combine: combine(exp1 ) collect: collect(exp1,x) convert: convert(exp1,opt ) > convert(sin(x),exp); > convert(sinh(x),exp); > convert(exp(i*x),trig);

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1 1 Maple exp Maple solve( ), diff( ), int( ), 1: simplify: lhs, rhs: subs: expand: numer, denom: assume: factor: coeff: assuming: normal: nops, op assign: combine: about: collect: anames( user ): sort: restart,a:= a : convert: rem( ), quo( ), limit( ) expand: expand(exp1 ) factor: factor(exp1 ) normal: normal(exp1 )

2 combine: combine(exp1 ) collect: collect(exp1,x) convert: convert(exp1,opt ) > convert(sin(x),exp); > convert(sinh(x),exp); > convert(exp(i*x),trig); > convert(1/(x-1)/(x+3),parfrac); 1/2 i ( e ix (e ix ) 1) 1/2 e x 1/2 (e x ) 1 cos (x) + i sin (x) 1/4 (x 1) 1 1/4 (x + 3) 1 2: convert opt polynom trig exp parfrac rational simplify: simplify(exp1 ), simplify(exp1, ) > exp1:=3*sin(x)^3-sin(x)*cos(x)^2; exp1 := 3 (sin (x)) 3 sin (x) (cos (x)) 2 > simplify(exp1); ( 4 (cos (x)) 2 3 ) sin (x) > simplify(exp1,{cos(x)^2=1-sin(x)^2}); sin (x) + 4 (sin (x)) 3 sort: sort(exp1 ), sort(exp1,[x,y]), sort(exp1, [x],opts);opts=tdeg,plex,ascending,or descending (,, ) > exp1:=x^3+4*x-3*x^2+1: > sort(exp1); > sort(exp1,[x],ascending); x 3 3 x x x 3 x 2 + x 3

3 > exp2:=x^3-3*x*y+4*x^2+y^2: > sort(exp2); > sort(exp2,[x]); > sort(exp2,[y],descending); x 3 + 4x 2 3xy + y 2 x 3 + 4x 2 3yx + y 2 y 2 3xy + x 3 + 4x 2 lhs, rhs: lhs(exp1 =exp2 ) numer, denom: numer(exp1 /exp1 ) coeff: coeff(exp1,x^2) op,nops:, op(exp1 ), nops(exp1 ) subs: subs(,exp1 ) > exp1:=x^2-4*x+4; > subs(x=a+2,exp1); assume: assume( ) exp1 := x 2 4 x + 4 (a + 2) 2 4 a 4 assuming: exp1 assuming > exp1:=x^2-4*x+4; > sqrt(exp1); > sqrt(exp1) assuming x>2; exp1 := x 2 4 x + 4 ( 2 + x) x additionally: assume assign: solve about: assume restart,a= a : anames( user ):

4 series: series(exp1,x,4) > series(exp(x),x); 1 + x + 1/2 x 2 + 1/6 x 3 + 1/24 x x5 + O (x 6 ) > series(sin(x),x=pi/3,2); 1/ /2 x 1/6 π + O ( (x 1/3 π) 2) > convert(%,polynom); 1/ /2 x 1/6 π : > a 1; > a b; seq: for-loop > seq(i,i=0..3); a1 ab 0, 1, 2, 3 map: > map(sin,[seq(a i,i=0..3)]); add, mul: [sin (a0 ), sin (a1 ), sin (a2 ), sin (a3 )] sum, product: > add(x^i,i=1..3); > add(x^i,i=1..n); x + x 2 + x 3 Error, unable to execute add > mul(x^i,i=1..3); mul(x^i,i=1..n); x 6 Error, unable to execute add

5 > sum(x^i,i=1..3); sum(x^i,i=1..n); x + x 2 + x 3 x n+1 x 1 > product(x^i,i=1..3); > product(x^i,i=1..n); x x 1 x 6 ni=1 x i limit: > limit(exp(-x),x=infinity); > limit(tan(x),x=pi/2,left); > limit(tan(x),x=pi/2,complex); 0 + I 1.2 Maple restart restart plot Plotting error,empty plot xe ( β cx2 1 + β gx 3) dx (1) Maple

6 > f1:=unapply(x*exp(-beta*c*x^2)*(1+beta*g*x^3),x); f1 := x xe β cx2 (1 + β gx 3 ) > int(f1(x),x=-infinity..infinity); ( g PIECEWISE [3/4 ) π β c 2, csgn (βc) = 1], [, otherwise] β c βc (csgn(βc)=1) (otherwise) Maple restart > restart; > f1:=unapply(x*exp(-beta*c*x^2)*(1+beta*g*x^3),x); f1 := x xe β c x2 (1 + β gx 3 ) > plot(f1(x),x= ); Warning, unable to evaluate the function to numeric values in the region; see the plotting command s help page to ensure the calling sequence is correct Error, empty plot > f1(10); 10 e 100 β c ( β g) beta,c,g > c:=1; g:=0.01; beta:=0.1; c := 1 g := 0.01 β := 0.1 > plot(f1(x),x= );..

