1 filename=mathformula tex 1 ax 2 + bx + c = 0, x = b ± b 2 4ac, (1.1) 2a x 1 + x 2 = b a, x 1x 2 = c a, (1.2) ax 2 + 2b x + c = 0, x = b ± b 2

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1 filename=mathformula58.tex ax + bx + c =, x = b ± b 4ac, (.) a x + x = b a, x x = c a, (.) ax + b x + c =, x = b ± b ac. a (.3). sin(a ± B) = sin A cos B ± cos A sin B, (.) cos(a ± B) = cos A cos B sin A sin B, (.) tan A ± tan B tan(a ± B) = tan A tan B, (.3) sin(a ± π ) = ± cos A, (.4) sin(a ± π) = sin A, (.5) cos(a ± π ) = sin A, (.6) cos(a ± π) = cos A, (.7) tan(a ± π ) = tan A, (.8) tan(a ± π) = tan A. (.9). sin A + sin B = sin( A + B sin A sin B = cos( A + B cos A + cos B = cos( A + B cos A cos B = sin( A + B tan A ± tan B = ) cos( A B ), (.) ) sin( A B ), (.) ) cos( A B ), (.) ) sin( A B ), (.3) sin(a ± B) cos A cos B. (.4)

2 3. sin A sin B = [cos(a + B) cos(a B)], (.5) sin A cos B = [sin(a + B) + sin(a B)], (.6) cos A sin B = [sin(a + B) sin(a B)], (.7) cos A cos B = [cos(a + B) + cos(a B)]. (.8) 3 3. n k= n k= n k= k = n(n + ), (3.) k = n(n + )(n + ), (3.) 6 k 3 = 4 n (n + ). (3.3) 3. n cos rx cos x + cos x + + cos nx r= = n+ nx cos( x) sin( sin( x) (3.4) n sin rx sin x + sin x + + sin nx r= n+ nx sin( x) sin( n r= n n r= n = sin( x ) (3.5) cos rx cos( nx) + + cos x + + cos nx = sin[(n + ) x ] sin( x ) (3.6) sin rx sin( nx) + + sin x + + sin nx =. (3.7)

3 x ( c f, g x df(x) dx lim f(x + x) f(x), x o x (4.) dc =, dx (4.) dx =, dx (4.3) d sin x = cos x, dx (4.4) d cos x = sin x, dx (4.5) d tan x = dx cos x, (4.6) de x dx = ex, (4.7) d log e x d ln x (= dx dx ) = x, (4.8) (4.9) 5 5. (pp.3-33,[]) π π exp( ax )dx = a ( exp( ax )dx = a ) (5.) exp( ax )x dx = π 4a a ( exp( ax )x dx = π a a ) (5.) exp( ax )x n (n )!! π dx =. (5.3) n+ an+ exp( ax )xdx = a. ( exp( ax )xdx = a. ) (5.4) exp( ax )x 3 dx = a. ( exp( ax )x 3 dx = a. ) (5.5) exp( ax )x n+ dx = n!. (5.6) an+ exp( α π 4ξ ) exp( ξ r )dξ = exp( αr). (5.7) r

4 4 cos(k r) exp( µr) dk = k + µ π. (5.8) r exp( µ r )dµ = π r. (5.9) 6 vector differential operators 6.. (gradient),,grad x, y ( e x, e y r, ϕ ( e r, e ϕ = e x x + e y y = e r r + e ϕ r ϕ (6.) (6.). (rotation),, rot(=curl) 3. (divergence),, div 4. ( ) (gradient),, grad x, y, z ( e x, e y, e z r, ϕ, θ ( e r, e ϕ, e θ = e x x + e y y + e z z = e r r + e θ r θ + e ϕ r sin θ ϕ (6.3) (6.4). (rotation),, rot(=curl) 3. (divergence),, div

5 5 4. ( ) 3 = x + y + z (6.5) (x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ) = r + r r + r sin θ = r r (r r ) + = r r r + r sin θ θ (sin θ θ ) + θ (sin θ θ ) + r sin θ θ (sin θ θ ) + r sin θ r sin θ ϕ (6.6) r sin θ ϕ (6.7) ϕ (6.8) A V φ(x, y, z), ψ(x, y, z) (ψ ϕ + ψ ϕ)dv = ψ dϕ da (7.) V A dn ( (ψ ϕ ϕ ψ)dv = ψ dϕ ) V A dn ϕdψ da (7.) dn dϕ/dn, dψ/dn ϕ, ψ 8 8. δ(x) = { (x = ) (x ). (8.) Heaviside s step function θ(x) = { (x < ) (x ). (8.)

