# Mathematical methods / Numerical codes
#Lectures on Quantum Mechanics
#http://www.uprh.edu/rbaretti
#http://www1.uprh.edu/rbaretti/MethodsofTheoreticalPhysics.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart2.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart3.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart4.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart5.htm
#http://www1.uprh.edu/rbaretti/methodsoftheoreticalphysicspart6.htm
#http://www1.uprh.edu/rbaretti/LQMIntro.htm
#http://www1.uprh.edu/rbaretti/LQMch1.htm
#http://www1.uprh.edu/rbaretti/LQMch2.htm
#http://www1.uprh.edu/rbaretti/LQMch3.htm
#http://www1.uprh.edu/rbaretti/LQMch4.htm
#http://www1.uprh.edu/rbaretti/LQMch5.htm
c1,c2,c3=var('c1,c2,c3')
solve([.175*c1+.248*c2+.524*c3==.903,.248*c1+.276*c2+.466*c3==.996,.524*c1+.466*c2
+.636*c3==1.99],c1,c2,c3)
# bi= 0.903E+00 0.996E+00 0.199E+01
#kij= 0.175E+00 0.248E+00 0.524E+00
# 0.248E+00 0.276E+00 0.466E+00
# 0.524E+00 0.466E+00 0.636E+00
#i1,i2=var('i1,i2')
#solve([200*i1-100*i2==1.5,-100*i1+200*i2==5],i1,i2)
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# www.geocities.com/serienumerica4
# bi= 0.903E+00 0.996E+00 0.199E+01
#c kij= 0.175E+00 0.248E+00 0.524E+00
#c 0.248E+00 0.276E+00 0.466E+00
R=RealField(20)
var('c1,c2');
solve([.175*c1+.248*c2==.903,.248*c1+.276*c2==.996],c1,c2)
[[c1 == (-555/3301), c2 == (12411/3301)]] [[c1 == (-555/3301), c2 == (12411/3301)]] |
n(-555/3301) , n (12411/3301)
(-0.168130869433505, 3.75976976673735) (-0.168130869433505, 3.75976976673735) |
# bi= 0.903E+00 0.996E+00 0.199E+01
#c kij= 0.175E+00 0.248E+00 0.524E+00
#c 0.248E+00 0.276E+00 0.466E+00
c1,c2=var('c1,c2')
solve([.175*c1+.248*c2==.903,.248*c1+.276*c2==.996],c1,c2)
#R(c1),R(c2)
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# bi= 0.927E+00 -0.135E+00 0.840E+01
# kij=
# 0.120E+01 -0.937E-01 0.775E+01
# -0.937E-01 -0.518E+00 0.571E+01
# 0.775E+01 0.571E+01 0.962E+01
c1,c2,c3=var('c1,c2,c3')
solve([1.20*c1-0.937E-01*c2+0.775E+01*c3==.927, -0.937E-01*c1 -0.518E+00*c2+0.571E+01*c3==
-0.135E,0.775E+01*c1+0.571E+01*c2+9.62*c3==8.40],c1,c2,c3)
#R(c1),R(c2),R(c3)
Syntax Error:
-0.135E,0.775E+01*c1+0.571E+01*c2+9.62*c3==8.40],c1,c2,c3)
Syntax Error:
-0.135E,0.775E+01*c1+0.571E+01*c2+9.62*c3==8.40],c1,c2,c3)
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continuation?
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#deltat= 1.
#b(i)= 0.100E+01 0.500E+00 0.333E+00
#b(i)= 0.667E+00 0.500E+00 0.400E+00 0.333E+00
# 0.333E+00 0.478E+00 0.342E+00 0.123E+00
# 0.333E+00 0.342E+00 0.176E+00 0.451E-01
# 0.333E+00 0.245E+00 0.902E-01 0.166E-01
# 0.333E+00 0.176E+00 0.463E-01 0.611E-02
f1,f2,f3,f4=var('f1,f2,f3,f4');
solve([.333*f1+.478*f2+.342*f3+.123*f3==.667,.33*f1+.34*f2+.17*f3+.045*f4==.5,.33*f1+.24*f2+.091*f3+.0166*f4==.4,.33*f1+.176*f2+.0463*f3+.00611*f4==.333],f1,f2,f3,f4)
[[f1 == 133596523/317946768, f2 == 237949891/211964512, f3 == -1085025/52991128, f4 == -9928125/26495564]] [[f1 == 133596523/317946768, f2 == 237949891/211964512, f3 == -1085025/52991128, f4 == -9928125/26495564]] |
R=RealField(20)
R(133596523/317946768) ; R(-1085025/52991128 );R(-9928125/26495564);
0.42019 -0.020476 -0.37471 0.42019 -0.020476 -0.37471 |
i1,i2=var('i1,i2')
solve([.333*i1+.478*i2==1.667,.33*i1+.34*i2==.5],i1,i2)
[[i1 == -5463/742, i2 == 12787/1484]] [[i1 == -5463/742, i2 == 12787/1484]] |
#sage: A=matrix([[1,1],[2,3]])
#sage: A.eigenvalues()
#[0.2679491924311228?, 3.732050807568878?]
