x ) σ = 1 N x < x >) 15) N 1 i=1 x ) σ = 1 N x < x >) 16) N 1 i=1 18

19 x x x + dx P x)dx P x) x P x) = φx) b a P x) 17) P x)dx = 1 x [a, b] 18) < x > = σ = b a b a xp x)dx 19) x < x >) P x)dx =< x > < x > ) < x < x >) > = < x > < x > 1) a) b) 1.. ν λ E p E = hν ) p = h λ 3) ) h E = pc 4. 19

20 43. 1 α β α 1 β N N N N! N N! N! 44. 1/, 3/,...) x y z S z z S, S + 1,..., S 1, S S + 1 S + 1 S ln 1 = k B lns + 1) 45. a) b) 1/ S m V

21 φ ) h m + V φ = Eφ 4) E h h π φr) r E E = E 1, E, E 3,... 5) E 1 E E ) 47. ) 1 φx) = π 1/4 d exp ikx x d 7) φ x)φx)dx = 1 ) exp x dx 8) πd d Gauss e αx dx = πα 1/ 9) α = 1/d ) 1 πd exp x d ) dx = 1 πd πd = 1 3) < x > < x > < p > < p > < x >= φ x)xφx)dx = 31) x= < x >= φ x)x φx)dx = 1 ) x exp x dx 3) πd d x e αx dx = π α 3/ 33) 1

22 α = 1/d ) < x >= 1 πd x exp x d ) dx = 1 π πd d3 = d 34) < x > < x < x >) >=< x > < x > = d < p > < p > p = i h x < p >= φ x) x φx)dx = < p >= 35) φ x)pφx)dx = i h φ x) φx)dx 36) x x φx) = ik x )φx) 37) d φ x)ik x d )φx)dx = 1 πd ik x d ) exp ) x dx = ik d 38) < p >= i h ik = hk 39) φ x)p φx)dx = h φ x) φx)dx 4) x x φx) = x x φx) = x ik x d )φx) = ikik x d )φx) 1 d φ x d ik x d )φx) = k ik x d 1 ) d + x φx) 41) d 4 φ x) x φx)dx = = φ x) k ik x d 1 ) d + x φx)dx k 1 d ) + 1 d 4 d d 4 = k 1 d 4) < p >= h k 1 ) = h d d + hk) 43) < p > < p < p >) >=< p > < p > = h d 44)

23 < x >< p >= d h d = h 45) x p h 46) 48. I = e ax dx 47) y I = I = e ay dy 48) e ax +y ) dxdy 49) x y x = r cos θ 5) y = r sin θ 51) r = x + y 5) r θ π J dxdy = Jdrdθ 53) x x r θ J = y y = cos θ r sin θ sin θ r cos θ = r 54) r θ dxdy = rdrdθ 55) r θ t = r dt = rdr I = = π = π a π dr dθre ar = π re at 1 r dt = π 1 re ar dr 56) [ e at dt = π 1 ] a e at 57) 58) I π e ax dx = a 3 59)

24 I x e ax dx 6) I = d da I 61) I = x e ax dx = 1 π 6) a 3 I 4 = I n x 4 e ax dx = d da I = 3 π 63) 4 a 5 x n e ax dx = d da I n 64) I = e ax dx = 1 e ax dx = 1 π a 65) 49. [ z d dz + b z) d ] dz a wz) = 66) Kummer a b b wz) = cma, b, z) 67) c M confluent hypergeometric function) Kummer Ma, b, z) = 1 + a b z 1! a) n aa + 1)...a + n 1) = aa + 1) z + bb + 1)! +... = a) n z n n= b) n n! a + n 1)! a 1)! Γz) n = 68) Γa + n), a) = 1 69) Γa) Γn + 1) = n! 7) b n n 66) w w = c 1 Ma, b, z) + c z 1 b Ma b + 1, b, z) 71) 4

25 c 1 c z Ma, b, z) = Γb) Γb a) eiɛπa z a ga, a b + 1, z) + Γb) Γa) ez z a b g1 a, b a, z) 7) ga, b, z) = ɛ = n= { 1, π/ < arg z < 3π/ 1, 3π/ < arg z π/ a) n b) n n!z n = 1 + ab z 73) + aa + 1)bb + 1) z ) n 69) a) n+1 = M n, b, z) n a n Ma, b, z) z e z Ma, b, z) = e z Mb a, b, z) 75) a = n w Laguerre 5. M n, k + 1, z) = n! k + 1) n L k nz) 76) [ z d dz + k + 1 z) d ] dz + n wz) =, k, n =, 1,,..., ) 77) w = L k nz) = [n + k)!] M n, k + 1, z) = n!k! n 1) s [n + k)!] s= n s)!k + s)!s! z 78) e zt 1 t 1 t) = t n k+1 n + k)! Lk nz), t < 1) 79) n= e z z k L k nz)l k mz)dz = [n + k)!]3 δ nm 8) n! 5

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.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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