7 x 10-1 > c:= c ; g:= g ; beta:= beta ; c := c g := g β := β > int(f1(x),x); 1 1/2 + β g e β cx2 β c 1/2 x3 e β cx2 + 3/2 β c 1/2 x=-alpha..alpha > int(f1(x),x=-alpha..alpha); 1/4 g alpha ( 4 α 3 e β cα2 β c β c+6 α e β cα2 β c 2 β c xe β cx2 β c + 1/4 β c 3 πerf ( ) πerf β cx β c ( )) β cα > limit(int(f1(x),x=-alpha..alpha),alpha=infinity); ( 4 α 3 e β cα2 β c β c+6 α e β cα2 β c 3 πerf β c ( )) β cα β 1 c 1 lim α 1/4 g β c 2 β c beta*c>0 (assume) > assume(beta*c>0); > limit(int(f1(x),x=-alpha..alpha),alpha=infinity); 3/4 πg β c 2 β c

8 1.3 ( ) ( ) Maple (1) (2) > ex1:=(x-3)^4; ex1 := (x 3) 4 > ex2:=x^4-12*x^3+54*x^2-108*x+81; > expand(ex1-ex2); ex2 := x 4 12 x x x expand (thermal expansion)

9 x U(x) = cx 2 gx 3 (2) x 3 x x = x exp ( βu(x)) dx exp ( βu(x)) dx (3) β 1/(k B T ) x = 3g 4βc 2 (4) Taylor Maple > restart; > U:=c*x^2-g*x^3; > eu:=expand(exp(-beta*u)); U := cx 2 gx 3 eu := eβ gx3 e β cx2 > ex:=convert(series(numer(eu),x,4),polynom); > f1:=ex/denom(eu); ex := 1 + β gx 3 f1 := 1+β gx3 e β cx2 > den:=int(f1,x=-infinity..infinity); ( den := PIECEWISE [ ) π, csgn (β c) = 1], [, otherwise] β c > num:=int(x*f1,x=-infinity..infinity);

10 ( g num := PIECEWISE [3/4 ) π β c 2, csgn (βc) = 1], [, otherwise] β c βc > 0 βc > 0 (assume) > assume(beta*c>0): > num/den; 3/4 g β c A A V0 B C 0 a V (x) = 0 h2 d 2 ϕ(x) = εϕ(x) (5) 2m dx 2 x 0 ϕ(x) = A exp(ikx) + B exp( ikx) (6) x a ϕ(x) = C exp(ikx). k = 2mε/ h 2 ε V 0 ε V 0 κ = 2m(ε V 0 )/ h 2 0 x a ϕ(x) = F exp(iκx) + G exp( iκx) (7)

11 x = 0 x = a x = 0 ϕ(x) A + B = F + G (8) x = 0 ϕ (x) k(a B) = κ(f G) x = a x = a ϕ(x) F exp(iκa) + G exp( iκa) = C exp(ika) ϕ (x) κf exp(iκa) κg exp( iκa) = kc exp(ika) 5 4 F, G B/A C/A 2 B A 2 C A = = [ 4k 2 κ 2 ] 1 [ 1 + (k 2 κ 2 ) 2 sin 2 = 1 + 4ε(ε V ] 1 0) κa V0 2 sin 2 (9) κa [ 1 + (k2 κ 2 ) 2 sin 2 ] 1 [ κa = 1 + V 0 2 sin 2 ] 1 κa 4k 2 κ 2 4ε(ε V 0 ) 0 < ε < V 0 α = 2m(V 0 ε)/ h 2 0 x a ϕ(x) = F exp(αx) + G exp( αx) (10) [ 2 C = 1 + V 0 2 sinh 2 ] 1 αa (11) A 4ε(ε V 0 ) mv 0 a 2 / h 2 = 8 E/V 0 < 1 C/A 2 C/A > restart; > psi1:=x->a*exp(i*k*x)+b*exp(-i*k*x); psi1 := x Ae ikx + Be ikx > psi2:=x->e*exp(i*kappa*x)+f*exp(-i*kappa*x); > psi3:=x->c*exp(i*k*x); psi2 := x Ee iκ x + F e iκ x psi3 := x Ce ikx