A=matrix([[1,1,1],[1,3,5],[1,5,4]])
A.eigenvalues()
[-1.527069054684481?, 0.7457280539128812?, 8.781341000771600?] [-1.527069054684481?, 0.7457280539128812?, 8.781341000771600?] |
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C,D,k2,k,a=var('C,D,k2,k,a')
solve([e^(-k2*a)*C+e^(k2*a)*D==e^(I*k*a),-k2*e^(-k2*a)*C+k2*e^(k2*a)*D==I*k*e^(I*k*a)],C,D)
[[C == 1/2*(-I*k + k2)*e^(I*a*k + a*k2)/k2, D == 1/2*(I*k + k2)*e^(I*a*k - a*k2)/k2]] [[C == 1/2*(-I*k + k2)*e^(I*a*k + a*k2)/k2, D == 1/2*(I*k + k2)*e^(I*a*k - a*k2)/k2]] |
#sage: k.<I> = QuadraticField(-1)
#sage: I.conjugate()
#-I
#sage: (I/(1+I)).conjugate()
#-1/2*I + 1/2
#sage: z6=CyclotomicField(6).gen(0)
#sage: (2*z6).conjugate()
#-2*zeta6 + 2
#sage: K.<j,b> = QQ[sqrt(-1), sqrt(2)]
#sage: j.conjugate()
#(3+2*I).conjugate()
#f=1+3*I ;
#(1+3*I)*( 1+I*3 ).conjugate()
f=1+3*I;f*f.conjugate()
A=(1/4*k2*k)
10 10 |
#sage: A=matrix([[1,1],[2,3]])
#sage: A.eigenvalues()
A=matrix([[1,2],[2,3]] );
A.eigenvalues()
[-0.2360679774997897?, 4.236067977499789?] [-0.2360679774997897?, 4.236067977499789?] |
A=matrix([[1,2],[2,4]]);
A.eigenvalues()
#[[0,4],[3,1]]
#A=matrix([[6,-1,1],[-1,3,-2],[1,-2,3]] );
#A.eigenvalues()
#A.eigenvectors()
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# 0.984E+01 -0.797E-07 0.570E+01
#-0.797E-07 0.303E+02 0.312E-06
# 0.570E+01 0.312E-06 0.561E+02
# A= matrix([[9.84,0 ],[0,30.3]]);
A=matrix([[9.84,0,5.70],[0,30.3,0],[5.70,0,56.1]] );
A.eigenvalues()
[56.7919835446169, 30.3000000000000, 9.14801645538306] [56.7919835446169, 30.3000000000000, 9.14801645538306] |
# http://www.sagemath.org/doc/tutorial/interfaces.html#section-maxima
sage: A.echelon()
matrix([1,1/2,1/3,1/4],[0,0,0,0],[0,0,0,0],[0,0,0,0])
sage: A.eigenvalues()
#[[0,4],[3,1]]
#sage: A.eigenvectors()
#[[[0,4],[3,1]],[1,0,0,-4],[0,1,0,-2],[0,0,1,-4/3],[1,2,3,4]]
Syntax Error:
sage: A.echelon()
Syntax Error:
sage: A.echelon()
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A=matrix([[3,2],[2,4]]);
A.eigenvalues()
A.eigenvectors()
Traceback (click to the left of this block for traceback) ... AttributeError: 'sage.matrix.matrix_integer_dense.Matrix_integer_de' object has no attribute 'eigenvectors' Traceback (most recent call last):
File "<stdin>", line 1, in <module>
File "/home/sage/sagenb/sage_notebook/worksheets/reibaretti/32/code/9.py", line 9, in <module>
A.eigenvectors()
File "", line 1, in <module>
AttributeError: 'sage.matrix.matrix_integer_dense.Matrix_integer_de' object has no attribute 'eigenvectors'
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sage: MS = MatrixSpace(QQ, 3, 3)
sage: A = MS([[1,1,0],[1,2,0],[0,0, 3]])
sage: A
[1 1 0]
[1 2 0]
[0 0 3]
#sage: A.eigenspaces()
A.eigenvalues()