6 6 δ(x) dθ(x) dx. { /ε ( x < ε/) δ ε (x) = ( x > ε/). (8.3) (8.4) δ(x) = lim ε δ ε (x) (8.5). sin αx δ(x) = lim α πx 3. e ε k (8.6) δ ε (x) = e ε k e ikx dk = ε π π x + ε, (8.7) δ(x) = lim δ ε (x) = e ikx dk (8.8) ε π δ(x a) = e ik(x a) dk. (8.9) π π δ(k k ) = j k l (kr) j l (k r) r dr, (8.) j l (x) Bessel function δ(x)dx =. (8.) f(x) x = a, x n f(x)δ(x a)dx = f(a), (8.) (8.3) f(x)δ(x a) = f(a)δ(x a), (8.3) xδ(x) =, (8.4) δ (x) = δ ( x), (δ (x) d δ(x)), dx (8.5) δ(ax) = δ(x) (a ), a (8.6) n δ(f(x)) = f (x i ) δ(x x i), (8.7) i= (f(x) f(x i ) =, i =,,, n, (8.8) f (x i ) df(x i) ), (8.9) dx f(x)[ dn dn δ(x)]dx = ( )n f(). (8.) dxn dxn

7 7 8. [] δ(r r ) = δ(x x )δ(y y ) (8.) = δ(r r )δ(ϕ ϕ ) (8.) r x = r cos ϕ, y = r sin ϕ, r = x + y, tan ϕ = y x. (8.3) δ(r r ) = δ(x x )δ(y y )δ(z z ). (8.4) r = (r, θ, ϕ) x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = r cos θ, (8.5) r = x + y + z, tan ϕ = y x, tan θ = x + y (8.6) z 3 δ(r r ) = δ(r r )δ(cos θ cos θ )δ(ϕ ϕ ) r (8.7) r = (ρ, ϕ, z) = δ(r r )δ(θ θ )δ(ϕ ϕ ) r sin θ (8.8) x = ρ cos ϕ, y = ρ sin ϕ, z, (8.9) ρ = x + y, tan ϕ = y (8.3) x 3 δ(r r ) = δ(ρ ρ )δ(ϕ ϕ )δ(z z ) ρ (8.3) 3 ( r ) = 4πδ(r), (r x + y + z ). (8.3)

8 f(x) F (k) F (k) f(x)e ikx dx. (9.) π F (k) f(x) F (k)e ikx dk = [ e ik(x x) f(x )dk]dx = f(x). (9.) π π / π /(π) Gauss. f(x) exp( x a ), a > F (k) = π f(x)e ikx dx (9.3) = a exp( a k ). (9.4) 4 a /a I, p.3 π e ax dx = (9.5) a pp.5-53 e (x+ib) dx = π, b > (9.6) ax + ikx = a(x + ik a ) k 4a. (9.7). f(x) exp( x ), a > a F (k) = π f(x)e ikx dx (9.8) = a exp( a k ). (9.9) a /a

9 f(r) F (k) F (k) ( f(r)e ik r d 3 r, (9.) π) 3 = ( f(x, y, z)e i(xk x+yk y +zk z ) dxdydz (9.) π) 3 f(r) = ( F (k)e ik r d 3 k (9.) π) 3 (9.3)

10 ± [9]. Γ(x) = e t t x dt, (.) Γ(x + ) = x Γ(x), (.) π Γ(x) Γ( x) = sin(πx), (.3) Γ(x + ) Γ( x) = π cos(πx), (.4) Γ(x) = x π Γ(x) Γ( + x). (.5) Γ() =, Γ() =, Γ(3) =, Γ(n + ) = n!. (.6) Γ(/) = π π, Γ(3/) =, Γ(5/) = 3 π 4, (.7) (n )!! (n)! Γ(n + /) = π = π. (.8) n n n!. Hermite polynomial. Schiff ([5, 6, 7, 8, 9, 3])