12 1 mv 0 a 2 /h 2 =8 0.8 C/A E/V 0 x = 0, x = a 0 1 > eq1:=psi1(0)=psi2(0); eq1 := A + B = E + F > eq2:=simplify(subs(x=0,diff(psi1(x),x))=subs(x=0,diff(psi2(x),x))); eq2 := iak ibk = ieκ if κ > eq3:=psi2(a)=psi3(a); eq3 := Ee iκ a + F e iκ a = Ce ika > eq4:=simplify(subs(x=a,diff(psi2(x),x))=subs(x=a,diff(psi3(x),x))); eq4 := iκ (Ee iκ a F e iκ a ) = icke ika 4 {A,B,C,E,F} > solve(eq1,eq2,eq3,eq4,a,b,c,e,f); C = C, B = 1/4 ( κ 2 κ 2 (e iκ a ) 2 k 2 + k 2 (e iκ a ) 2) Ce ika iκ a κ k, a C ( κ + k) eika+iκ F = 1/2, ( κ A = 1/(4κ k) C e ika iκ a κ 2 e ika iκ a κ ( 2 e iκ a) 2 + e ika iκ a k 2 + e ika iκ a k ( 2 e iκ a) 2 +2 e ika iκ a kκ + 2 e ika+iκ a kκ 2 e ika+iκ a k 2), a (κ + k) Ceika iκ E = 1/2 κ > assign(%); assign A/C (conjugate) kappa, k,a Maple (12)

13 > assume(kappa,real);assume(k,real),assume(a,real); conjugate (trig) convert > CC:=convert(A/C,trig): > CC1:=combine(conjugate(CC)*CC); > #CC1:=convert(CC,trig); CC1 := 1/8 k4 +1/8 κ 4 +3/4 κ 2 k 2 1/8 k 4 cos(2 κ a)+1/4 κ 2 k 2 cos(2 κ a) 1/8 κ 4 cos(2 κ a) κ 2 k 2 > C_num:=simplify(expand(numer(CC1)), > cos(kappa*a)^2=1-sin(kappa*a)^2, > cos(k*a)^2=1-sin(k*a)^2); C num := 8 κ 2 k 2 + (2 k 4 4 κ 2 k κ 4 ) (sin (κ a)) 2 > C_den:=denom(CC1); > saa:=sin(kappa*a); C den := 8 κ 2 k 2 saa := sin (κ a) > CC2:=collect(C_num/C_den,saa); CC2 := 1 + 1/8 (2 k4 4 κ 2 k 2 +2 κ 4 )(sin(κ a)) 2 κ 2 k 2 k,kappa,a > NN:=8; > a2:= a2 ; > a2:=solve(m*v0*a2/h2=nn,a2); > kappa2:=2*m*(epsilon-v0)/h2; > k2:=2*m*epsilon/h2; NN := 8 a2 := a2 a2 := 8 h2 mv0 kappa2 := 2 m(ɛ V0 ) h2 k2 := 2 mɛ h2 > CC3:=simplify(subs(k=sqrt(k2),kappa=sqrt(kappa2), > coeff(cc2,saa^2))); CC3 := 1/4 V0 2 (V0 ɛ)ɛ > CC4:=simplify(subs(epsilon=x*V0,CC3)); CC4 := 1/4 1 ( 1+x)x > a2a2:=simplify(subs(epsilon=x*v0,sqrt(kappa2*a2))); a2a2 := x

14 > CC5:=1+CC4*sin(a2a2)^2; CC5 := 1 + 1/4 (sin(4 1+x)) 2 ( 1+x)x > f1:=unapply(cc5,x); > plot(1/f1(x),x=0..10); f1 := x 1 + 1/4 (sin(4 1+x)) 2 ( 1+x)x 1. ( ) (i) (ii) 2. 0 < ε < V 0 sinh sin cosh 2 (α a) = 1 + sinh 2 (α a) (13)

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Maple Copyright 005 Shigeto R. Nishitani Department of Informatics, Kwansei Gakuin University, Sanda, Japan e-mail : nishitani@ksc.kwansei.ac.jp 7 5 3 i PowerBookG4 Maple 9 L A T E X Maple Maple 9 Waterloo