[1 1 0] [1 2 0] [0 0 3] [1 1 0] [1 2 0] [0 0 3] |
A=matrix(QQ,[[1,1,0],[1,2,0],[0,0,3]]);
#A.eigenspaces()
A.eigenvalues()
[3, 0.3819660112501051?, 2.618033988749895?] [3, 0.3819660112501051?, 2.618033988749895?] |
K=matrix(QQ,[[1,2],[2,3]]);
K.eigenvalues()
[-0.2360679774997897?, 4.236067977499789?] [-0.2360679774997897?, 4.236067977499789?] |
#i,j, kij= 1 1 0.83338E-01
#i,j, kij= 1 2 -0.17320E+00
#i,j, kij= 1 3 -0.37957E-03
#i,j, kij= 1 4 0.18915E-01
#i,j, kij= 2 2 -0.82829E-01
#i,j, kij= 2 3 -0.27667E-01
#i,j, kij= 2 4 -0.31605E-03
#i,j, kij= 3 3 0.59531E-03
#i,j, kij= 3 4 -0.28169E-01
#i,j, kij= 4 4 0.61117E-03
K=matrix(QQ,[[8.33E-2,-1.73E-1],[-1.73E-1,-8.28E-2]]);
K.eigenvalues()
[-0.1916518043166870?, 0.1921518043166870?] [-0.1916518043166870?, 0.1921518043166870?] |
var('x ,f');f=(x-8.33E-2)*(x+8.28E-2) -(1.73e-1)^2 ;
y=plot(f,x,-.3,1);
show(y)
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#i,j, kij= 1 1 0.10119E+00
#i,j, kij= 1 2 -0.15749E+00
#i,j, kij= 1 3 0.71977E-01
#i,j, kij= 2 2 -0.57507E-01
#i,j, kij= 2 3 0.82789E-02
#i,j, kij= 3 3 0.25756E-01
K=matrix(QQ,[[0.10119E+00,-0.15749E+00,0.71977E-01],[-0.15749E+00,-0.57507E-01,0.82789E-02],[ 0.71977E-01,0.82789E-02 ,0.25756E-01 ]]);
K.eigenvalues()
[-0.1655127048306282?, 0.01919336910404436?, 0.2157583357265839?] [-0.1655127048306282?, 0.01919336910404436?, 0.2157583357265839?] |
a=1/(0.2157583357);
n(a)
4.63481513590485 4.63481513590485 |
a=1/.01919336910404436;
n(a)
52.1013270040893 52.1013270040893 |
K=matrix(QQ,[[0.10119E+00,0.60232E-09 ],[ 0.60232E-09,0.25002E-01 ]]);
K.eigenvalues()
[0.02500200000000000?, 0.10119000000000001?] [0.02500200000000000?, 0.10119000000000001?] |
n(1/pi^2); n(1/(4*pi^2))
0.101321183642338 0.0253302959105844 0.101321183642338 0.0253302959105844 |
#i,j, kij= 1 1 0.10119E+00
#i,j, kij= 1 2 0.60232E-09
#i,j, kij= 1 3 -0.34367E-02
#i,j, kij= 2 2 0.25002E-01
#i,j, kij= 2 3 -0.34627E-08
#i,j, kij= 3 3 0.10933E-01
K=matrix(QQ,[[0.10119E+00,0.60232E-09,-0.34367E-02 ],[ 0.60232E-09,0.25002E-01,-0.34627E-08 ],[ -0.34367E-02, -0.34627E-08 , 0.10933E-01 ]]);
K.eigenvalues()
[0.01080233055180890?, 0.02500200000000083?, 0.1013206694481903?] [0.01080233055180890?, 0.02500200000000083?, 0.1013206694481903?] |
n(1/pi^2) , n(1/(4*pi^2)) ,n(1/(9*pi^2))
(0.101321183642338, 0.0253302959105844, 0.0112579092935931) (0.101321183642338, 0.0253302959105844, 0.0112579092935931) |
var('x');
phi1=(30)^(1/2)*x*(1-x);
phi2=2*(210)^(1/2)*( x^2*(1-x) -(1-x)*x/2 );
#phi3=(17640)^(1/2)*( x^3*(1-x) - ((30)^(1/2)/105)*phi1 - ( 1/(2*(210)^(1/2)))*phi2 );
phi3=(17640)^(1/2)*( x^3*(1-x) );
y=plot(phi3,x,0,1);
show(y)
Syntax Error:
y=plot(phi3,x,0,1);
Syntax Error:
y=plot(phi3,x,0,1);
|
phin=x^2-3*x^2+5;phin