11 ([5]) ( d dx x d dx + n)h n(x) =, (.9) H n (x) = ( ) n e x dn dx n e x, (.) : H (x) =, (.) H (x) = x, (.) H (x) = 4x, (.3) H 3 (x) = 8x 3 x, (.4) H 4 (x) = 6x 4 48x +, (.5) H 5 (x) = 3x 5 6x 3 + x, (.6) : H n (x) n x n, (as x ), (.7) : e tx t = H n (x) tn n!, (.8) n= H n+ (x) = xh n (x) nh n, (.9) d dx H n(x) = nh n (x), (.). III[4] H n (x)h n e x dx = δ nn n n! π, (.) : H n ( x) = ( ) n H n (x). (.) ( d dx x d dx + n)h n(x) =, (.3) H n (x) = ( ) n e x / dn dx n e x, (.4) : H (x) =, (.5) H (x) = x, (.6) H (x) = x, (.7) H 3 (x) = x 3 3x, (.8) H 4 (x) = x 4 6x + 3, (.9) H 5 (x) = x 5 x 3 + 5x, (.3) : H n () = ( ) n (n )!!, (.3) H n+ () =, (.3) H n+() = ( ) n (n + )!!, (.33) : e tx t / = H n (x) tn n!, (.34) n= H n+ (x) = xh n (x) nh n, (.35) d dx H n(x) = nh n (x), (.36) H n (x)h n e x / dx = δ nn n! π. (.37)

12 .3 associated Laguerre polynomial) 3. Schiff [5] (a) (b) : e tx/( t) ( t) = n= L n (x) tn n!, (.38) [x d d + ( x) dx dx + n]l n(x) =, (.39) L n+ (x) = (n + x)l n (x) n L n (x),(.4) d dx L n(x) n d dx L n (x) = nl n (x), (.4) (.4) L α n(x) dα dx α L n(x), (.43) [x d dx : ( t)α e tx/( t) ( t) α+ = : d + (α + x) dx + (n α)]lα n(x) (.44) =, L α n(x) tn n!, (.45) n=α x α e x L α m(x)l α (n!) 3 n(x)dx = δ mn (.46) (n α)!. III[4], [9] (a) L n (x) L α= n (x). (.47) (b) [x d d + (α + x) dx dx + n]lα n(x) =, (.48) : L α (x) =, (.49) L α (x) = (α + ) x, (.5) L α (x) = x (α + )x + (α + )(α + ) (.5) L n n (x) = ( ) n xn n!, (.5)

13 3 : exp( tx t ) ( t) α+ = n= L α n(x)t n, (.53) nl α n(x) + (x n α + )L α n (x) (.54) +(n + α )L α n (x) =, (.55) x d dx Lα n(x) = nl α n(x) (n + α)l α n (x), (.56) 3. Morse and H. Feshbach[] (a) L α n(x)l α n e x x α dx = δ nn (α + n)!, (.57) n! (b) L α n(x) = (n + α)! n! e x x α d n dx n (xn+α e x ), (.58) [x d d + (α + x) dx dx + n]lα n(x) =, (.59) tx exp( t : ( t) = L α n(x) α+ (n + α)! tn, (.6) n= (x α n )L α (n + ) n(x) = (α + n + ) Lα n+(x) (α + n) L α n (x) (.6) x d dx Lα n(x) = (x α)l α n(x) + (n + )L α n+(x), (.6) : x α e x L α m(x)l α [(n + α)!] 3 n(x)dx = δ mn (.63) n!.4 Legendre polynomial [( x ) d dx x d dx + l(l + )]P l(x) =, (.64) P l (x) = d l l l! dx l (x ) l, (.65) : P (x) =, (.66) P (x) = x, (.67) P (x) = (3x ), (.68) P 3 (x) = (5x3 x), (.69) P 4 (x) = 8 (35x4 3x + 3), (.7)

14 4 P l () = ( ) l (l)! l (l!), (.7) P l+ () =, (.7) P l () =, (.73) P l ( ) = ( ) l, (.74) : lp l (x) = (l )xp l (x) (l )P l, (.75) : (x ) d dx P l(x) = l[xp l (x) P l (x)], (.76) l(l + ) = l + [P l+(x) P l (x)], (.77) = (l + )[P l+ (x) xp l (x)], (.78) P l (x)p l dx = δ ll l +, (.79) x k P l (x)dx = for k =,,,, l. (.8).5 Legendre associated polynomial.6 spherical harmonics Y lm (θ, ϕ) ( ) m+ m P m l (cos θ) π : sin θdθ π l + (l m )! 4π (l + m )! P m l (cos θ) e imϕ (.8) dϕ Ylm(θ, ϕ) Y l m (θ, ϕ) = δ ll δ mm. (.8) Y (θ, ϕ) = 4π, (.83) Y,+ (θ, ϕ) = 3 π sin θeiϕ, (.84) Y, (θ, ϕ) = 3 cos θ, (.85) π Y, (θ, ϕ) = 3 π sin θe iϕ, (.86) Y,+ (θ, ϕ) = π sin θe iϕ = π ( cos θ)eiϕ, (.87) Y,+ (θ, ϕ) = 3 5 π cos θ sin θeiϕ = π sin θeiϕ, (.88)

