= n a X 2 n 2 2 EnE 2 ma3. ( /24 pts) Suppose an infinite square well potential is given by c D (a) (4 pts) Sketch the potential. (b) (12 pts) As we saw in class, the general solution to the infinite square well is given by 2/(2) = Asin (kx) + Bcos (ka), where K = VmE h Use the boundary condition v(-D) = v(D) = 0 to show that A = 0 and K = (2n-1)T 2D (HINT: Consider trigonometric functions of negative angles.) (c) (8 pts) Normalize the wave function to determine B. Discuss this result in the context of the normalization constant we derived in class together for the particle in an infinite square well.Helpful Integrals and Mathematical Relationships Commutator Relationships 1. [A+B,C] : [14,0] + [3,0] 2. [A, A] = 0 3. [A,B] = [B,A] 4. [A, [3, 0]] + [3, [0, A]] + [0, [A,B]] = 0 Gamma Functions I'(n) = (TL 7 1)!, and hence 7(1) 2 1, m2) = 1, 7(3) 2 2, I'(4) = 6, l"(5) : 24, For positive half-integers, the function values are given e: 712\" F(%)=%, 27 or equivalently, for non-negative integer values 01 n: 2n 1 1! 2n ! Fe +0 = ' 8) = 583177 (2)\" (001! P i _ n : 7T 2 7r (2 ) (2n 1)\" f (27;)! f where 71!! denotes the double factorial. In particular, r (g) : W? x 1.772453 850 905 516 0273 = % as 0.886 226 925 452 758 0137 15 F 71' x 3.323 350 970 447 842 5512 ) = 5/7 :8 1.329340 3881791370205 telq balm Definite Integrals Vie * da = -VT (see also Gamma function) e at cos ba da = a a2 + 62 e - -of sin ba da = b a2 + 62 8 e ar sin bx dx = tan-1 8 e -ax e-bx dx = In e ax? da = for a > 0 (the Gaussian integral) e at cos ba da = e (an* +batc) da . erfc - b a 2 a' Where erfc(p) = - 2 JT . e (as' thatc) da = xhe at da = r(n + 1) anti iVal for a > 0 2n (2n - 1)! TT (2n)! TT 2a on+1 a2n+1 n!22+1 a2n+1 for a > 0 , n = 1, 2, 3 ... (where !! is the double factorial) 1 2a 2 for a > 0 n! 2 anti for a > 0 , n = 0, 1, 2... (m+1 me out da = 2 2a 2Table of Integrals* Basic Forms Integrals with Logarithms (1) IVar + bda = 15a2 (-262 + abx + 3a 2 ) ar + 6 (26) In arda = r lnar - I -dx = In /x] (42) (2) Inat do = - (In az)? Judo = wv - fvdu Vi(ax + b)da = Aat/2 [(2ax + 6) Var(ax + b) (43) (3) -62 In avi + Va(az + b)|] (27) ar + 6 dx = - In laz + b ) (4) In(az + b)de = (x+ 2 ) In(az + b) - z, a $ 0 (44) Integrals of Rational Functions VI' (ax + 6)dx = 12a - 8027 + In(23 + a?) dx = xin(23 + a?) + 2atan-1 2 - 2x (45) 63 (5) 8as/2 In avi + va(ax + b)| (28) (z +a)" dx = (I + a )nti In(22 - a ) dx = zin(x2 - a?) +aln 2+d - 2x (46) n+1 , n * - 1 (6) x(x +a)"da = (1 + a)nti ((n + 1)x - a) In ( az? + ba + c) da = 1 Vac -be tan -1 2az + 6 ( n + 1 ) ( n + 2 ) (7 ) ( 29 ) Viac - b - 22 + 20 + x ) In ( az ? + ba + c ) (47) It 2 da = tan "'x (8 ) [ vaz - andx = ,avaz-28+ 20 tan-1- at aeda = _ tan -1 2 (9 ) (30) xin(ax + b)da = 20 - (10) + 2 (27 - -2 ) In ( az + 6 ) ( 48 ) ( 31 ) Jarta da = x - atan-1 X (11) (32) (12 ) (2 - 12 ) In ( a ? - 6'2 7 ) (49) 1 Vaz _ madu = sin-1 2 (33) VAac - be tan-1 2ax + 6 Integrals with Exponentials Viac - b2 (13) (34 ) eat da = beaz (50) (14) ( 35 ) (15 ) 2a3/2 erf (ivaI) , where erf(x) = Vet at (51) ( 36 ) xe" do = (1 -1)e" av lac -2 tan -1 2ax + b (52) VAac - 62 (16) Integrals with Roots Vax? + br + cdr = 6 + 2at Var' + bat c (53) Vx - adz = =(x - a)3/2 4ac - b- 873/2 In 2ar + 6 + 2va(ax2 + batc) (37) xedo = (12 - 2x +2) e" (17) (54) (18) [zeat da = ( 2 - 22 + 2) cam (55) Va_ada = -2va -z IVax? +bec= Agas/2 (2Vavar' + bat c (19 rede = (13 - 323 + 6x - 6) e" (56) x (-362 + 2abx + 8a(c + az?)) [ava - ade = Za(x - a) 3/2 + 2(x - a) 5/2 (57) (20) +3(63 - 4abc) In 6 + 2ar + 2vavax2 + ba + c) (38) anti [[1 +n, - az], ( 21 ) I Vax' +ba + da = In 20z + 6+ 2Va(azz + ba + c) (58 (ar + 6) 3/2 dx = 2 (ax + 6) 5/2 where I'(a, z) = / ta-le- dt ( 22 ) (39) eax? dr = _ive 2 cerf (izva) (59) (23) Var' +bat da = - Var' +batc e-as' da = Vn 2vaelf (IVa) 60) (Val,da = -Vz(a-1) - atan-1 Vi(a -1) c - a ( 24 ) 2a3/2 In 2ax + 6 + 2va(ax2 + bx + c) (40) re-and dx = - - 2ge (61) ( Valda = Vz(a +x) - aln [Vi + Vz + a] (25) ( a2 + 12 ) 3/2 - a2 Va - + 1 2 (41) Sxe-andx = 1 " erflava) -2. (62 )Integrals with Trigonometric Functions sec' x dx = - secrtanz + , In |seca + tan.) (84) e costde = ,e (sin x + cos I) (106) sin arde = - - cos ax (63) sin? ardi = ? sin 2an sec I tan rdr = sec r (64) (85) ez cos ardi = 22 4 62 eat (asin ax + bcos ax) (107) 4a sec r tan ade = - sec x (86) sin " andr = re" sin adz = ,e" (cost - rcost + r sinI) (108) sec" rtandr = - sec" x, n # 0 -cosaz 2F1 1, 1-n, 3 (87) , 2 cos ax (65) re" cos ada = ze" (Icost - sini + Ising) (109) sin andr = _3cos at Cos 3ax cscada = In tan - = In |cscr - cot z| + C (88) 4a 12a (66) Integrals of Hyperbolic Functions cos ardx = _ sin ax (67) csc arde = - - cot ax (89) cos' andr = [ + Si 4a (68 ) csc' xde = - - cot rosci + , In |csci - cot z) (90) cosh andr = - sinh ar 110) cost ardi = -- Cosite arx csc" x cot adi = - - csc" I, n # 0 (91) eat cosh badr = a(1 + p) e az 1 + p 1 3 + P, cos2 ax "' 2' 2 (69) sec x cscudx = In | tan x] (92) a2 - 62 [a cosh br - bsinhbr] a # b (111) Aa a = b cos' andr = 3 sin ax _ sin 3ar Products of Trigonometric Functions and 4a 12a (70) Monomial sinh ardr = _ cosh ax (112) cos ar sin bedx _ cos [(a - b)x] _cos[(a + b)x] 2 (a - b) 2 (a + 6 ) I cos Idr = cos r + I sin I (93) eat sinh badr = (71) I cos ucds =- ear a2 cus un + 2 sin us (94) sin [(2a - b)x] e Laz a2 - 62 [-6cosh br + a sinh bx] a * b sin' ar cos brax = - (113) 4(2a - b) a = b sin [(2a + b)x] 12 cos rdi = 2x cos r + (x2 - 2) sina (95) sin br 2b 4(2a + b) (72) I' cos ardr = 2x cosaI ar - 2 sinar ex tanh brdr = sin? r cos adx = = sin3 x (73) a 3 (96) e(a+2b) 2 a + 26) 2F1 1+ 2, 1,2 + 5, -262 cos' ax sin badr = cos[(2a - b)x] cos br I" costde = - (anti [r(n + 1, -iz) ez 2 F1 5, 1, 1E,-26x] a * b (114) 4(2a - b) 2b ear - 2 tan '[eazy cos (2a + b)x] +(-1)"T(n + 1, iz)] (97) a =b 4(2a + b) (74) tanhardx = _ In cosh ar (115) cos' ar sin ardi = -2 cos' ax (75) I" cosaada = , (ia)'-~ [(-1)"I(n + 1, -iaz) -r(n + 1, ira)] (98) cos ar cosh badx = sin? arcos' budz = I _ sin 2at sin[2(a - b)I] a2 + 62 la sin ar cosh br 8a 16(a - b) +bcos ar sinh bx] 116) sin 26x sin[2(a + b)x] I sin adr = -rcosr + sin x (99) 8b 16(a + b) 76) I sin arde = _ I cos at sin ax (100) cos ar sinh badx = 2 sin' ar cos' andr = sin 4ax a a2 a2 + 62 6 cos ar cosh br + 32a (77) a sin ar sinh bx] (117) tan andr = - - In cos ar I' sin edx = (2 - z?) cost + 2x sin x (101) (78) tan? ardr = -r + - tan ar sin ar cosh brdr = 2 (79) x' sin ardi = 2 - 4212 a2 + 62 - a cos ar cosh but - cos ax + 2x sin ax a 2 (102) bsin ar sinh br (118) tan" ardx = tan" at a(1 + n) x" sinada = -- (i)" [I(n+ 1, -iz) - (-1)"[(n+ 1, -iz)] sin ax sinh budz = 2 2 , - tan ar) a2 + 62 [b cosh be sin ax- 80 103) a cos ar sinh bx] (119) Products of Trigonometric Functions and tan' arde = _ Incosar + 2- sec ax (81 ) Exponentials sinh ar cosh ardx = - [-2ax + sinh 2ax] (120) sec ada = In | secr + tanz| = 2tanh-1 (tan ? ) (82) e" sin adr = _ e* (sin x - cos I) (104) sec' arda = 1 tan ax sinh ar cosh badr = be (83) coz sin ardi = = 62 _ a2 6 cosh br sinh ar a2 + be (bsinax - acosar) (105) -a cosh ar sinh bax] 121)