-2*x^2 + 5 -2*x^2 + 5 |
var('x');phi1=1;
phi3a=(17640)^(1/2)*( x^3*(1-x)) -( (30)^(1/2)/105 )*(30)^(1/2) x*(1-x) );
phi3=phi3a - (17640)^(1/2)*x^3;
plot(phi3,x,0,1);
show(y)
Syntax Error:
phi3=phi3a - (17640)^(1/2)*x^3;
Syntax Error:
phi3=phi3a - (17640)^(1/2)*x^3;
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a3=1/0.01080233055180890 ;n(a3) , n(9*pi^2)
(92.5726161779548, 88.8264396098042) (92.5726161779548, 88.8264396098042) |
#solve([.333*i1+.478*i2==1.667,.33*i1+.34*i2==.5],i1,i2)
c2,c3=var('c2,c3')
c1=1;lambda= 9.8696544;mu=1/lambda; k11=0.10119; k12= 0.60232e-09; k13=-0.34367e-02; k21=k12; k22= 0.25002e-01;k23=-0.34627e-08 ; b1=-(k11-mu) ;b2=-k21;
solve([k12*c2+k13*c3==b1,(k22-mu)*c2+ k23*c3==b2],c2,c3)
Syntax Error:
c1=1;lambda= 9.8696544;mu=1/lambda; k11=0.10119; k12= 0.60232e-09;
k13=-0.34367e-02; k21=k12; k22= 0.25002e-01;k23=-0.34627e-08 ;
b1=-(k11-mu) ;b2=-k21;
Syntax Error:
c1=1;lambda= 9.8696544;mu=1/lambda; k11=0.10119; k12= 0.60232e-09; k13=-0.34367e-02; k21=k12; k22= 0.25002e-01;k23=-0.34627e-08 ; b1=-(k11-mu) ;b2=-k21;
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c1,c2=var('c1,c2')
a11=1.75e-1;a12=0.248;
solve([a11*c1+a12*c2==.903,.248*c1+.276*c2==.996],c1,c2)
[[c1 == (-555/3301), c2 == (12411/3301)]] [[c1 == (-555/3301), c2 == (12411/3301)]] |
a11=3 ; a22=5;
phi1(x)=a11*x ; phi2(x)=a22*x^2;
integral(phi1(x)*phi2(x),x,0,2)
60 60 |
var('x ,f');f=(1/8)*(1-x^2)*x ;
y=plot(f,x,-1,1);
show(y)
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var('a,b');
#(27/32)a + (243/128)b + 729/128
# 3456/625*a + 108864/3125*b + 746496/3125
solve([(27/32)*a+ (243/128)*b==-729/128, (3456/625)*a+(108864/3125)*b==-746496/3125],a,b)
[[a == (27/2), b == -9]] [[a == (27/2), b == -9]] |
c0,c1,c2=var('c0,c1,c2');
#b0,b1,b2= 0.159E+01 -0.120E+00 0.576E-02
b0=0.159E+01;b1=-0.120E+00;b2=0.576E-02;
#k00,k01,k02,k11,k12,,k22= 0.132E+01 0.122E+00 0.896E-02 0.469E-01 0.47
#3E-02 0.120E-02
k00=0.132E+01;k01=0.122E+00;k10=k01;k02=0.896E-02;k20=k02;k11=0.469E-01;k12=0.473E-02;k21=k12;k22=0.120E-02;
solve([k00*c0+ k01*c1+k02*c2==b0,k10*c0+ k11*c1+k12*c2 ==b1,k20*c0+ k21*c1+k22*c2 ==b2 ],c0,c1,c2);
[[c0 == (186938801/92978556), c1 == (-260766320/23244639), c2 == (790475750/23244639)]] [[c0 == (186938801/92978556), c1 == (-260766320/23244639), c2 == (790475750/23244639)]] |
n(790475750/23244639 )
34.0067982987389 34.0067982987389 |
c0,c1=var('c0,c1');
#b0,b1,b2=0.796E+00 0.368E+00
b0=.417 ;b1=.278 ;
k00=.250 ;k01=.167;k10=k01;k11=.111;
solve([k00*c0+ k01*c1==b0,k10*c0+ k11*c1==b1 ],c0,c1);
[[c0 == 1, c1 == 1]] [[c0 == 1, c1 == 1]] |
n( 13544/3119 ) , n (-21442/3119)
(4.34241744148766, -6.87463930747034) (4.34241744148766, -6.87463930747034) |