15 5 Y, (θ, ϕ) = 5 4 π (3 cos θ ) = 5 ( + 3 cos θ), (.89) 8 π Y, (θ, ϕ) = 3 5 π cos θ sin θe iϕ = π sin θe iϕ, (.9) Y, (θ, ϕ) = π sin θe iϕ = π ( cos θ)e iϕ, (.9) (.9) l + Y lm (, ) = 4π δ m, (.93) l + Y lm (, ϕ) = 4π δ m. (.94).7 [4].7. d du ν (z ) + ( z dz dz z )u = d u dz + du ν + ( )u =. (.95) z dz z J ν (z), N ν (z), H ν () (z), H ν () (z) H ν () (z),h ν () (z) 3 : J ν (z) = ( z ( ) n ( z )ν )n, (z ) (.96) n!γ(ν + n + ) n= : N ν (z) = Y ν (z) = cos νπj ν(z) J ν (z) (.97) sin νπ : J n (z) = ( ) n J n (z), N n (z) = ( ) n N n (z), ( n) (.98) (ν + )π : J ν (z) = {P (z) cos[z ] zπ 4 (ν + )π Q(z) sin[z ]} (.99) 4 = t k cos( kπ k= z + νπ + π ), (.) 4 P (z) + ( ) k (4ν )(4ν 3 ) (4ν (4k ) ) k= (k)!(8z) k Q(z) ( ) k (4ν )(4ν 3 ) (4ν (4k + ) ) k= (k + )!(8z) k+ t k = (k ) ν t k, t = (.) kz zπ z ν.96. [9]

16 6.7. z d dz du ν (z ) ( + dz z )u = d u dz + du z dz ν ( + )u =. (.) z I ν (z) = e νπi/ J ν (iz) ( π < arg z < π/) (.3) = e 3νπi/ J ν (iz) (π/ < arg z < π) (.4) = ( z ) (z/) n [z ] (.5) n!γ(ν + n + ) n=

17 7. Â, ˆB, Ĉ A, B AB Ψ (AB)Ψ = A(BΨ) (.) AB Ψ B BΨ χ A Aχ AA = A, AAA = A 3, (.) A f(a) x f(x) {c n ; n =,,,, } f(x) = c + c x + c x + = c n x n. (.3) n= A f(a) f(a) = c + c A + c A + = c n A n. (.4) n= A e A = + A +! A + = n= n! An (.5) commutator [A, B] AB BA. (.6) A, B

18 8. Â, ˆB Â = /Â, ˆB = / ˆB. Â ˆB = Â ( ˆB Â) ˆB = ˆB ( ˆB Â) Â (.7) Â ˆB = ˆB ˆB Â Â Â ˆB = Â ( ˆB Â) ˆB. (.8) Â ˆB = ˆB ˆB Â ˆB Â Â = ˆB ( ˆB Â) Â. (.9). Â ˆB = Â ( + ˆB ) Â ˆB (.),ˆ ˆÂ = Âˆ = Â ˆ = Â Â = Â (Â ˆB) + ˆB (.) Â. (Â ˆB). [4] A, B, C [A, B] AB BA, (.) [AB, C] = A [B, C] + [A, C] B (.3)

19 e A Be A = B + [B, A] +! [[B, A], A] + [[[B, A], A], A] +, (.4) 3! e A Be A = B + [A, B] +! [A, [A, B]] + [A, [A, [A, B]]] +. (.5) 3! if [[A, B], A] = [[A, B], B] = e A+B = e A e B e [A,B]/, (.6) = e B e A e [A,B]/ (.7) Campbell-Hausdorf X, Y log[e X e Y ] = (X + Y ) + [X, Y ] + ([X, [X, Y ]] + [Y, [Y, X]]) + (.8) e A+B+C = e A e Z ( ), (.9) Z (B + C) [A, B + C] [[A, B + C], A + B + C] + [] P.M. Morse and H. Feshbach, Method of Theoretical Physics, McGraw-Hill, 953,vol.I. [], I,. [3], II,. [4], III,. [5] L.I. Schiff,Quantum Mechanics,third edition, McGraw Hill,968,pp [6] 99 [7] I 984 p.4. [8] 984 pp [9] 99 [] 99 [] 984 pp.49-5.

20 [] 984 p.3. [3] J. Schwinger,Quantum Mechanics, Springer,,pp.8-9. [4] 977 [5] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, (translated from the Russian by Scripta Technica, INC.) Academic Press,977

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