x,y=var('x,y')
integral(exp(y)*exp(y*x),y,0,1)
e^(x + 1)/(x + 1) - 1/(x + 1) e^(x + 1)/(x + 1) - 1/(x + 1) |
b0,b1,b2= 0.237E+01 0.130E+01 0.187E+00
k00,k01,k02= 0.132E+01 0.718E+00 0.911E-01
k10,k11,k12= 0.718E+00 0.400E+00 0.634E-01
k20,k21,k22= 0.911E-01 0.634E-01 0.278E-01
Syntax Error:
k10,k11,k12= 0.718E+00 0.400E+00 0.634E-01
Syntax Error:
k10,k11,k12= 0.718E+00 0.400E+00 0.634E-01
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c0,c1,c2,c3=var('c0,c1,c2,c3'); b0,b1,b2,b3=var(' b0,b1,b2,b3 ');
k00,k01,k02,k03= (' k00,k01,k02,k03 ' );
k10,k11,k12,k13= (' k10,k11,k12,k13 ' );
k20,k21,k22,k23= (' k20,k21,k22,k23 ' );
k30,k31,k32,k33= ('k30,k31,k32,k33 ' );
b0=0.237E+01; b1=0.130E+01; b2= 0.913E+00;b3=.706 ;
k00= 0.132E+01; k01= 0.718E+00;k02=.500;k03=.385;
k10=0.718E+00;k11=0.400E+00; k12= 0.282;k13=.218 ;
k20=0.500; k21=.282; k22=0.199;k23=.155 ;
k30=.385; k31=.218 ; k32=.155;k33=.121 ;
solve([k00*c0+ k01*c1+k02*c2+k03*c3==b0,k10*c0+ k11*c1+k12*c2+k13*c3==b1,k20*c0+ k21*c1+k22*c2+k23*c3==b2,k30*c0+ k31*c1+k32*c2+k33*c3==b3 ],c0,c1,c2,c3);
Traceback (click to the left of this block for traceback) ... ValueError: too many values to unpack Traceback (most recent call last): k20,k21,k22,k23= (' k20,k21,k22,k23 ' );
ValueError: too many values to unpack
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(0.454769990737882, 4.97530101883297, -3.60512503859216) (0.454769990737882, 4.97530101883297, -3.60512503859216) |
a=1;
f(x)=a*exp(-x);
#integral(f(x),x,0,.75)
y=plot(f(x),x,0,4)
show(y)
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#A=matrix([[9.84,0,5.70],[0,30.3,0],[5.70,0,56.1]] );
#A.eigenvalues()
A=matrix([[-0.434047461,-0.0411980152],[-0.0411980152, 0.0384942591 ]])
A.eigenvalues()
[0.0420591673690439, -0.437612369269044] [0.0420591673690439, -0.437612369269044] |
A=matrix([[ -0.427069992 ,-0.0412402749 ],[-0.0412402749 , 0.0452692211 ]])
A.eigenvalues()
[0.0488429008622748, -0.430643671762275] [0.0488429008622748, -0.430643671762275] |
A=matrix([[ -0.42706 ,-0.0412402 ],[-0.0412402 , 0.045269 ]]);A
es=A.eigenvectors_right();es
[(0.0488427429990885, [ ], 1), (-0.430633742999088, [ ], 1)] [(0.0488427429990885, [ ], 1), (-0.430633742999088, [ ], 1)] |
A=matrix([[ 1 ,3 ],[3 , 1 ]]);A
es=A.eigenvectors_right();es
[(4, [ (1, 1) ], 1), (-2, [ (1, -1) ], 1)] [(4, [ (1, 1) ], 1), (-2, [ (1, -1) ], 1)] |
n( (-0.434047461+0.4306436717)/-0.0411980152)
0.0826202253549345 0.0826202253549345 |
c1=1
c2=8.26E-2
g1(p)=(2/pi)^(3/4)*exp(-p^2 )
g2(p)= 2.978485*exp(-2*p^2 ) -1.621282*exp(-p^2 )
phi(p)=c1*g1(p)+c2*g2(p)
norm=integral(phi(p)^2*4*pi*p^2,p,0,oo);n(norm)
1.00682273129350 1.00682273129350 |
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